Algebra Commons^™

Cohen–Macaulay Dimension For Coherent Rings, Rebecca Egg

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation presents a homological dimension notion of Cohen-Macaulay for non-Noetherian rings which reduces to the standard definition in the case that the ring is Noetherian, and is inspired by the homological notion of Cohen-Macaulay for local rings developed by Gerko. Under this notion, both coherent regular rings (as defined by Bertin) and coherent Gorenstein rings (as defined by Hummel and Marley) are Cohen-Macaulay.

This work is motivated by Glaz's question regarding whether a notion of Cohen-Macaulay exists for coherent rings which satisfies certain properties and agrees with the usual notion when the ring is Noetherian. Hamilton and Marley gave …

Takens Theorem With Singular Spectrum Analysis Applied To Noisy Time Series, Thomas K. Torku

Electronic Theses and Dissertations

The evolution of big data has led to financial time series becoming increasingly complex, noisy, non-stationary and nonlinear. Takens theorem can be used to analyze and forecast nonlinear time series, but even small amounts of noise can hopelessly corrupt a Takens approach. In contrast, Singular Spectrum Analysis is an excellent tool for both forecasting and noise reduction. Fortunately, it is possible to combine the Takens approach with Singular Spectrum analysis (SSA), and in fact, estimation of key parameters in Takens theorem is performed with Singular Spectrum Analysis. In this thesis, we combine the denoising abilities of SSA with the Takens …

Introductory And Intermediate Algebra Mth 100x, Joanna Burkhardt

Library Impact Statements

No abstract provided.

Rigidity Of The Frobenius, Matlis Reflexivity, And Minimal Flat Resolutions, Douglas J. Dailey

Department of Mathematics: Dissertations, Theses, and Student Research

Let R be a commutative, Noetherian ring of characteristic p >0. Denote by f the Frobenius endomorphism, and let R^(e) denote the ring R viewed as an R-module via f^e. Following on classical results of Peskine, Szpiro, and Herzog, Marley and Webb use flat, cotorsion module theory to show that if R has finite Krull dimension, then an R-module M has finite flat dimension if and only if Tor_i^R(R^(e),M) = 0 for all i >0 and infinitely many e >0. Using methods involving the derived category, we show that one only needs vanishing for dim R +1 consecutive values of …

Conjugacy Geodesics In Coxeter Groups, Aaron Calderon

UCARE: Research Products

Take a square and flip it over the vertical axis, rotate it 90 degrees counterclockwise and then flip it again over the vertical axis. This sequence is the same as a 90 degree clockwise rotation but takes more steps to demonstrate the same symmetry. In general, the question of when a sequence of symmetries has minimal length is hard to answer and is dependent on the chosen generating set (in our toy example, rotation by 90 degrees and reflection). By realizing sequences of symmetries as paths in a group's Cayley graph, the problem becomes one about the set of shortest …

College Algebra (College Of Coastal Georgia), German Vargas, Jose Lugo, Laura Lynch, Jamil Mortada, Treg Thompson, Victor Vega

Mathematics Grants Collections

This Grants Collection for College Algebra was created under a Round Two ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

College Algebra (Fort Valley State University), Josephine Davis, Samuel Cartwright, Shadreck Chitsonga, Bhavana Burell, Ian Toppin

Mathematics Grants Collections

This Grants Collection for College Algebra was created under a Round Two ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

College Algebra (University Of North Georgia), Minsu Kim, Hashim Saber, Bikash Das, Thomas Hartfield

Mathematics Grants Collections

This Grants Collection for College Algebra was created under a Round Four ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

On Emmy Noether And Her Algebraic Works, Deborah Radford

All Student Theses and Dissertations

In the early 1900s a rising star in the mathematics world was emerging. I will discuss her life as a female mathematician and the struggles she faced being a rebel in her time. I will also take an in depth look at some of her contributions to the mathematics and science community . Her work in algebra and more specifically, ring theory, are said to be foundations for much of the work done since then. Her developments in abstract algebra helped to unify topology, geometry, logic and linear algebra. Also, Noether's theorem is a widely used theorem in physics along …

Using Ipads And Video-Based Instruction To Teach Algebra To High School Students With Disabilities, Elias Clinton, Tom J. Clees

National Youth Advocacy & Resilience Conference

This presentation targets a study in which four high school students with disabilities were taught to solve algebraic equations using iPads and video-based instruction. All students showed immediate increases in accurate responding following the introduction of the video-based intervention. This presentation provides practitioners with a flexible technology-based intervention for students with disabilities in need of grade-level academic instruction. The intervention could be used across a variety of subjects and academic tasks.

Review: A C*-Algebra Approach To Complex Symmetric Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Impartial Avoidance Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

Mathematics Faculty Publications

We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.

Supplemental Instruction For Developmental Mathematics: Two-Year Summary, Olen Dias, Alice W. Cunningham, Loreto Porte

Publications and Research

Supplemental instruction—using trained peer tutors to conduct additional class sessions in a group-work format—has been in use for over forty years. However, its success in developmental mathematics has been inconclusive. In the two years since institution of the strategy for developmental mathematics students at Hostos Community College, overall results (n = 5403 students) show significantly improved course pass rates to at least a 99% confidence level. Although no significant course retention differences have yet appeared, academic success itself promotes future retention. The program has proved beneficial for the College’s developmental mathematics students and is being expanded. Future research including …

Jay Leno And Abstract Algebra, Adam Glesser, Martin Bonsangue

Journal of Humanistic Mathematics

The Jay Leno skit Jaywalking, showing ordinary people struggling to answer basic questions, is both entertaining and applicable to teaching. This article describes how an instructor can strengthen students' conceptual understanding by creating an element of confusion, or "cognitive dissonance," in the students' minds using Jaywalking-style interactions in the classroom.

Dramathizing Functions: Building Connections Between Mathematics And Arts, Gunhan Caglayan

Journal of Humanistic Mathematics

This article focuses on connections between mathematics and performance arts (drama). More specifically we offer an exposition of a segment of college algebra mathematics (an introduction to functions), with an approach primarily emphasizing the aesthetic aspects of mathematical learning, teaching, and performing.

On The Equivalence Of Probability Spaces, Daniel Alpay, Palle Jorgensen, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

For a general class of Gaussian processes W, indexed by a sigma-algebra F of a

general measure space (M,F, _), we give necessary and sufficient conditions for the validity

of a quadratic variation representation for such Gaussian processes, thus recovering _(A),

for A 2 F, as a quadratic variation of W over A. We further provide a harmonic analysis

representation for this general class of processes. We apply these two results to: (i) a computation

of generalized Ito-integrals; and (ii) a proof of an explicit, and measure-theoretic

equivalence formula, realizing an equivalence between the two approaches to Gaussian processes,

one …

Review: Transitivity And Bundle Shifts, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Lattices From Hermitian Function Fields, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

Pomona Faculty Publications and Research

We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of …

Graph Cohomology, Matthew Lin

HMC Senior Theses

What is the cohomology of a graph? Cohomology is a topological invariant and encodes such information as genus and euler characteristic. Graphs are combinatorial objects which may not a priori admit a natural and isomorphism invariant cohomology ring. In this project, given any finite graph G, we constructively define a cohomology ring H*(G) of G. Our method uses graph associahedra and toric varieties. Given a graph, there is a canonically associated convex polytope, called the graph associahedron, constructed from G. In turn, a convex polytope uniquely determines a toric variety. We synthesize these results, and describe the …

Convexity Of Neural Codes, Robert Amzi Jeffs

HMC Senior Theses

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to …