Algebra Commons^™

Residuated Maps, The Way-Below Relation, And Contractions On Probabilistic Metric Spaces., M. Ryan Luke

Electronic Theses and Dissertations

In this dissertation, we will examine residuated mappings on a function lattice and how they behave with respect to the way-below relation. In particular, which residuated $\phi$ has the property that $F$ is way-below $\phi(F)$ for $F$ in appropriate sets. We show the way-below relation describes the separation of two functions and how this corresponds to contraction mappings on probabilistic metric spaces. A new definition for contractions is considered using the way-below relation.

Cayley Graphs Of Groups And Their Applications, Anna Tripi

Graduate Theses/Dissertations

Cayley graphs are graphs associated to a group and a set of generators for that group (there is also an associated directed graph). The purpose of this study was to examine multiple examples of Cayley graphs through group theory, graph theory, and applications. We gave background material on groups and graphs and gave numerous examples of Cayley graphs and digraphs. This helped investigate the conjecture that the Cayley graph of any group (except Z_2) is hamiltonian. We found the conjecture to still be open. We found Cayley graphs and hamiltonian cycles could be applied to campanology (in particular, to the …

Solving A System Of Linear Equations Using Ancient Chinese Methods, Mary Flagg

Linear Algebra

No abstract provided.

Blow-Up Algebras, Determinantal Ideals, And Dedekind-Mertens-Like Formulas, Alberto Corso, Uwe Nagel, Sonja Petrović, Cornelia Yuen

Mathematics Faculty Publications

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine …

College Algebra, Trigonometry, And Precalculus (Clayton), Chaogui Zhang, Scott Bailey, Billie May, Jelinda Spotorno, Kara Mullen

Mathematics Grants Collections

This Grants Collection for College Algebra, Trigonometry, and Precalculus was created under a Round Five 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

Foundations For College Algebra, Da'mon Andrews, Antre' Drummer

Mathematics Grants Collections

This Grants Collection for Biochemistry was created under a Round Seven 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

Multi-Type Display Calculus For Semi-De Morgan Logic, Giuseppe Greco, Fei Liang, M. Andrew Moshier, Alessandra Palmigiano

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce a proper multi-type display calculus for semi De Morgan logic which is sound, complete, conservative, and enjoys cut-elimination and subformula property. Our proposal builds on an algebraic analysis of semi De Morgan algebras and applies the guidelines of the multi-type methodology in the design of display calculi.

Discovery Learning Plus Direct Instruction Equals Success: Modifying American Math Education In The Algebra Classroom, Sean P. Ferrill Mr.

Honors Projects

In light of both high American failure rates in algebra courses and the significant proportion of innumerate American students, this thesis examines a variety of effective educational methods in mathematics. Constructivism, discovery learning, traditional instruction, and the Japanese primary education system are all analyzed to incorporate effective education techniques. Based on the meta-analysis of each of these methods, a hybrid method has been constructed to adapt in the American Common Core algebra classroom.

Elimination For Systems Of Algebraic Differential Equations, Richard Gustavson

Dissertations, Theses, and Capstone Projects

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the ``algebraic data" of …

Solving Algorithmic Problems In Finitely Presented Groups Via Machine Learning, Jonathan Gryak

Dissertations, Theses, and Capstone Projects

Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this dissertation, we seek to extend these techniques to finitely presented non-free groups, in particular to polycyclic and metabelian groups that are of interest to non-commutative cryptography.

As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using decision trees, random forests, and N-tuple neural network models are evaluated for several non-free groups. …

Conquering Worrisome Word Problems – Algebra Success, Vicki-Lynn Holmes, Karla Spence, Jane Finn, Shelia Mcgee Ingram, Libbey Horton

Faculty Publications

High school students can struggle with word problems in upper level math classes. Causes for this struggle could include lower reading comprehension, limited mathematic vocabulary, and difficulty changing words to algebraic expressions. This article proposes three techniques to help teachers instruct these struggling students that include (a) organization by difficulty of comprehension and computation (b) scaffolding and (c) utilizing the Explain, Practice and Assess (EPA) strategy.

Representation And Decomposition Of An Intuitionistic Fuzzy Matrix Using Some (Α, Α') Cuts, T. Muthuraji, S. Sriram

Applications and Applied Mathematics: An International Journal (AAM)

The aim of this paper is to study the properties of various (α, α') cuts on Intuitionistic Fuzzy Matrices. Here we introduce different kinds of cuts on Intuitionistic Fuzzy Sets. We discuss some properties of the cuts with some other existing operators on Intuitionistic Fuzzy Matrix. Finally some representation and decomposition of an Intuitionistic Fuzzy Matrix using (α, α') cuts are given.

On (Semi)Topological Bcc-Algebras, F. R. Setudeh, N. Kouhestani

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we introduce the notion of (semi)topological BCC-algebras and derive here conditions that imply a BCC-algebra to be a (semi)topological BCC-algebra. We prove that for each cardinal number α there is at least a (semi)topological BCC-algebra of order α: Also we study separation axioms on (semi)topological BCC-algebras and show that for any infinite cardinal number α there is a Hausdorff (semi)topological BCC-algebra of order α with nontrivial topology.

Some Relations On Generalized Rice's Matrix Polynomials, Ayman Shehata

Applications and Applied Mathematics: An International Journal (AAM)

The main aim of this paper is to obtain certain properties of generalized Rice’s matrix polynomials such as their matrix differential equation, generating matrix functions, an expansion for them. We have also deduced the various families of bilinear and bilateral generating matrix functions for them with the help of the generating matrix functions developed in the paper and some of their applications have also been presented here.

On Circulant-Like Rhotrices Over Finite Fields, P. L. Sharma, Shalini Gupta, Mansi Rehan

Applications and Applied Mathematics: An International Journal (AAM)

Circulant matrices over finite fields are widely used in cryptographic hash functions, Lattice based cryptographic functions and Advanced Encryption Standard (AES). Maximum distance separable codes over finite field GF2 have vital a role for error control in both digital communication and storage systems whereas maximum distance separable matrices over finite field GF2 are used in block ciphers due to their properties of diffusion. Rhotrices are represented in the form of coupled matrices. In the present paper, we discuss the circulant- like rhotrices and then construct the maximum distance separable rhotrices over finite fields.

Some Properties Of Certain Mixed Type Special Matrix Functions And Polynomials, Ahmed A. Al-Gonah, Fatima M. Al-Samadi

Applications and Applied Mathematics: An International Journal (AAM)

By using certain operational methods, the authors introduce some new mixed type special matrix functions and polynomials. Some properties of these matrix functions and polynomials are established.

Simple And Semi-Simple Artinian Rings, Ulyses Velasco

Electronic Theses, Projects, and Dissertations

The main purpose of this paper is to examine the road towards the structure of simple and semi-simple Artinian rings. We refer to these structure theorems as the Wedderburn-Artin theorems. On this journey, we will discuss R-modules, the Jacobson radical, Artinian rings, nilpotency, idempotency, and more. Once we reach our destination, we will examine some implications of these theorems. As a fair warning, no ring will be assumed to be commutative, or to have unity. On that note, the reader should be familiar with the basic findings from Group Theory and Ring Theory.

The Recognition Problem For Table Algebras And Reality-Based Algebras, Allen Herman, Mikhail Muzychuk, Bangteng Xu

EKU Faculty and Staff Scholarship

Given a finite-dimensional noncommutative semisimple algebra A over C with involution, we show that A always has a basis B for which ( A , B ) is a reality-based algebra. For algebras that have a one-dimensional representation δ , we show that there always exists an RBA-basis for which δ is a positive degree map. We characterize all RBA-bases of the 5-dimensional noncommutative semisimple algebra for which the algebra has a positive degree map, and give examples of RBA-bases of C ⊕ M n ( C ) for which the RBA has a positive degree map, for all n …

Beurling-Lax Type Theorems In The Complex And Quaternionic Setting, Daniel Alpay, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give a generalization of the Beurling–Lax theorem both in the complex and quaternionic settings. We consider in the first case functions meromorphic in the right complex half-plane, and functions slice hypermeromorphic in the right quaternionic half-space in the second case. In both settings we also discuss a unified framework, which includes both the disk and the half-plane for the complex case and the open unit ball and the half-space in the quaternionic setting.

On Rings Of Invariants For Cyclic P-Groups, Daniel Juda

Graduate Theses and Dissertations

This thesis studies the ring of invariants R^G of a cyclic p-group G acting on k[x_1,\ldots, x_n] where k is a field of characteristic p >0. We consider when R^G is Cohen-Macaulay and give an explicit computation of the depth of R^G. Using representation theory and a result of Nakajima, we demonstrate that R^G is a unique factorization domain and consequently quasi-Gorenstein. We answer the question of when R^G is F-rational and when R^G is F-regular.

We also study the a-invariant for a graded ring S, that is, the maximal graded degree of the top local cohomology module of S. …