Algebra Commons^™

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.

Beurling-Lax Type Theorems And Cuntz Relations, Daniel Alpay, Fabrizio Colombo, Irene Sabadini, Baruch Schneider

Mathematics, Physics, and Computer Science Faculty Articles and Research

We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.

Free Complexes Over The Exterior Algebra With Small Homology, Erica Hopkins

Department of Mathematics: Dissertations, Theses, and Student Research

Let M be a graded module over a standard graded polynomial ring S. The Total Rank Conjecture by Avramov-Buchweitz predicts the total Betti number of M should be at least the total Betti number of the residue field. Walker proved this is indeed true in a large number of cases. One could then try to push this result further by generalizing this conjecture to finite free complexes which is known as the Generalized Total Rank Conjecture. However, Iyengar and Walker constructed examples to show this generalized conjecture is not always true.

In this thesis, we investigate other counterexamples of …

Algorithms Related To Triangle Groups, Bao The Pham

LSU Doctoral Dissertations

Given a finite index subgroup of $\PSL_2(\Z)$, one can talk about the different properties of this subgroup. These properties have been studied extensively in an attempt to classify these subgroups. Tim Hsu created an algorithm to determine whether a subgroup is a congruence subgroup by using permutations \cite{hsu}. Lang, Lim, and Tan also created an algorithm to determine if a subgroup is a congruence subgroup by using Farey Symbols \cite{llt}. Sebbar classified torsion-free congruence subgroups of genus 0 \cite{sebbar}. Pauli and Cummins computed and tabulated all congruence subgroups of genus less than 24 \cite{ps}. However, there are still some problems …

Injective And Projective Semimodules Over Involutive Semirings, Peter Jipsen, Sara Vanucci

Mathematics, Physics, and Computer Science Faculty Articles and Research

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval [d,1] to be a subalgebra of an involutive residuated lattice, where d is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for …

Irreducibility And Galois Groups Of Random Polynomials, Hanson Hao, Eli Navarro, Henri Stern

Rose-Hulman Undergraduate Mathematics Journal

In 2015, I. Rivin introduced an effective method to bound the number of irreducible integral polynomials with fixed degree d and height at most N. In this paper, we give a brief summary of this result and discuss the precision of Rivin's arguments for special classes of polynomials. We also give elementary proofs of classic results on Galois groups of cubic trinomials.

Disjointness Of Linear Fractional Actions On Serre Trees, Henry W. Talbott

Rose-Hulman Undergraduate Mathematics Journal

Serre showed that, for a discrete valuation field, the group of linear fractional transformations acts on an infinite regular tree with vertex degree determined by the residue degree of the field. Since the p-adics and the polynomials over the finite field of order p act on isomorphic trees, we may ask whether pairs of actions from these two groups are ever conjugate as tree automorphisms. We analyze permutations induced on finite vertex sets, and show a permutation classification result for actions by these linear fractional transformation groups. We prove that actions by specific subgroups of these groups are conjugate only …

Directed Graphs Of The Finite Tropical Semiring, Caden G. Zonnefeld

Rose-Hulman Undergraduate Mathematics Journal

The focus of this paper lies at the intersection of the fields of tropical algebra and graph theory. In particular the interaction between tropical semirings and directed graphs is investigated. Originally studied by Lipvoski, the directed graph of a ring is useful in identifying properties within the algebraic structure of a ring. This work builds off research completed by Beyer and Fields, Hausken and Skinner, and Ang and Shulte in constructing directed graphs from rings. However, we will investigate the relationship (x, y)→(min(x, y), x+y) as defined by the operations of tropical algebra and applied to tropical semirings.

Computable Model Theory On Loops, Josiah Schmidt

All NMU Master's Theses

We give an introduction to the problem of computable algebras. Specifically, the algebras of loops and groups. We start by defining a loop and group, then give some of their properties. We then give an overview of comptability theory, and apply it to loops and groups. We conclude by showing that a finitely presented residually finite algebra has a solvable word problem.

Modernization Of Scienttific Mathematics Formula In Technology, Iwasan D. Kejawa Ed.D, Prof. Iwasan D. Kejawa Ed.D

Department of Mathematics: Faculty Publications

Abstract
Is it true that we solve problem using techniques in form of formula? Mathematical formulas can be derived through thinking of a problem or situation. Research has shown that we can create formulas by applying theoretical, technical, and applied knowledge. The knowledge derives from brainstorming and actual experience can be represented by formulas. It is intended that this research article is geared by an audience of average knowledge level of solving mathematics and scientific intricacies. This work details an introductory level of simple, at times complex problems in a mathematical epidermis and computability and solvability in a Computer Science. …

Noncommutative Tensor Triangular Geometry And Its Applications To Representation Theory, Kent Barton Vashaw

LSU Doctoral Dissertations

One of the cornerstones of the representation theory of Hopf algebras and finite tensor categories is the theory of support varieties. Balmer introduced tensor triangular geometry for symmetric monoidal triangulated categories, which united various support variety theories coming from disparate areas such as homotopy theory, algebraic geometry, and representation theory. In this thesis a noncommutative version will be introduced and developed. We show that this noncommutative analogue of Balmer's theory can be determined in many concrete situations via the theory of abstract support data, and can be used to classify thick tensor ideals. We prove an analogue of prime ideal …

On The Generalization Of Interval Valued Fuzzy Generalized Bi-Ideals In Ordered Semigroups, Muhammad S. Ali Khan, Saleem Abdullah, Kostaq Hila

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a new general form than interval valued fuzzy generalized bi-ideals in ordered semigroups is introduced. The concept of interval valued fuzzy generalized bi-ideals is initiated and several properties and characterizations are provided. A condition for an interval valued fuzzy generalized bi-ideal to be an interval valued fuzzy generalized bi-ideal is obtained. Using implication operators and the notion of implication-based an interval valued fuzzy generalized bi-ideal, characterizations of an interval valued fuzzy generalized bi-ideal and an interval valued fuzzy generalized bi-ideal are considered.

Hamacher Operations Of Fermatean Fuzzy Matrices, I. Silambarasan

Applications and Applied Mathematics: An International Journal (AAM)

The purpose of this study is to extend the Fermatean fuzzy matrices to the theory of Hamacher operations. In this paper, the concept of Hamacher operations of Fermatean fuzzy matrices are introduced and some desirable properties of these operations, such as commutativity, idempotency, and monotonicity are discussed. Further, we prove DeMorgan’s laws over complement for these operations. Furthermore, the scalar multiplication and exponentiation operations of Fermatean fuzzy matrices are constructed and their algebraic properties are investigated. Finally, some properties of necessity and possibility operators of Fermatean fuzzy matrices are proved.

On The Representation Of Boolean Magmas And Boolean Semilattices, Peter Jipsen, M. Eyad Kurd-Misto, James Wimberley

Mathematics, Physics, and Computer Science Faculty Books and Book Chapters

A magma is an algebra with a binary operation ·, and a Boolean magma is a Boolean algebra with an additional binary operation · that distributes over all finite Boolean joins. We prove that all square-increasing (x ≤ x²) Boolean magmas are embedded in complex algebras of idempotent (x = x²) magmas. This solves a problem in a recent paper [3] by C. Bergman. Similar results are shown to hold for commutative Boolean magmas with an identity element and a unary inverse operation, or with any combination of these properties.

A Boolean semilattice is …

N-Fold Matrix Factorizations, Eric Hopkins

Department of Mathematics: Dissertations, Theses, and Student Research

The study of matrix factorizations began when they were introduced by Eisenbud; they have since been an important topic in commutative algebra. Results by Eisenbud, Buchweitz, and Yoshino relate matrix factorizations to maximal Cohen-Macaulay modules over hypersurface rings. There are many important properties of the category of matrix factorizations, as well as tensor product and hom constructions. More recently, Backelin, Herzog, Sanders, and Ulrich used a generalization of matrix factorizations -- so called N-fold matrix factorizations -- to construct Ulrich modules over arbitrary hypersurface rings. In this dissertation we build up the theory of N-fold matrix factorizations, proving analogues of …

Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell

Theses

This paper explores and elaborates on a method of solving Pell’s equation as introduced by Norman Wildberger. In the first chapters of the paper, foundational topics are introduced in expository style including an explanation of Pell’s equation. An explanation of continued fractions and their ability to express quadratic irrationals is provided as well as a connection to the Stern-Brocot tree and a convenient means of representation for each in terms of 2×2 matrices with integer elements. This representation will provide a useful way of navigating the Stern-Brocot tree computationally and permit us a means of computing continued fractions without the …

Algebraic Structures And Variations: From Latin Squares To Lie Quasigroups, Erik Flinn

All NMU Master's Theses

In this Master's Thesis we give an overview of the algebraic structure of sets with a single binary operation. Specifically, we are interested in quasigroups and loops and their historical connection with Latin squares; considering them in both finite and continuous variations. We also consider various mappings between such algebraic objects and utilize matrix representations to give a negative conclusion to a question concerning isotopies in the case of quasigroups.

Frobenius And Homological Dimensions Of Complexes, Taran Funk

Department of Mathematics: Dissertations, Theses, and Student Research

Much work has been done showing how one can use a commutative Noetherian local ring R of prime characteristic, viewed as algebra over itself via the Frobenius endomorphism, as a test for flatness or projectivity of a finitely generated module M over R. Work on this dates back to the famous results of Peskine and Szpiro and also that of Kunz. Here I discuss what work has been done to push this theory into modules which are not necessarily finitely generated, and display my work done to weaken the assumptions needed to obtain these results.

Adviser: Tom Marley

Symmetric Presentation Of Finite Groups, And Related Topics, Marina Michelle Duchesne

Electronic Theses, Projects, and Dissertations

We have discovered original symmetric presentations for several finite groups, including 2²:^.(2⁴:(2^.S₃)), M₁₁, 3:(PSL(3,3):2), S₈, and 2^.M₁₂. We have found homomorphic images of several progenitors, including 2^*18:((6x2):6), 2^*24:(2^.S₄), 2^*105:A₇, 3^*3:_m(2³:3), 7^*8:_m(PSL(2,7):2), 3^*4:_m(4²:2²), 7^*5:(2xA₅), and 5^*6:_mS₅. We have provided the isomorphism type of …

On The Hamiltonicity Of Subgroup Lattices, Nicholas Charles Fleece

Graduate Theses/Dissertations

In this paper we discuss the Hamiltonicity of the subgroup lattices of

different classes of groups. We provide sufficient conditions for the

Hamiltonicity of the subgroup lattices of cube-free abelian groups. We also

prove the non-Hamiltonicity of the subgroup lattices of dihedral and

dicyclic groups. We disprove a conjecture on non-abelian p-groups by

producing an infinite family of non-abelian p-groups with Hamiltonian

subgroup lattices. Finally, we provide a list of the Hamiltonicity of the

subgroup lattices of every finite group up to order 35 barring two groups.

Proper Sum Graphs, Austin Nicholas Beard

Graduate Theses/Dissertations

The Proper Sum Graph of a commutative ring with identity has the prime ideals as vertices, with two ideals adjacent if their sum is a proper ideal. This thesis expands upon the research of Dhorajia. We will cover the groundwork to understanding the basics of these graphs, and gradually narrow our efforts into the minimal prime ideals of the ring.

On Elliptic Curves, Montana S. Miller

Graduate Theses/Dissertations

An elliptic curve over the rational numbers is given by the equation y² = x³+Ax+B. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.

Classification Of Cayley Rose Window Graphs, Angsuman Das, Arnab Mandal

Theory & Applications of Graphs

Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, Kovacs, Kutnar and Marusic classified the edge-transitive rose window graphs and Dobson, Kovacs and Miklavic characterized the vertex transitive rose window graphs. In this paper, we classify the Cayley rose window graphs.

Normality Properties Of Composition Operators, Grace Weeks, Hallie Kaiser, Katy O'Malley

Celebration of Scholarship 2021

We explore two main concepts in relation to truncated composition matrices: the conditions required for the binormal and commutative properties. Both of these topics are important in linear algebra due to their connection with diagonalization.

We begin with the normal solution before moving onto the more complex binormal solutions. Then we cover conditions for the composition matrix to commute with the general matrix. Finally, we end with ongoing questions for future work.

Factoring: Difference Of Squares, Thomas Lauria

Open Educational Resources

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

Matrix Product Structure Of A Permuted Quasi Cyclic Code And Its Dual, Perian Perdhiku

Dissertations

In my Dissertation I will work mostly with Permuted Quasi Cyclic Codes. They are a generalization of Cyclic Codes, one of the most important families of Linear Codes in Coding Theory. Linear Codes are very useful in error detection and correction. Error Detection and Correction is a technique that first detects the corrupted data sent from some transmitter over unreliable communication channels and then corrects the errors and reconstructs the original data. Unlike linear codes, cyclic codes are used to correct errors where the pattern is not clear and the error occurs in a short segment of …

Lecture 00: Opening Remarks: 46th Spring Lecture Series, Tulin Kaman

Mathematical Sciences Spring Lecture Series

Opening remarks for the 46th Annual Mathematical Sciences Spring Lecture Series at the University of Arkansas, Fayetteville.

On Leibniz Cohomology, Jorg Feldvoss, Friedrich Wagemann

University Faculty and Staff Publications

In this paper we prove the Leibniz analogue of Whitehead's vanishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we obtain the second Whitehead lemma for Leibniz algebras. Moreover, we compute the cohomology of several Leibniz algebras with ad joint or irreducible coefficients. Our main tool is a Leibniz analogue of the Hochschild-Serre spectral sequence, which is an extension of the dual of a spectral sequence of Pirashvili for Leibniz homology from symmetric bimodules to arbitrary bimodules.

Some Model Theory Of Free Groups, Christopher James Natoli

Dissertations, Theses, and Capstone Projects

There are two main sets of results, both pertaining to the model theory of free groups. In the first set of results, we prove that non-abelian free groups of finite rank at least 3 or of countable rank are not A-homogeneous. We then build on the proof of this result to show that two classes of groups, namely finitely generated free groups and finitely generated elementary free groups, fail to form A-Fraisse classes and that the class of non-abelian limit groups fails to form a strong A-Fraisse class.

The second main result is that if a countable group is elementarily …

Total Differentiability And Monogenicity For Functions In Algebras Of Order 4, I. Sabadini, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we discuss some notions of analyticity in associative algebras with unit. We also recall some basic tool in algebraic analysis and we use them to study the properties of analytic functions in two algebras of dimension four that played a relevant role in some work of the Italian school, but that have never been fully investigated.