Algebra Commons^™

Partially-Ordered Multi-Type Algebras, Display Calculi And The Category Of Weakening Relations, Peter Jipsen, Fei Liang, M. Andrew Moshier, Apostolos Tzimoulis

Mathematics, Physics, and Computer Science Faculty Articles and Research

"We define partially-ordered multi-type algebras and use them as algebraic semantics for multi-type display calculi that have recently been developed for several logics, including dynamic epistemic logic [7], linear logic[10], lattice logic [11], bilattice logic [9] and semi-De Morgan logic [8]."

Equations Of Multi-Rees Algebras, Babak Jabbar Nezhad

Graduate Theses and Dissertations

In this thesis we describe the defining equations of certain multi-Rees algebras. First, we determine the defining equations of the multi-Rees algebra $R[I^{a_1}t_1,\dots,I^{a_r}t_r]$ over a Noetherian ring $R$ when $I$ is an ideal of linear type. This generalizes a result of Ribbe and recent work of Lin-Polini and Sosa. Second, we describe the equations defining the multi-Rees algebra $R[I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where $R$ is a Noetherian ring containing a field and the ideals are generated by a subset of a fixed regular sequence.

Determining The Determinant, Danny Otero

Linear Algebra

No abstract provided.

Webwork Problems For Linear Algebra, Hashim Saber, Beata Hebda

Mathematics Ancillary Materials

This set of problems for Linear Algebra in the open-source WeBWorK mathematics platform was created under a Round Eleven Mini-Grant for Ancillary Materials Creation. The problems were created for an implementation of the CC-BY Lyrix open textbook A First Course in Linear Algebra. Also included as an additional file are the selected and modified Lyryx Class Notes for the textbook.

Topics covered include:

• Linear Independence
• Linear Transformations
• Matrix of a Transformation
• Isomorphisms
• Eigenvalues and Eigenvectors
• Diagonalization
• Orthogonality

Finding Meaning In A Multivariable World: A Conceptual Approach To An Algebra-Based Second Course In Statistics, Karen Mcgaughey, Beth Chance, Nathan L. Tintle, Soma Roy, Todd Swanson, Jill Vander Stoep

Faculty Work Comprehensive List

Although the teaching of the first course in statistics has improved dramatically in recent years, there has been less focus on a similarly conceptual-based second course aimed at non-majors. We present a curriculum for the second course, designed to expand statistical literacy across disciplines, which focuses on conceptual understanding of multivariable relationships through data visualization, study design, the role of confounding variables, reduction of unexplained variation, and simulation-based inference, rather than the mathematically-based discourse often used in the second course. Our curriculum uses a student-centered pedagogical approach, utilizing guided discovery activities based on real-world case studies, facilitated by student-focused technology …

Invariant Of Noncommutative Algebras And Poisson Geometry, Bach Van Nguyen

LSU Doctoral Dissertations

In this dissertation, we describe the structure of discriminant of noncommutative algebras using the theory of Poisson quantization and ring theoretic properties of Poisson algebra. In particular, under appropriate conditions, we express the discriminant of specialization of K[q^{+-1}]-algebras as product of Poisson prime elements in some Poisson central subalgebra. In addition, we provide methods for computing noncommutative discriminant in various settings using results obtained for specialization of K[q^{+-1}]-algebras. Further, to demonstrate, we explicitly compute the discriminant of algebra of quantum matrices and quantum Schubert cell algebras specializing at roots of unity. This dissertation is part of the collaboration with Trampel …

Dimers On Cylinders Over Dynkin Diagrams And Cluster Algebras, Maitreyee Chandramohan Kulkarni

LSU Doctoral Dissertations

This dissertation describes a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well-studied case of dimer models on a disc. We prove that all Berenstein--Fomin--Zelevinsky quivers for Schubert cells in a symmetric Kac--Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers.

Homomorphism Of Fuzzy Multigroups And Some Of Its Properties, P. A. Ejegwa

Applications and Applied Mathematics: An International Journal (AAM)

In a way, the notion of fuzzy multigroups is an application of fuzzy multisets to the theory of group. The concept of fuzzy multigroups is a new algebraic structure of uncertainty which generalizes fuzzy groups. Fuzzy multigroup is a multiset of X x [0; 1] satisfying some set of axioms, where X is a classical group. In this paper, we propose the concept of homomorphism in fuzzy multigroups context. Some homomorphic properties of fuzzy multigroups are explicated. Again, we show that the homomorphic image and homomorphic preimage of fuzzy multigroups are also fuzzy multigroups. Finally, we present some homomorphic properties …

Study Of Pseudo Bl–Algebras In View Of Left Boolean Lifting Property, B. Barani Nia, A. B. Saeid

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we define left Boolean lifting property (right Boolean lifting property) LBLP (RBLP) for pseudo BL–algebra which is the property that all Boolean elements can be lifted modulo every left filter (right filter) and next, we study pseudo BL-algebra with LBLP (RBLP). We show that Quasi local, local and hyper Archimedean pseudo BL–algebra that have LBLP (RBLP) has an interesting behavior in direct products. LBLP (RBLP) provides an important representation theorem for semi local and maximal pseudo BL–algebra.

Symmetric Presentations, Representations, And Related Topics, Adam Manriquez

Electronic Theses, Projects, and Dissertations

The purpose of this thesis is to develop original symmetric presentations of finite non-abelian simple groups, particularly the sporadic simple groups. We have found original symmetric presentations for the Janko group J₁, the Mathieu group M₁₂, the Symplectic groups S(3,4) and S(4,5), a Lie type group Suz(8), and the automorphism group of the Unitary group U(3,5) as homomorphic images of the progenitors 2^*60 : (2 x A₅), 2^{*60 :} A₅, 2^*56 : (2³ : 7), and 2^*28 : (PGL(2,7):2), respectively. We have also discovered the groups 2 …

Simple Groups, Progenitors, And Related Topics, Angelica Baccari

Electronic Theses, Projects, and Dissertations

The foundation of the work of this thesis is based around the involutory progenitor and the finite homomorphic images found therein. This process is developed by Robert T. Curtis and he defines it as 2^{*n} :N {pi w | pi in N, w} where 2^{*n} denotes a free product of n copies of the cyclic group of order 2 generated by involutions. We repeat this process with different control groups and a different array of possible relations to discover interesting groups, such as sporadic, linear, or unitary groups, to name a few. Predominantly this work was produced from transitive groups …

Supporting English Language Learners Inside The Mathematics Classroom: One Teacher’S Unique Perspective Working With Students During Their First Years In America, Amy Marie Fendrick

Research and Evaluation in Education, Technology, Art, and Design

Reflecting upon my personal experiences teaching mathematics to English Language Learners (ELL) in a public high school in Lincoln, Nebraska, this essay largely focuses on the time I spent as the only Accelerated Math teacher in my school building. From 2012 – 2017, I taught three different subjects at this high school: Advanced Algebra, Algebra, and Accelerated Math. This essay highlights why I chose to become a math and ELL teacher, as well as the challenges, issues, struggles, and successes I experienced during my time teaching. I focus on the challenges I faced teaching students who did not share my …

Matrix Methods Of Data Analysis, Kaiye Yu

Renée Crown University Honors Thesis Projects - All

Nowadays, data is playing a bigger part than ever before in the history. In order to get more useful information, methods involving matrix are powerful. There are different algorithms that are able to help one to learn the information they need from data. In this study, there are two main algorithms that I will focus on. One is Pagerank algorithm, a traditional algorithm that was applied to searching engine decades ago in Google. However, Pagerank algorithm has certain limits in providing information like covariance between different factors. Thus, another method is also studied, which is principal component analysis (PCA), while …

Resolutions Of Finite Length Modules Over Complete Intersections, Seth Lindokken

Department of Mathematics: Dissertations, Theses, and Student Research

The structure of free resolutions of finite length modules over regular local rings has long been a topic of interest in commutative algebra. Conjectures by Buchsbaum-Eisenbud-Horrocks and Avramov-Buchweitz predict that in this setting the minimal free resolution of the residue field should give, in some sense, the smallest possible free resolution of a finite length module. Results of Tate and Shamash describing the minimal free resolution of the residue field over a local hypersurface ring, together with the theory of matrix factorizations developed by Eisenbud and Eisenbud-Peeva, suggest analogous lower bounds for the size of free resolutions of finite length …

Nonassociative Right Hoops, Peter Jipsen, Michael Kinyon

Mathematics, Physics, and Computer Science Faculty Articles and Research

The class of nonassociative right hoops, or narhoops for short, is defined as a subclass of right-residuated magmas, and is shown to be a variety. These algebras generalize both right quasigroups and right hoops, and we characterize the subvarieties in which the operation x ^^ y = (x/y)y is associative and/or commutative. Narhoops with a left unit are proved to be integral if and only if ^ is commutative, and their congruences are determined by the equivalence class of the left unit. We also prove that the four identities defining narhoops are independent.

An Investigation Into The Properties Of Quaternions: Their Origin, Basic Properties, Functional Analysis, And Algebraic Characteristics, James Miller

Masters Essays

No abstract provided.

Cayley Graphs Of Psl(2) Over Finite Commutative Rings, Kathleen Bell

Masters Theses & Specialist Projects

Hadwiger's conjecture is one of the deepest open questions in graph theory, and Cayley graphs are an applicable and useful subtopic of algebra.

Chapter 1 will introduce Hadwiger's conjecture and Cayley graphs, providing a summary of background information on those topics, and continuing by introducing our problem. Chapter 2 will provide necessary definitions. Chapter 3 will give a brief survey of background information and of the existing literature on Hadwiger's conjecture, Hamiltonicity, and the isoperimetric number; in this chapter we will explore what cases are already shown and what the most recent results are. Chapter 4 will give our decomposition …

Linear Algebra (Ung), Hashim Saber, Beata Hebda, Piotr Hebda, Benkam Bobga

Mathematics Grants Collections

This Grants Collection for Linear Algebra 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

Some Studies On Algebraic Integers In Q(I,√3) By Using Coset Diagram, Florentin Smarandache, Saima Anis, Seok-Zun Song, Young Bae Jun

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we studied the action of Picard modular group PSL(2,Z[i])

On Lattices Of Z-Ideals Of Function Rings, Themba Dube, Oghenetega Ighedo

Mathematics, Physics, and Computer Science Faculty Articles and Research

An ideal I of a ring A is a z-ideal if whenever a, b ∈ A belong to the same maximal ideals of A and a ∈ I, then b ∈ I as well. On the other hand, an ideal J of A is a d-ideal if Ann²(a) ⊆ J for every a ∈ J. It is known that the lattices Z(L) and D(L) of the ring 𝓡L of continuous real-valued functions on a frame L, consisting of z-ideals and d-ideals of 𝓡 …

Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

Let S be a Riemann surface and G a large subgroup of Aut(S) (Aut(S) may be unknown). We are particularly interested in regular n-gonal surfaces, i.e., the quotient surface S/G (and hence S/Aut(S)) has genus zero. For various H the ramification information of the branched coverings S/K -> S/H may be captured in a matrix. The ramification information, in particular strong branching, may be then be used in analyzing the structure of Aut(S). The ramification information is conjugation invariant so the matrix's rows and columns may be indexed by conjugacy classes of subgroups. The only required …

Monomial Progenitors And Related Topics, Madai Obaid Alnominy

Electronic Theses, Projects, and Dissertations

The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M₁₁, HS × D₅, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L₂(149) as homomorphic images of the monomial progenitors 11*⁴ :_m (5 :4), 5*⁶ :_m S₅ and 149*² :_m D₃₇. We have also discovered 2 …

Progenitors, Symmetric Presentations, And Related Topics, Joana Viridiana Luna

Electronic Theses, Projects, and Dissertations

Abstract

A progenitor developed by Robert T. Curtis is a type of infinite groups formed by the semi-direct product of a free group m∗n and a transitive permutation group of degree n. To produce finite homomorphic images we had to add relations to the progenitor of the form 2∗n : N. In this thesis we have investigated several permutations progenitors and monomials, 2∗12 : S4, 2∗12 : S4 × 2, 2∗13 : (13 : 4), 2∗30 : ((2• : 3) : 5), 2∗13 :13,2∗13 :(13:2),2∗13 :(13:S3),53∗2 :m (13:4),7∗8 :m (32 :8),and 53∗4 :m (13 : 4). We have discovered that …

Progenitors, Symmetric Presentations And Constructions, Diana Aguirre

Electronic Theses, Projects, and Dissertations

Abstract

In this project, we searched for new constructions and symmetric presentations of important groups, nonabelian simple groups, their automorphism groups, or groups that have these as their factor groups. My target nonabelian simple groups included sporadic groups, linear groups, and alternating groups. In addition, we discovered finite groups as homomorphic images of progenitors and proved some of their isomorphism type and original symmetric presentations. In this thesis we found original symmeric presentations of M12, J1 and the simplectic groups S(4,4) and S(3,4) on various con- trol groups. Using the technique of double coset enumeration we constucted J2 as a …

Italian Folk Multiplication Algorithm Is Indeed Better: It Is More Parallelizable, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, many ethnic groups had their own versions of arithmetic algorithms. Nowadays, most of these algorithms are studied mostly as pedagogical curiosities, as an interesting way to make arithmetic more exciting to the kids: by applying to their patriotic feelings -- if they are studying the algorithms traditionally used by their ethic group -- or simply to their sense of curiosity. Somewhat surprisingly, we show that one of these algorithms -- a traditional Italian multiplication algorithm -- is actually in some reasonable sense better than the algorithm that we all normally use -- namely, it is easier to parallelize.

College Algebra Through Problem Solving (2018 Edition), Danielle Cifone, Karan Puri, Debra Maslanko, Ewa Dabkowska

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.

Categories Of Residuated Lattices, Daniel Wesley Fussner

Electronic Theses and Dissertations

We present dual variants of two algebraic constructions of certain classes of residuated lattices: The Galatos-Raftery construction of Sugihara monoids and their bounded expansions, and the Aguzzoli-Flaminio-Ugolini quadruples construction of srDL-algebras. Our dual presentation of these constructions is facilitated by both new algebraic results, and new duality-theoretic tools. On the algebraic front, we provide a complete description of implications among nontrivial distribution properties in the context of lattice-ordered structures equipped with a residuated binary operation. We also offer some new results about forbidden configurations in lattices endowed with an order-reversing involution. On the duality-theoretic front, we present new results on …

Quiver Varieties And Crystals In Symmetrizable Type Via Modulated Graphs, Vinoth Nandakumar, Peter Tingley

Mathematics and Statistics: Faculty Publications and Other Works

Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac–Moody algebra. The underlying set consists of the irreducible components of Lusztig’s quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac–Moody algebras by replacing Lusztig’s preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.

On Spectral Theorem, Muyuan Zhang

Honors Theses

There are many instances where the theory of eigenvalues and eigenvectors has its applications. However, Matrix theory, which usually deals with vector spaces with finite dimensions, also has its constraints. Spectral theory, on the other hand, generalizes the ideas of eigenvalues and eigenvectors and applies them to vector spaces with arbitrary dimensions. In the following chapters, we will learn the basics of spectral theory and in particular, we will focus on one of the most important theorems in spectral theory, namely the spectral theorem. There are many different formulations of the spectral theorem and they convey the "same" idea. In …

Parametric Polynomials For Small Galois Groups, Claire Huang

Honors Theses

Galois theory, named after French mathematician Evariste Galois in 19th-century, is an important part of abstract algebra. It brings together many different branches of mathematics by providing connections among fields, polynomials, and groups.

Specifically, Galois theory allows us to attach a finite field extension with a finite group. We call such a group the Galois group of the finite field extension. A typical way to attain a finite field extension to compute the splitting field of some polynomial. So we can always start with a polynomial and find the finite group associate to the field extension on its splitting field. …