Algebra Commons^™

Colorings And Sudoku Puzzles, Katelyn D. May

Rose-Hulman Undergraduate Mathematics Journal

Map colorings refer to assigning colors to different regions of a map. In particular, a typical application is to assign colors so that no two adjacent regions are the same color. Map colorings are easily converted to graph coloring problems: regions correspond to vertices and edges between two vertices exist for adjacent regions. We extend these notions to Shidoku, 4x4 Sudoku puzzles, and standard 9x9 Sudoku puzzles by demanding unique entries in rows, columns, and regions. Motivated by our study of ring and field theory, we expand upon the standard division algorithm to study Gr\"obner bases in multivariate polynomial rings. …

New Theorems For The Digraphs Of Commutative Rings, Morgan Bounds

Rose-Hulman Undergraduate Mathematics Journal

The digraphs of commutative rings under modular arithmetic reveal intriguing cycle patterns, many of which have yet to be explained. To help illuminate these patterns, we establish a set of new theorems. Rings with relatively prime moduli a and b are used to predict cycles in the digraph of the ring with modulus ab. Rings that use Pythagorean primes as their modulus are shown to always have a cycle in common. Rings with perfect square moduli have cycles that relate to their square root.

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 …

Some Generalizations Of Classical Integer Sequences Arising In Combinatorial Representation Theory, Sasha Verona Malone

Masters Theses & Specialist Projects

There exists a natural correspondence between the bases for a given finite-dimensional representation of a complex semisimple Lie algebra and a certain collection of finite edge-colored ranked posets, laid out by Donnelly, et al. in, for instance, [Don03]. In this correspondence, the Serre relations on the Chevalley generators of the given Lie algebra are realized as conditions on coefficients assigned to poset edges. These conditions are the so-called diamond, crossing, and structure relations (hereinafter DCS relations.) New representation constructions of Lie algebras may thus be obtained by utilizing edge-colored ranked posets. Of particular combinatorial interest are those representations whose corresponding …

Numerical Computations Of Vortex Formation Length In Flow Past An Elliptical Cylinder, Matthew Karlson, Bogdan Nita, Ashwin Vaidya

Department of Mathematics Faculty Scholarship and Creative Works

We examine two dimensional properties of vortex shedding past elliptical cylinders through numerical simulations. Specifically, we investigate the vortex formation length in the Reynolds number regime 10 to 100 for elliptical bodies of aspect ratio in the range 0.4 to 1.4. Our computations reveal that in the steady flow regime, the change in the vortex length follows a linear profile with respect to the Reynolds number, while in the unsteady regime, the time averaged vortex length decreases in an exponential manner with increasing Reynolds number. The transition in profile is used to identify the critical Reynolds number which marks the …

Spectral Sequences For Almost Complex Manifolds, Qian Chen

Dissertations, Theses, and Capstone Projects

In recent work, two new cohomologies were introduced for almost complex manifolds: the so-called J-cohomology and N-cohomology [CKT17]. For the case of integrable (complex) structures, the former cohomology was already considered in [DGMS75], and the latter agrees with de Rham cohomology. In this dissertation, using ideas from [CW18], we introduce spectral sequences for these two cohomologies, showing the two cohomologies have natural bigradings. We show the spectral sequence for the J-cohomology converges at the second page whenever the almost complex structure is integrable, and explain how both fit in a natural diagram involving Bott-Chern cohomology and the Frolicher spectral sequence. …

Length Neutrosophic Subalgebras Of Bck/Bci-Algebras, Florentin Smarandache, Young Bae Jun, Madad Khan, Seok-Zun Song

Branch Mathematics and Statistics Faculty and Staff Publications

the notion of (i, j, k)-length neutrosophic subalgebras in BCK/BCI-algebras is introduced, and their properties are investigated. Characterizations of length neutrosophic subalgebras are discussed by using level sets of interval neutrosophic sets. Conditions for level sets of interval neutrosophic sets to be subalgebras are provided.

On Product Of Smooth Neutrosophic Topological Spaces, Florentin Smarandache, Kalaivani Chandran, Swathi Sundari Sundaramoorthy, Saeid Jafari

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we develop the notion of the basis for a smooth neutrosophic topology in a more natural way. As a sequel, we define the notion of symmetric neutrosophic quasi-coincident neighborhood systems and prove some interesting results that fit with the classical ones, to establish the consistency of theory developed. Finally, we define and discuss the concept of product topology, in this context, using the definition of basis.

Transitioning From The Abstract To The Concrete: Reasoning Algebraically, Andrea Lynn Martin

Graduate Theses/Dissertations

Why are students not making a smooth transition from arithmetic to algebra? The purpose of this study was to understand the nature of students’ algebraic reasoning through tasks involving generalizing. After students’ algebraic reasoning had been analyzed, the challenges they encountered while reasoning were analyzed. The data was collected through semi-structured interviews with six eighth grade students and analyzed by watching recorded interviews while tracking algebraic reasoning. Through data analysis of students’ algebraic reasoning, three themes emerged: 1) it was possible for students to reach stage two (informal abstraction) and have an abstract understanding of the mathematical pattern even if …

Universal Localizations Of Certain Noncommutative Rings, Tyler B. Bowles

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A common theme throughout algebra is the extension of arithmetic systems to ones over which new equations can be solved. For instance, someone who knows only positive numbers might think that there is no solution to x + 3 = 0, yet later learns x = -3 to be a feasible solution. Likewise, when faced with the equation 2x = 3, someone familiar only with integers may declare that there is no solution, but may later learn that x = 3/2 is a reasonable answer. Many eventually learn that the extension of real numbers to complex numbers unlocks solutions …

Math Active Learning Lab: Math 98 Notebook, Gwennie Byron, Michele Iiams, Department Of Mathematics, University Of North Dakota

Open Educational Resources

This course notebook has been designed for students of Math 98 (Intermediate Algebra) at the University of North Dakota. It has been designed to help you get the most out of the ALEKS resources and your time.

• Topics in the Notebook are organized by weekly learning module.
• Space for notes from ALEKS learning pages, e-book and videos directs you to essential concepts.
• Examples and “You Try It” problems have been carefully chosen to help you focus on these essential concepts.
• Completed Notebook is an invaluable tool when studying for exams.

A General Model Of Neutrosophic Ideals In Bck/Bci-Algebras Based On Neutrosophic Points, Florentin Smarandache, Hashem Bordbar, Rajab Ali Borzooei, Young Bae Jun

Branch Mathematics and Statistics Faculty and Staff Publications

More general form of (∈, ∈∨q)-neutrosophic ideal is introduced, and their properties are investigated.

Math Active Learning Lab: Math 103 College Algebra Notebook, Gwennie Byron, Department Of Mathematics, University Of North Dakota

Open Educational Resources

This course notebook has been designed for students of Math 103 (College Algebra) at the University of North Dakota. It has been designed to help you get the most out of the ALEKS resources and your time.

• Topics in the Notebook are organized by weekly learning module.
• Space for notes from ALEKS learning pages, e-book and videos directs you to essential concepts.
• Examples and “You Try It” problems have been carefully chosen to help you focus on these essential concepts.
• Completed Notebook is an invaluable tool when studying for exams.

Exact And Strongly Exact Filters, M. A. Moshier, A. Pultr, A. L. Suarez

Mathematics, Physics, and Computer Science Faculty Articles and Research

A meet in a frame is exact if it join-distributes with every element, it is strongly exact if it is preserved by every frame homomorphism. Hence, finite meets are (strongly) exact which leads to the concept of an exact resp. strongly exact filter, a filter closed under exact resp. strongly exact meets. It is known that the exact filters constitute a frame FiltE(L) somewhat surprisingly isomorphic to the frame of joins of closed sublocales. In this paper we present a characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters FiltsE(L).

On Pseudo-Spectral Factorization Over The Complex Numbers And Quaternions, Daniel Alpay, Fabrizio Colombo, Izchak Lewkowicz, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

This paper is a continuation of the research of our previous work[5] and considers quaternionic generalized Carathéodory functions and the related family of generalized positive functions. It is addressed to a wide audience which includes researchers in complex and hypercomplex analysis, in the theory of linear systems, but also electric engineers. For this reason it includes some results on generalized Carathéodory functions and their factorization in the classic complex case which might be of independent interest. An important new result is a pseudo-spectral factorization and we also discuss some interpolation problems in the class of quaternionic generalized positive functions.

Families Of Homogeneous Licci Ideals, Jesse Keyton

Graduate Theses and Dissertations

This thesis is concered with the graded structure of homogeneous CI-liaison. Given two homogeneous ideals in the same linkage class, we want to understand the ways in which you can link from one ideal to the other. We also use homogeneous linkage to study the socles and Hilbert functions of Artinian monomial ideals.

First, we build off the work of C. Huneke and B. Ulrich on monomial liaison. They provided an algorithm to check the licci property of Artinian monomial ideals and we use their method to characterize when two Artinian monomial ideals can be linked by monomial regular sequences. …

A Real World Example Of Solving A Quadratic Equation In Movie Cgi, Cynthia J. Huffman Ph.D.

Faculty Submissions

It is important to expose students to the beauty and usefulness of mathematics. Since computer graphics are familiar to most students due to video games and movies, they make a great source for motivating topics in mathematics. This activity shows students an application of solving quadratic equations to computing the line of sight to spherical objects in computer graphics.

College Algebra Notes And Exercises (Gcsu), Rabia Shahbaz, Janice Alves

Mathematics Ancillary Materials

Developed as part of a Round 13 Mini-Grant, these updated supplementary materials for Stitz-Zeager Open Source Mathematics and the LibGuides Open Course for College Algebra at GCSU include notes and exercises on equations, inequalities, functions, polynomial and rational functions, and exponential and logarithmic functions are included in one .zip file.

Topics In Gravitational Wave Physics, Aaron David Johnson

Graduate Theses and Dissertations

We begin with a brief introduction to gravitational waves. Next we look into the origin of the Chandrasekhar transformations between the different equations found by perturbing a Schwarzschild black hole. Some of the relationships turn out to be Darboux transformations. Then we turn to GW150914, the first detected black hole binary system, to see if the nonlinear memory might be detectable by current and future detectors. Finally, we develop an updated code for computing equatorial extreme mass ratio inspirals which will be open sourced as soon as it has been generalized for arbitrary inclinations.

Conjugacy Separability And Cyclic Conjugacy Separability Of Certain Hnn Extensions, Generalised Free Products And Tree Products, Hui Min Lim

Student Works (2020-2029)

In this thesis, we study two interrelated strong residually finite properties of groups, namely conjugacy separability and cyclic conjugacy separability. We extend them to certain HNN extensions, generalized free products and tree products where the associated subgroups and amalgamated subgroups are not necessarily cyclic. In the first part of the thesis, we consider HNN extensions. We begin by establishing two criteria, one for conjugacy separability and another for cyclic conjugacy separability. Using these two criteria we establish conditions for HNN extensions where the associated subgroups are central or they are a finite extension of a central subgroup or cyclic to …

On The Extension Of Positive Definite Kernels To Topological Algebras, Daniel Alpay, Ismael L. Paiva

Mathematics, Physics, and Computer Science Faculty Articles and Research

We define an extension of operator-valued positive definite functions from the real or complex setting to topological algebras and describe their associated reproducing kernel spaces. The case of entire functions is of special interest, and we give a precise meaning to some power series expansions of analytic functions that appears in many algebras.

Combining Transformation Of Graphs With Solutions To Absolute Value Inequalities, Ryan D. Fox

Colorado Mathematics Teacher

I present how transformations can be applied to support students’ solving linear inequalities involving absolute value. In particular, the horizontal dilations/compressions and translations of graphical representations of distances from zero along a number line are important tools to emphasize a visual representation of the solutions to absolute value inequalities.

The Structure Of Generalized Bi-Algebras And Weakening Relation Algebras, Nikolaos Galatos, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras …

Model Theory Of Groups And Monoids, Laura M. Lopez Cruz

Dissertations, Theses, and Capstone Projects

We first show that arithmetic is bi-interpretable (with parameters) with the free monoid and with partially commutative monoids with trivial center. This bi-interpretability implies that these monoids have the QFA property and that finitely generated submonoids of these monoids are definable. Moreover, we show that any recursively enumerable language in a finite alphabet X with two or more generators is definable in the free monoid. We also show that for metabelian Baumslag-Solitar groups and for a family of metabelian restricted wreath products, the Diophantine Problem is decidable. That is, we provide an algorithm that decides whether or not a given …

Ideal Theory In Bck/Bci-Algebras In The Frame Of Hesitant Fuzzy Set Theory, G. Muhiuddin, Habib Harizavi, Young Bae Jun

Applications and Applied Mathematics: An International Journal (AAM)

Several generalizations and extensions of fuzzy sets have been introduced in the literature, for example, Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy sets and fuzzy multisets, etc. Using the Torra’s hesitant fuzzy sets, the notions of Sup-hesitant fuzzy ideals in BCK/BCI-algebras are introduced, and its properties are investigated. Relations between Sup-hesitant fuzzy subalgebras and Sup-hesitant fuzzy ideals are displayed, and characterizations of Sup-hesitant fuzzy ideals are discussed.

Assessing Student Understanding While Solving Linear Equations Using Flowcharts And Algebraic Methods, Edima Umanah

Electronic Theses, Projects, and Dissertations

Solving linear equations has often been taught procedurally by performing inverse operations until the variable in question is isolated. Students do not remember which operation to undo first because they often memorize operations with no understanding of the underlying meanings. The study was designed to help assess how well students are able to solve linear equations. Furthermore, the lesson is designed to help students identify solving linear equations in more than one-way. The following research questions were addressed in this study: Does the introduction of multiple ways to think about linear equations lead students to flexibly incorporate appropriate representations/strategies in …

Hyperbolic Triangle Groups, Sergey Katykhin

Electronic Theses, Projects, and Dissertations

This paper will be on hyperbolic reflections and triangle groups. We will compare hyperbolic reflection groups to Euclidean reflection groups. The goal of this project is to give a clear exposition of the geometric, algebraic, and number theoretic properties of Euclidean and hyperbolic reflection groups.

Pemahaman Ungkapan Algebra Dalam Kalangan Murid Tingkatan Dua, Hamid Azizah

Student Works (2020-2029)

Pemahaman tentang konsep ungkapan algebra penting bagi murid sekolah menengah kerana ia merupakan asas kepada pembelajaran matematik di peringkat menengah dan peringkat pengajian tinggi memandangkan kesukaran dalam pembelajaran algebra menjadi semakin kritikal. Kajian ini adalah untuk mengenal pasti pemahaman murid Tingkatan Dua tentang ungkapan algebra dengan berlandaskan konstruktivisme radikal. Penyelidikan ini menggunakan kajian kes sebagai reka bentuk kajian, yang membabitkan data berbentuk kualitatif. Peserta kajian membabitkan enam murid dari tiga kategori pencapaian akademik dengan menggunakan teknik persampelan bertujuan. Data kajian terkumpul melalui tiga sesi temu duga klinikal dengan menggunakan perakam video, catatan murid dan pengkaji. Data kajian dianalisis dengan menggunakan …

Acoustic Versus Electromagnetic Field Theory: Scalar, Vector, Spinor Representations And The Emergence Of Acoustic Spin, Lucas Burns, Konstantin Y. Bliokh, Franco Nori, Justin Dressel

Mathematics, Physics, and Computer Science Faculty Articles and Research

We construct a novel Lagrangian representation of acoustic field theory that describes the local vector properties of longitudinal (curl-free) acoustic fields. In particular, this approach accounts for the recently-discovered nonzero spin angular momentum density in inhomogeneous sound fields in fluids or gases. The traditional acoustic Lagrangian representation with a scalar potential is unable to describe such vector properties of acoustic fields adequately, which are however observable via local radiation forces and torques on small probe particles. By introducing a displacement vector potential analogous to the electromagnetic vector potential, we derive the appropriate canonical momentum and spin densities as conserved Noether …

"Fireworks And Quadratic Functions”, Kelly W. Remijan

Teacher Resources

No abstract provided.