Algebra Commons^™

A Comparison Of Assessment Experiences Between Standards-Based Practices And Traditional Practices Within Secondary Mathematics Classrooms, Emily Mayes

Honors Theses

The purpose of this research was to compare assessment experiences and find ways to improve those experiences for students in two mathematics classrooms: one classroom that employs Standards-Based Grading and one classroom that uses traditional grading practices. The research examines students’ perceptions regarding their level of preparation, their anxiety levels surrounding assessment, the validity of assessments, and using assessments and grading practices to give accurate indications of student progress in their learning, given the students’ perceptions. Students in both settings voluntarily and anonymously participated in completing pre- and post-assessment free-response surveys which asked questions about students’ assessment experiences. This research …

Boolean Group Structure In Class Groups Of Positive Definite Quadratic Forms Of Primitive Discriminant, Christopher Albert Hudert Jr.

Departmental Honors & Graduate Capstone Projects

It is possible to completely describe the representation of any integer by binary quadratic forms of a given discriminant when the discriminant’s class group is a Boolean group (also known as an elementary abelian 2-group). For other discriminants, we can partially describe the representation using the structure of the class group. The goal of the present project is to find whether any class group with 32 elements and a primitive positive definite discriminant is a Boolean group. We find that no such class group is Boolean.

Counting Hamming-Graceful Labelings Of Paths, Ashka Dalal

Mathematical Sciences Technical Reports (MSTR)

Let Γ be a graph of m edges and n vertices. A Hamming-graceful labeling of Γ labels vertices with binary strings of length m and the edge labels are induced by the Hamming distance between vertex labels. It is known that all paths have Hamming-graceful labelings, thus the question arises, how many possible labelings exist for a path of a given size. We develop an algebraic way to generate labelings, conjecture a method for counting, prove this for small examples, and verify larger examples using a Python program.

On Distortion Of Surface Groups In Right-Angled Artin Groups, Lucas Bridges

Mathematical Sciences Undergraduate Honors Theses

Surfaces have long been a topic of interest for scholars inside and outside of mathe- matics. In a topological sense, surfaces are spaces which appear flat on a local scale. Surfaces in this sense have a restricted set of properties, including the behavior of loops around a surface, codified in the fundamental group.

All but 3 surface groups have been shown to embed into a class of groups called right-angled Artin groups. The method through which these embeddings are created places large restrictions on all homomorphisms from surface groups to right-angled Artin groups.

One such restriction on these homomorphisms is …

Tasks For Learning Trigonometry, Sydnee Andreasen

All Graduate Reports and Creative Projects, Fall 2023 to Present

Many studies have been done using task-based learning within different mathematics courses. Within the field of trigonometry, task-based learning is lacking. The following research aimed to create engaging, mathematically rich tasks that meet the standards for the current trigonometry course at Utah State University and align with the State of Utah Core Standards for 7th through 12th grades. Four lessons were selected and developed based on the alignment of standards, the relevance to the remainder of the trigonometry course, and the relevance to courses beyond trigonometry. The four lessons that were chosen and developed were related to trigonometric ratios, graphing …

Flipped Classroom For Linear Algebra At Undergraduate Level, M. Thulasidas

Research Collection School Of Computing and Information Systems

In this article, we describe our experience in developing an undergraduate Linear Algebra course tailored to highlight its relevance and applicability in Computer Science. Over the course of three years, the course transitioned from a traditional direct-instruction format to a flipped-classroom design, resulting in positive student learning outcomes. This article covers the course design philosophy, its syllabus, learning objectives, and the incorporation of both quantitative and qualitative student feedback in shaping the course. Furthermore, the article shares the insights gleaned from our experience, which can serve as best practices for instructors aiming to deliver a successful Linear Algebra course for …

(R2082) Two New Operations And Extended Modal Operators On Bipolar Pythagorean Fuzzy Matrices, S. Sriram, K. Sivaranjani

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, two novel binary operations concerning bipolar Pythagorean fuzzy matrices are delineated. Several algebraic properties, such as commutativity and associativity, are explored. Additionally, extended modal operators for Bipolar Pythagorean fuzzy matrices are introduced. Subsequently, these methodologies are applied to a decision-making scenario wherein a scoring matrix is formulated and alternatives are ranked according to their cumulative score values.

A Post-Quantum Mercurial Signature Scheme, Madison Mabe

All Theses

This paper introduces the first post-quantum mercurial signature scheme. We also discuss how this can be used to construct a credential scheme, as well as some practical applications for the constructions.

Art And Math Via Cubic Polynomials, Polynomiography And Modulus Visualization, Bahman Kalantari

LASER Journal

Throughout history, both quadratic and cubic polynomials have been rich sources for the discovery and development of deep mathematical properties, concepts, and algorithms. In this article, we explore both classical and modern findings concerning three key attributes of polynomials: roots, fixed points, and modulus. Not only do these concepts lead to fertile ground for exploring sophisticated mathematics and engaging educational tools, but they also serve as artistic activities. By utilizing innovative practices like polynomiography—visualizations associated with polynomial root finding methods—as well as visualizations based on polynomial modulus properties, we argue that individuals can unlock their creative potential. From crafting captivating …

Classification Of Topological Defects In Cosmological Models, Abigail Swanson

Departmental Honors & Graduate Capstone Projects

In nature, symmetries play an extremely significant role. Understanding the symmetries of a system can tell us important information and help us make predictions. However, these symmetries can break and form a new type of symmetry in the system. Most notably, this occurs when the system goes through a phase transition. Sometimes, a symmetry can break and produce a tear, known as a topological defect, in the system. These defects cannot be removed through a continuous transformation and can have major consequences on the system as a whole. It is helpful to know what type of defect is produced when …

Representation Theory And Burnside's Theorem, Nathan Fronk

Senior Seminars and Capstones

In this paper we give a brief introduction to the representation theory of finite groups, and by extension character theory. These tools are extensions of group theory into linear algebra, that can then be applied back to group theory to prove propositions that are based entirely in group theory. We discuss the importance of simple groups and the Jordan-Hölder theorem in order to prepare for the statement of Burnside’s pq theorem. Lastly, we provide a proof of Burnside’s theorem that utilizes the character theory we covered earlier in the paper.

Rsa Algorithm, Evalisbeth Garcia Diazbarriga

ATU Scholars Symposium

I will be presenting about the RSA method in cryptology which is the coding and decoding of messages. My research will focus on proving that the method works and how it is used to communicate secretly.

The Lowest Discriminant Ideal Of Cayley-Hamilton Hopf Algebras, Zhongkai Mi

LSU Doctoral Dissertations

Discriminant ideals are defined for an algebra R with central subalgebra C and trace tr : R → C. They are indexed by positive integers and more general than discriminants. Usually R is required to be a finite module over C. Unlike the abundace of work on discriminants, there is hardly any literature on discriminant ideals. The levels of discriminant ideals relate to the sums of squares of dimensions of irreducible modules over maximal ideals of C containing these discriminant ideals. We study the lowest level when R is a Cayley-Hamilton Hopf algebra, i.e. C is also a Hopf subalgebra, …

Extensions Of Algebraic Frames, Papiya Bhattacharjee

Algebra Seminar

A frame is a complete lattice that satisfies a strong distributive law, known as the frame law. Frames are also known as Pointfree Topology, as every topology is a frame. Even though the concept of frames originated from topology, the idea has expanded to many other areas of mathematics and frames are now studied in their own merit. Given two frame L and M, we say M is an extension of L if L is a subframe of M. In this talk we will discuss different types of frames extensions, such as Rigid extension, r-extension, and r*-extension between two frames. …

On Properties Of Pair Operations, Sarah Jane Poiani

Mathematics & Statistics ETDs

For any closure operation $\cl$ and interior operation $\ri$ on a class of $R$-modules, we develop the theory of $\cl$-prereductions and $\ri$-postexpansions. A pair operation is a generalization of closure and interior operations. Using Epstein, R.G. and Vassilev's duality \cite{ERGV-nonres}, we show that these notions are in fact dual to each other. We discuss the relationship between the core and hull and prereductions and postexpansions. We further the thematic notion of duality and seek to understand how it arises in the context of properties pair operations can be endowed with and focus on inner product spaces and properties demonstrated by …

Structure Of A Class Of Ordinary Differential Equations, Letao Chang

SACAD: Scholarly Activities

In the realm of mathematics, an ordinary differential equation (ODE) denotes a particular type of differential equation contingent solely upon a single independent variable. Similar to other differential equations, an ODE involves one or more functions as its unknowns and encompasses derivatives of these functions. The designation "ordinary" serves to distinguish these equations from partial differential equations, which may pertain to multiple independent variables.

Game 'Make 24', Seunghyeok Jang

SACAD: Scholarly Activities

• Basic numerical skills are a must-have in today’s world. However, children are not picking up the four basic numerical skills adequately.

• To improve their mathematical skills, they need a way to learn the numerical skills easily.

• "Make 24" is a game for young children who are having a difficult time with basic numerical operations. The game helps children improve their numerical skills by playing this game.

A Note On Umbilic Points At Infinity, Brendan Guilfoyle

Department of Mathematics Publications

In this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed in Guilfoyle and Ortiz-Rodríguez (Math Proc R Ir Acad 123A(2), 63–94, 2023). A geometric …

A Note On Umbilic Points At Infinity, Brendan Guilfoyle

Publications

In this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed in Guilfoyle and Ortiz-Rodríguez (Math Proc R Ir Acad 123A(2), 63–94, 2023). A geometric …

The Modular Generalized Springer Correspondence For The Symplectic Group, Joseph Dorta

LSU Doctoral Dissertations

The Modular Generalized Springer Correspondence (MGSC), as developed by Achar, Juteau, Henderson, and Riche, stands as a significant extension of the early groundwork laid by Lusztig's Springer Correspondence in characteristic zero which provided crucial insights into the representation theory of finite groups of Lie type. Building upon Lusztig's work, a generalized version of the Springer Correspondence was later formulated to encompass broader contexts.

In the realm of modular representation theory, Juteau's efforts gave rise to the Modular Springer Correspondence, offering a framework to explore the interplay between algebraic geometry and representation theory in positive characteristic. Achar, Juteau, Henderson, and Riche …