Algebra Commons^™

Determinants And Invertibility In Finite Modular Systems, Osasu Omobude

Discovery Day - Daytona Beach

This project investigates determinants and matrix invertibility in finite modular systems, focusing on matrices over Zn. Using the Hill cipher as context, it examines the algebraic conditions under which a matrix is invertible in modular arithmetic. In particular, the project studies how the determinant determines invertibility, showing that a matrix over Zn is invertible if and only if its determinant is coprime with n.   The project further compares invertibility over the real numbers with invertibility over modular systems, highlighting the distinction between prime moduli Zp and composite moduli. In the prime case, matrices behave similarly to those over fields, where …

The General Solution Analysis Of Homogeneous Linear Equations, Jacob Schwamb, Edward Whipple

Discovery Day - Daytona Beach

The general solution analysis of homogeneous linear equations are any systems of equations in which all constant terms are equal to zero is classified as a homogeneous linear equation. Some key characteristics of homogeneous linear equations are that there are “zero” solutions, where every system has at least a single solution where all variables are zero, also all solutions to any homogenous linear equation is linearly independent, along with having preserved homogeneity, where if any variable (x) may be added to the system, then any scalar multiple of the variable is also a solution. The General solution of any homogeneous …

Modelling Quadcopter Thrust And Torque Using Lu-Decomposition, Diya Patil

Discovery Day - Daytona Beach

3 & 4 motor systems have been in the world of aviation in many different forms as the field grows and evolves. To understand the complexities of an Unmanned Aerial Vehicle (UAV) and its stability, assessing the amount of thrust put into each motor can help generate the torque produced despite factors such as multidirectional movement. While a UAV does this multiple times a second, producing a simplified version of this calculation can aid in simpler models simulating UAV movement. Due to the popularity of the quadcopter drone, a simple algorithm depicting thrust through each spinning motor can aid in …

Underclosed Posets, Richard Ngo

McNair Summer Research Program

Underclosed complexes are a recent generalization of interval graphs to higher dimensions. Motivated by underclosed complexes, we define and study underclosed posets. Order ideals of these posets correspond to pure underclosed complexes. We classify which principal order ideals are rank-symmetric (and in fact are self-dual).

Rockin’ Rover On The Rainbow Road, Michael Kolta, Lawrence Burgee, Ying Yuan

Transformations

This paper presents a progressive series of age-appropriate lesson plans for grades K-12 that all use the same interdisciplinary activity to educate students about Science, Technology, Engineering, Art, and Mathematics (STEAM) simultaneously. Technology from Texas Instruments (TI) was employed including a TI Nspire graphing calculator that can run Python programs, a TI Innovator Hub, and a TI Rover. The TI Rover is a small, robotic car that has sensors and is controlled by the calculator via the Hub hardware interface. A Python program was developed that uses the color sensor in the Rover to detect the color on colored paper …

A Categorical Framework For Modeling Genetic Drift, Taylor G. Mendes

Rose-Hulman Undergraduate Mathematics Journal

Genetic drift describes changes in allele frequencies that arise from chance sampling in finite populations. This paper develops a categorical framework for organizing the structural features of drift. Population states are modeled as objects, evolutionary transitions as morphisms, reversible transitions as groupoid morphisms, and structure-preserving comparisons between models as functors. Group actions are used to describe deterministic evolutionary operators such as mutation and selection, while orbits and fixed points identify reachable allele-frequency states and stable absorbing outcomes. Universal properties are then used to describe drift as a coherence condition connecting stochastic transitions with deterministic evolutionary maps. The resulting framework complements …

Trattato Dell’Alcibra Amuchabile (Anonimo): A Guided Translation, Gary Towsley, Olympia Nicodemi

Geneseo Authors

The Trattato dell’Alcibra Amuchabile is a pre-modern algebra text from c. 1365. It is written in a Tuscan dialect of Italian and is situated in the abbacus school tradition, schools that taught the mathematics needed for a mercantile society. Like all the algebra written in Italy at the time, it was inherited from al-Khwarizmi and, like his, written with no symbols—no x’s, y’s, plus signs, etc. It was what is sometimes called “rhetorical algebra.”  There is very little source  material available in English from this important era in the history of algebra. This book helps fill that gap. …

Conditionals And Modalities In Constructive Quantum Logics, Juan P. Aguilera, Guillaume Massas

Mathematics, Physics, and Computer Science Faculty Articles and Research

We investigate logics that generalize both intuitionistic logic and quantum logic. In earlier work, we introduced Ex-logic, an extension of Holliday's fundamental logic that coincides with the intersection of orthologic and the implication-free fragment of intuitionistic logic. In this paper, we add an implication connective to Ex-logic and axiomatize iEx-logic, the intersection of full intuitionistic logic and orthomodular logic with the implication connective interpreted as the Sasaki hook. As a consequence, we obtain a characterization of the lattice of logics extending iEx-logic as the product of the lattice of intermediate logics and the lattice of orthomodular logics. We also explore …

(R2177) Further Study On Intuitionistic Fuzzy Matrices, S. Sriram, A. Anitha

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we present a comprehensive study of the distributive law associated with two newly introduced operations addition and multiplication defined on Intuitionistic Fuzzy Matrices (IFMs). The proposed operations are systematically examined to investigate their algebraic properties, with particular emphasis on the behavior of the distributive law under these novel definitions. Furthermore, modal and extended modal operators are employed to establish several theoretical results that reinforce the mathematical foundation of the proposed framework. The validity and practical relevance of these results are further demonstrated through applications in decision-making scenarios, highlighting their effectiveness in addressing complex problems characterized by uncertainty.

(R2185) Operations On Bipolar Complex Neutrosophic Matrices And Its Application In Un’S Sdg-14 & Sdg-3 Using Python, N. Krishnapraveen, T. Muthuraji

Applications and Applied Mathematics: An International Journal (AAM)

Decision-making in sustainability oriented environments frequently involves bipolar evaluations, indeterminate information and phase dependent uncertainty that cannot be adequately represented by existing neutrosophic matrix models. To address this limitation, this study introduces a Bipolar Complex Neutrosophic Matrix (BCNM) framework that integrates bipolar semantics with complex valued uncertainty in a coherent algebraic structure. Fundamental operations and structural properties are rigorously established to ensure mathematical consistency. To facilitate practical multi criteria decision analysis (MCDA), novel score, accuracy, and hybrid aggregation operators are developed. The computational feasibility of the proposed approach is analyzed, demonstrating linear complexity with respect to the number of alternatives …

(R2187) Analysis Of Neurological Impairments In Hospitalized Patients Using Cubic Neutrosophic Sets, B. Anitha, M. Lavanya

Applications and Applied Mathematics: An International Journal (AAM)

This study introduces an MCDM-based framework for identifying neurological diseases in hospitalized patients using symptom-based evaluations. A team of interns, guided by the chief doctor, was responsible for determining each patient’s precise condition from the presented neurological symptoms. To enhance diagnostic accuracy, the interns employed the TOPSIS and WASPAS methods to assess and rank the potential disease options. The combined analysis yielded a clear identification of the highest ranked disease for every patient, highlighting the effectiveness of these MCDM techniques in supporting clinical decision making.

Quantitatively Hyper-Positive Real Rational Functions Iii, Daniel Alpay, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

Hyper-Positive Real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of Hyper-positive functions, related through nested inclusions, are introduced. Structurally, this family of functions turns out to be matrix-convex and closed under inversion. A state-space characterization of these functions through a corresponding Kalman-Yakubovich-Popov Lemma, is given. Technically, the classical Linear Matrix Inclusions, associated with passive systems, are here substituted by Quadratic Matrix Inclusions.

Course Portfolio: Elements Of Physics Phys 151, Evan A. Rich

UNL Faculty Course Portfolios

This course portfolio documents the instructional design, teaching methods, and ongoing assessment efforts for PHYS 151: Elements of Physics, an algebra-based introductory physics course at the University of Nebraska-Lincoln. The course serves a broad undergraduate population, including architecture, construction management, and life science majors. The portfolio describes the teaching framework that integrates pre-lecture video preparation, active in-class engagement through iClicker questions, and collaborative weekly recitation sections, all unified around a structured six-step problem-solving approach. A central concern of the course is building students’ self-efficacy in physics, particularly among those with math anxiety or limited preparation. Two assessments are reported: a …

Gauss Composition And Orthogonal Modular Forms On Binary Lattices, Haochen Wu

Dartmouth College Ph.D Dissertations

We revisit Gauss composition over a general base scheme, with a focus on orthogonal groups. We show that the Clifford and norm functors provide a discriminant-preserving equivalence of categories between binary quadratic modules and pseudoregular modules over quadratic algebras. This perspective synthesizes the constructions of Kneser and Wood, reconciling algebraic and geometric approaches and clarifying the role of orientations and the natural emergence of narrow class groups.

As an application, we restrict to lattices and show that binary orthogonal eigenforms correspond to Hecke characters. Using theta series, we show the explicit connection between Hilbert modular forms and orthogonal modular forms …

Quiver Of Affine Monoid Of A Vector Space Over Finite Field, James Junie Chen Cleary

Dissertations, Theses, and Capstone Projects

In this paper, we study the quiver of the complex monoid algebra CAFF(n, q). There are n + 1 maximal subgroups of AFF(n, q), each isomorphic to AGL(k, q) for some 0 ≤ k ≤ n. Every irreducible representation of CAFF(n, q) arises from a character of CAGL(k, q) for a suitable k. Thus, we study two different approaches to classifying the characters of CAGL(k, q). Next, we compute the full quiver Q(CAFF(n, q)). Finally, we show that this quiver is a disjoint union of straight-line paths and that its basic algebra has radical square zero. Hence, it has finite …

College Algebra With Review, Lanee Young Ph.D., Jayme Goetz

All Open Educational Resources

This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the thousands of other texts available within this powerful platform, it is freely available for reading, printing, and "consuming." The LibreTexts mission is to bring together students, faculty, and scholars in a collaborative effort to provide an accessible, and comprehensive platform that empowers our community to develop, curate, adapt, and adopt openly licensed resources and technologies; through these efforts we can reduce the financial burden born from traditional educational resource costs, ensuring education is more accessible for students and communities worldwide. Most, but …

Relation Subspaces In Vertex Operator Algebras: Residue Generators For O_N^\Circ(V) And Intersections With (L(−1) +L(0))V, Junghyun Kim

Undergraduate Research Journal

We study the relation subspace 𝑂◦𝑛 (𝑉) that appears in the definition o f the level-𝑛 Zhu algebra 𝐴𝑛 (𝑉) = 𝑉/𝑂𝑛 (𝑉), where 𝑂𝑛 (𝑉) = 𝑂𝐿 (𝑉) + 𝑂◦𝑛 (𝑉) and 𝑂𝐿 (𝑉) =(𝐿(−1) + 𝐿(0))𝑉. Using residue calculus, we introduce operators 𝑅𝑛,𝑘 that encode the circle products 𝑢◦𝑛 𝑣 and prove explicit change-of-generators formulas between the standard generators (𝑢−𝑚1)◦𝑛 𝑣 and the residue generators 𝑢 𝑅𝑛,0𝑣, together with a binomial inversion. These identities provide a practical framework for computing 𝑂◦𝑛 (𝑉), especially in strongly generated VOAs. As progress toward understanding the overlap 𝑂◦𝑛 (𝑉) ∩ 𝑂𝐿 (𝑉), …

Numerical And Harmonic Analysis Of Simplex Number Parity, Hunter Dm Hannula

All NMU Master's Theses

The primary object of this thesis is the study of periodicity in the parity of simplex numbers by number-theoretic and harmonic methods. The regular d-simplex numbers are introduced geometrically, arithmetically, and combinatorially. We demonstrate that all sequences indexing even d-simplex numbers are defined by finitely many congruences in a single modulus, and are thus quasiperiodic. We therefore show that these "even index-sequences" are  particular elements in an affine space of functions, providing a natural decomposition result. We then introduce the discrete Fourier transform to construct the periodic parts of each index sequence, enabling the development of explicit forms for the …

Galois Action And Arithmetic In Algebraic Number Fields, Jared Kettinger

All Dissertations

This dissertation explores the arithmetic of numerous algebraic objects living within an algebraic number field from submonoids of the integers up to localizations of the ring of integers. We begin with a study of factorization in proper orders using an element-theoretic approach. In Chapter 2, by defining a natural generalization of the Davenport constant, we are able to determine the elasticity of certain orders whose integral closure is a unique factorization domain. In Chapter 3, using ideal-theoretic analogues, we are able to significantly broaden the scope of our results and the literature on factorization in orders. In particular, we give …

The Fundamental Group: A Geometric Perspective, Kayla M. Bittenbinder

All NMU Master's Theses

This thesis explores the deep mathematical connection between the flexible, continuous world of topology and the rigid, distance-preserving world of metric geometry. We begin by constructing the fundamental group of a topological space, using the concept of loops and loop homotopy to identify a global topological invariant such as a hole or puncture. We then transition to the geometric realm of metric spaces and isometries, demonstrating how the isometry group of a space algebraically encodes its rigid symmetries. To bridge these two distinct mathematical frameworks, we utilize the construction of the universal covering space. We pull back the metric from …

On The Algebraicity Of The Tic-Tac-Toe Matroid And Homogeneous Mixed Bowtie Systems, Benjamin R. Allen

Electronic Theses and Dissertations

This thesis is presented in two parts. First, we explore whether the class of algebraic matroids is closed under duality, a decades-old open question. We consider the Tic-Tac-Toe matroid as a potential candidate to answer the open question. The Tic-Tac-Toe matroid is known to satisfy many of the necessary conditions for a matroid to be algebraic and has a non-algebraic dual.  Second, we focus on decompositions of the complete mixed graph into mixed bowties. A complete mixed graph has between every pair of vertices an undirected edge and antiparallel arcs. A mixed bowtie is a graph consisting of two 3-cycles …

Ideals And Lattices In Number Fields, Sarah Ali Saeed Alyammahi

Thesis/ Dissertation Defenses

This thesis investigates algebraic number fields and their rings of integers, which generalize the ring of integers Z in Q. The study focuses on ideals, units, and ideal class groups, which describe the arithmetic structure of number fields and the failure of unique factorization. Key invariants such as the norm, trace, and discriminant are developed and applied, with particular emphasis on quadratic number fields and classical examples such as the Gaussian and Eisenstein integers. Some explicit computations of ideal class groups are carried out. The thesis also explores connections with lattice theory by interpreting rings of integers as lattices and …

A 4-Dimensional Rubik’S Cube You Can Hold: How It’S Possible And The Math Behind It, Eric J. Moon

SACAD: Scholarly Activities

This poster examines the physical 2x2x2x2, a hand-held realization of a 4-dimensional Rubik’s Cube invented by Melinda Green. Unlike most higher-dimensional twisty puzzles, which exist only as software simulations, this puzzle provides a physical model for exploring 4-dimensional rotation, symmetry, and solving methods. The poster introduces the structure of the puzzle, its canonical move system, and several algebraic ideas that help explain how scrambling and solving work.

From a mathematical perspective, the puzzle can be studied using group actions, commutators, conjugation, and combinatorial counting. In particular, the number of reachable states depends on corner permutations, corner orientations, parity restrictions, twist …

Non-Redundant Rovibrational Hamiltonians By Molien Generating Functions And Gröbner-Basis Reduction, Leandro Ajo

University Libraries Undergraduate Research Award

High-resolution molecular spectroscopy requires an effective Hamiltonian whose operator content is both complete, containing every term allowed by molecular symmetry and nonredundant, free of any algebraically dependent operators that would cause ill-conditioned parameter fits. Traditional derivations based on Van Vleck contact transformations satisfy neither criterion automatically. This paper develops a rigorous, algorithmic pipeline that guarantees both properties simultaneously. Starting from the permutation– inversion (PI) group GPI of a molecule (Longuet-Higgins, 1963), we apply Molien’s theorem (Molien, 1897) to the symplectic normal-coordinate representation to obtain the vibrational generating function Φvib(t); integrate over the Haar measure of SO(3) (Weyl, 1946) to obtain …

Weakening Relations Over Duals Of Limit Ordinals, Nikolaos Galatos, Isis A. Gallardo

Mathematics: Faculty Scholarship

We prove that all varieties generated by weakening relation algebras over duals of limit ordinals have a decidable equational theory. We also show that there are only countably-many such varieties and they form a chain that embeds in. The time warp algebra is isomorphic to one of these weakening relation algebras (over the dual of the naturals), so we obtain the main result of a recent publication as a special case. The algebras we study connect to the theory of relation algebras, to time warps and graded modalities, and to the algebraic semantics of substructural logics. Our methods are inspired …

Algebraic Invariants Of Multi-Virtual Links, Louis H. Kauffman, Sujoy Mukherjee, Petr Vojtěchovský

Mathematics: Faculty Scholarship

Multi-virtual knot theory was introduced in 2024 by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves, we discuss the equivalence of multi-virtual link diagrams, particularly those that have the same virtual projections. We introduce operator quandles (that is, quandles with a list of pairwise commuting automorphisms) and construct an infinite family of connected operator quandles in which at least one third of right translations are distinct and pairwise commute. Using our set of generating moves, we establish the operator quandle coloring invariant …

Path Odd-Covers Of Graphs, Steffen Borgwardt, Calum Buchanan, Eric Culver, Bryce Frederickson, Puck Rombach, Youngho Yoo

Mathematics: Faculty Scholarship

We introduce and study “path odd-covers,” a weakening of Gallai's path decomposition problem and a strengthening of the linear arboricity problem. The path odd-cover number  of a graph G is the minimum cardinality of a collection of paths whose vertex sets are contained in  and whose symmetric difference of edge sets is. We prove an upper bound on  in terms of the maximum degree Δ and the number of odd-degree vertices  of the form . This bound is only a factor of 2 from a rather immediate lower bound of the form . We also investigate some natural relaxations of …

Mckinsey-Tarski Algebras And Raney Extensions, G. Bezhanishvili, R. Raviprakash, A. L. Suarez, Joanne Walters-Wayland

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the T₀-hull of a Raney extension generalizes that of the T_D-hull of a frame.

Unitary Rational Functions: The Scaled Quaternion Case, Daniel Alpay, Ilwoo Choo, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the quaternions and the split quaternions. The methods involved are not a direct generalization of the complex or quaternionic settings, and in particular, the adjoint is not the classical adjoint and we use properties of real Hilbert spaces. This adjoint allows to define the counterpart of unitarity for matrix-rational functions, and we develop the corresponding theories of realizations and unitary factorizations. We also begin a …

Does Sequence Matter? Impact Of Redesigning Sequential Calculus Course On Students’ Learning Outcomes, Chantal Levesque-Bristol Dr., Wonki Lee Dr., Emily M. Bonem, Benjamin C. Wiles, Jennifer D. Moss, Wilella D. Burgess, Weiling Li

International Journal of Teaching and Learning in Higher Education

As colleges and universities increasingly transform their STEM courses through the adoption of more active, student-centered pedagogies, there is a growing need to understand the impact of these educational innovations on student outcomes. This study is conducted in the context of a university-wide faculty development and course redesign project, IMPACT (Instruction Matters, Purdue Academic Course Transformation). IMPACT supports faculty in implementing student-centered pedagogical practices and creating equitable and inclusive learning environments. This study focuses on a sequence of two introductory calculus courses (Calculus 1 and Calculus 2, hereafter CALC 1 and CALC 2) that were transformed as part of the …