Algebra Commons^™

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).

(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 …

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 …

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 …

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 …

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 …

Extensions Between Modules Defined By Lattice Paths In The Preprojective Algebra, Chloe Napier

Theses and Dissertations--Mathematics

In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. The main goal is to find a combinatorial description for when certain cluster variables are compatible, or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc, and Schröer that answer this question in the case of …

Catalan And Hyper-Catalan Numbers: Combinatorial Applications To Polynomial Equations, Leilani Natale

Williams Honors College, Honors Research Projects

In this paper, we study Catalan numbers and their generalization, hyper-Catalan numbers, and explore how these sequences arise naturally in the context of solving polynomial equations using infinite power series. We begin by introducing the Catalan numbers through their combinatorial interpretation as triangulations of convex polygons. Using this geometric definition, we derive a relation whose recursive structure leads to a quadratic functional equation. Interpreting this relation as a formal power series equation allows us to express solutions to quadratic equations as infinite power series whose coefficients are given by the Catalan numbers. This framework is then extended by allowing polygon …

A Leslie System For A Demographic Simulation: From An Actuarial Point Of View, David Kings

Electronic Theses and Dissertations

This thesis develops a discrete stochastic linear systems interpretation of age–stage demographic evolution grounded in Leslie operators and realized in a discrete-event simulation implemented with salabim. The central claim is that one annual cycle of the simulation constitutes a cone-preserving, stochastic affine transformation on a high- dimensional population state vector indexed by age, sex, marital status, household type, employment, and education, and that the composition of yearly operators yields a random matrix product whose top Lyapunov exponent is the stochastic counterpart of the Perron–Frobenius growth rate (Caswell, 2001; Tuljapurkar, 1997)[1, 2]. The actuarial bridge is constructed by mapping simulated survival …

On The Combinatorial Invariance For Kazhdan-Lusztig Polynomials, Grover C. Harrell Iii

College of Graduate Studies: Theses & Dissertations

This thesis will be a discussion on the Combinatorial Invariance Conjecture for Kazhdan Lusztig polynomials. The conjecture is widely suspected to be true; and there is an abun dance of computational evidence which supports it. Despite this, no complete proof has been discovered for more than forty years. We will explore some known results about the CIC, particularly those by Dyer, Incitti, Brenti, Caselli, and Marietti.

On Linear Invariants Of Hypergraphs, Clara Chaplin

Honors Theses

We introduce linear invariants of hypergraphs as a way to study hypergraphs by their tensor representations. Our primary research goal is to determine what information linear invariants capture about the hypergraphs they arise from. We first investigate the centroid, which is shown to determine the connected components of a hypergraph. Next, we study the derivations of a hypergraph, and use this linear invariant to define a quotient operator $Q_\mathrm{Der}$ on the collection of all hypergraphs. This operator is shown to be a closure operator in that $Q_\mathrm{Der}(Q_\mathrm{Der}(\mathcal{H}))=Q_\mathrm{Der}(\mathcal{H})$ for any hypergraph $\mathcal{H}$. We apply the operator $Q_\mathrm{Der}$ to synthetically generated hypergraphs, …

Moore Graphs, Trevor Saxton

Williams Honors College, Honors Research Projects

A Moore graph is a simple regular graph, with n vertices, degree d, and diameter k, that satisfies the Moore bound: n = 1 + d (d − 1)k − 1 d − 2 . There are graphs for which the bound is met and in which existence and uniqueness are known. For k = 2 it is known that the Moore bound is achieved for d = 2, 3, 7, with the case of d = 57 conjectured to exist. For k = 3 the bound is achieved for only d = 3 [5]. Due to the construction of …

Counting The Classes Of Projectively-Equivalent Pentagons On Finite Projective Planes Of Prime Order, Maxwell Hosler

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we examine the number of equivalence classes of pentagons on finite projective planes of prime order under projective transformations. We are interested in those pentagons in general position, meaning that no three vertices are collinear. We consider those planes which can be constructed from finite fields of prime order, and use algebraic techniques to characterize them by their symmetries. We are able to construct a unique representative for each pentagon class with nontrivial symmetries. We can then leverage this fact to count classes of pentagons in general. We discover that there are (1/10)((p+3)(p-3)+4 …

Categorical Chain Conditions For Étale Groupoid Algebras, Sunil Philip

Dissertations, Theses, and Capstone Projects

Let R be a unital commutative ring and G an ample groupoid. Using the topology of the groupoid G, Steinberg defined an étale groupoid algebra RG. These étale groupoid algebras generalize various algebras, including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for étale groupoid algebras. In this work, we characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple étale groupoid algebras, thereby generalizing existing results for Leavitt path algebras and introducing new results for inverse …

Cohen-Macaulay Type Of Open Neighborhood Ideals Of Unmixed Trees, Jounglag Lim

All Theses

Given a tree T and a field k, we define the open neighborhood ideal N(T) of T in k[V] to be the ideal generated by the open neighborhoods of all vertices in the graph. If T is unmixed with respect to the total domination problem, then it is known that N(T) is Cohen-Macaulay. Our goal is to compute the (Cohen-Macaulay) type of k[V]/N(T) using graph theoretical properties of T. We achieve this by using homological algebra and properties of monomial ideals. Along the way, we also provide a different characterization of unmixed trees and a generalization of the total dominating …

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.

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 …

Slₖ-Tilings And Paths In ℤᵏ, Zachery T. Peterson

Theses and Dissertations--Mathematics

An SLₖ-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher k by constructing a bijection between SLₖ-tilings and certain pairs of bi-infinite strips of vectors in ℤᵏ called paths. The key ingredient in the proof is the relation to Plucker friezes and …

Reducing Food Scarcity: The Benefits Of Urban Farming, S.A. Claudell, Emilio Mejia

Journal of Nonprofit Innovation

Urban farming can enhance the lives of communities and help reduce food scarcity. This paper presents a conceptual prototype of an efficient urban farming community that can be scaled for a single apartment building or an entire community across all global geoeconomics regions, including densely populated cities and rural, developing towns and communities. When deployed in coordination with smart crop choices, local farm support, and efficient transportation then the result isn’t just sustainability, but also increasing fresh produce accessibility, optimizing nutritional value, eliminating the use of ‘forever chemicals’, reducing transportation costs, and fostering global environmental benefits.

Imagine Doris, who is …

The Vulnerabilities To The Rsa Algorithm And Future Alternative Algorithms To Improve Security, James Johnson

Cybersecurity Undergraduate Research Showcase

The RSA encryption algorithm has secured many large systems, including bank systems, data encryption in emails, several online transactions, etc. Benefiting from the use of asymmetric cryptography and properties of number theory, RSA was widely regarded as one of most difficult algorithms to decrypt without a key, especially since by brute force, breaking the algorithm would take thousands of years. However, in recent times, research has shown that RSA is getting closer to being efficiently decrypted classically, using algebraic methods, (fully cracked through limited bits) in which elliptic-curve cryptography has been thought of as the alternative that is stronger than …

The Gamma-Signless Laplacian Adjacency Matrix Of Mixed Graphs, Omar Alomari, Mohammad Abudayah, Manal Ghanem

Theory & Applications of Graphs

The α-Hermitian adjacency matrix H_α of a mixed graph X has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number α. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as a generalization of the signless Laplacian adjacency matrix of graphs. We then study the spectral properties of the gamma-signless Laplacian adjacency matrix of a mixed graph. Lastly, we characterize when the signless Laplacian adjacency matrix of …

Representations From Group Actions On Words And Matrices, Joel T. Anderson

Master's Theses

We provide a combinatorial interpretation of the frequency of any irreducible representation of Sn in representations of Sn arising from group actions on words. Recognizing that representations arising from group actions naturally split across orbits yields combinatorial interpretations of the irreducible decompositions of representations from similar group actions. The generalization from group actions on words to group actions on matrices gives rise to representations that prove to be much less transparent. We share the progress made thus far on the open problem of determining the irreducible decomposition of certain representations of Sm × Sn arising from group actions on matrices.

Roots Of Quaternionic Polynomials And Automorphisms Of Roots, Olalekan Ogunmefun

Electronic Theses and Dissertations

The quaternions are an extension of the complex numbers which were first described by Sir William Rowan Hamilton in 1843. In his description, he gave the equation of the multiplication of the imaginary component similar to that of complex numbers. Many mathematicians have studied the zeros of quaternionic polynomials. Prominent of these, Ivan Niven pioneered a root-finding algorithm in 1941, Gentili and Struppa proved the Fundamental Theorem of Algebra (FTA) for quaternions in 2007. This thesis finds the zeros of quaternionic polynomials using the Fundamental Theorem of Algebra. There are isolated zeros and spheres of zeros. In this thesis, we …

Cohen-Macaulay Properties Of Closed Neighborhood Ideals, Jackson Leaman

All Theses

This thesis investigates Cohen-Macaulay properties of squarefree monomial ideals, which is an important line of inquiry in the field of combinatorial commutative algebra. A famous example of this is Villareal’s edge ideal [11]: given a finite simple graph G with vertices x1, . . . , xn, the edge ideal of G is generated by all the monomials of the form xixj where xi and xj are adjacent in G. Villareal’s characterization of Cohen-Macaulay edge ideals associated to trees is an often-cited result in the literature. This was extended to chordal and bipartite graphs by Herzog, Hibi, and Zheng in …

A Stronger Strong Schottky Lemma For Euclidean Buildings, Michael E. Ferguson

Dissertations, Theses, and Capstone Projects

We provide a criterion for two hyperbolic isometries of a Euclidean building to generate a free group of rank two. In particular, we extend the application of a Strong Schottky Lemma to buildings given by Alperin, Farb and Noskov. We then use this extension to obtain an infinite family of matrices that generate a free group of rank two. In doing so, we also introduce an algorithm that terminates in finite time if the lemma is applicable for pairs of certain kinds of matrices acting on the Euclidean building for the special linear group over certain discretely valued fields.

An Inquiry Into Lorentzian Polynomials, Tomás Aguilar-Fraga

HMC Senior Theses

In combinatorics, it is often desirable to show that a sequence is unimodal. One method of establishing this is by proving the stronger yet easier-to-prove condition of being log-concave, or even ultra-log-concave. In 2019, Petter Brändén and June Huh introduced the concept of Lorentzian polynomials, an exciting new tool which can help show that ultra-log-concavity holds in specific cases. My thesis investigates these Lorentzian polynomials, asking in which situations they are broadly useful. It covers topics such as matroid theory, discrete convexity, and Mason’s conjecture, a long-standing open problem in matroid theory. In addition, we discuss interesting applications to known …

Long Increasing Subsequences, Hannah Friedman

HMC Senior Theses

In my thesis, I investigate long increasing subsequences of permutations from two angles. Motivated by studying interpretations of the longest increasing subsequence statistic across different representations of permutations, we investigate the relationship between reduced words for permutations and their RSK tableaux in Chapter 3. In Chapter 4, we use permutations with long increasing subsequences to construct a basis for the space of ��-local functions.

Q-Polymatroids And Their Application To Rank-Metric Codes., Benjamin Jany

Theses and Dissertations--Mathematics

Matroid theory was first introduced to generalize the notion of linear independence. Since its introduction, the theory has found many applications in various areas of mathematics including coding theory. In recent years, q-matroids, the q-analogue of matroids, were reintroduced and found to be closely related to the theory of linear vector rank metric codes. This relation was then generalized to q-polymatroids and linear matrix rank metric codes. This dissertation aims at developing the theory of q-(poly)matroid and its relation to the theory of rank metric codes. In a first part, we recall and establish preliminary results for both q-polymatroids and …