Algebra Commons^™

Free Probability Theory Over The Scaled Hyperbolic Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers {D_t} _t∈R are constructed as sub-structures of scaled hypercomplex numbers {H_t} _t∈R under the scales (or, the moments) of the set R of real numbers.We show that if t < 0, then the classical free probability theory covers our free probability on {D_t} _{t< 0}; if t > 0, then our free probability on {D_t} _t>0 is represented by the free probability over the classical hyperbolic numbers D = D₁; and if t = 0, then the free probability on D₀ is actually over the …

(R2127) On 2-Absorbing Hesitant Primary Fuzzy Ideals Of Rings, B. Anitha, M. Vidhya

Applications and Applied Mathematics: An International Journal (AAM)

By presenting 2-absorbing hesitant primary fuzzy ideals, we begin the investigation of a generalisation of hesitant primary fuzzy ideals (HPRFI) in rings in this research. The concepts of a weakly completely 2-absorbing hesitant primary fuzzy ideal (WC2-AHPRFI) and a Weakly completely 2-absorbing hesitant fuzzy ideal (WC2-AHFI) are developed, and their structural features and attributes are examined. We introduce the idea of a 2-absorbing hesitant K-fuzzy ideal (2-AHK-FI), 2-absorbing hesitant K-primary fuzzy ideal (2-AHK-PRFI) and examine a few of its characteristics.

(R2094) On Mds Bisymmetric Rhotrices Using Self-Dual Bases And Conjugate Elements Of Finite Fields, Shalini Gupta, Ruchi Narang, Manpreet Singh

Applications and Applied Mathematics: An International Journal (AAM)

Bisymmetric matrices have wide range of applications in statistics, engineering problems, information theory and computer science including coding theory and cryptography. In cryptography, a rhotrix being a couple matrix doubles the security of the cryptosystem. Here, we construct maximum distance separable (MDS) bisymmetric rhotrices using self-dual bases and conjugate elements of finite fields. MDS rhotrices are very crucial for the designing of block ciphers and hash functions in cryptography.

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 …

The Kaczmarz Algorithm In Hilbert C*-Modules, Daniel Alpay, Chad Berner, Eric S. Weber

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert C*-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert C(X)-modules can be generated by the Kaczmarz …

Principal Quandles, Jesse Parrish

Electronic Theses and Dissertations

This thesis concerns principal quandles and, as a special case, Alexander quandles. A principal quandle is a coset quandle, Q(G,H, f), in which the subgroup H is trivial. If in addition the group G is abelian, then the coset quandle is called an Alexander quandle. The isomorphism types of Alexander quandles were classified [16], and some progress was made on describing the isomorphism types of principal quandles in [14] and [13]. The first part of this thesis applies the ideas used in Holmes’ paper to give a classification of the isomorphism types of principal quandles. The second …

Invitation To Polynomiography Via Chatgpt: For Teachers, Students And Artists, Bahman Kalantari

LASER Journal

Teachers, students, and artists in the United States and abroad who have encountered polynomiography, in lectures, demos, or software, consistently appreciate its educational value and artistic potential. While dedicated polynomiography programs require upkeep as systems evolve, AI chatbots now offer a practical, accessible alternative. This article invites readers to explore polynomiography with ChatGPT, broadening access beyond specialized tools. While the approach will not match the full range or polish of advanced software, it provides a powerful and flexible entry point with many possibilities.

At its core, polynomiography transforms polynomial equations, each encoding a finite set of points in the complex …

Dimension Of Subalgebras Of Fomin-Kirillov Algebras, Taro Ikeda, Sirous Homayouni

Annual Student Research Poster Session

One of the central open problems concerning the Fomin-Kirillov algebras was whether their dimension is finite or infinite. This question was recently resolved in one research by C. Barligea, where it was shown that FK(n) is infinite-dimensional for all n ≥ 6. In our work, we investigate the dimension of a certain subalgebra of FK(6).

(Si15-064) Permutation Pentanomials Over Finite Fields With Even Characteristic, Shalini Gupta, Sushil Kumar, Ashima .

Applications and Applied Mathematics: An International Journal (AAM)

Permutation polynomials over finite fields constitute an active area of research and play an important role in diverse domains, including finite geometry, combinatorial design, coding theory, and cryptography. The study of these polynomials has a long history, and many results have been obtained in recent years. This paper presents new classes of permutation pentanomials based on permutation over the unit circle of finite fields with even characteristic that contribute to the theoretical development of permutation polynomials.

How Many Symmetries Of The Regular N-Gon Are Even?, Thi Mai Khoi Ha

Rose-Hulman Undergraduate Mathematics Journal

One of the simplest classes of finite groups used as a source of counterexamples in a first course of modern algebra is the class of finite dihedral groups. Among the subgroups of dihedral group, finding subgroups of index 2 is of interest in part because these subgroups are normal subgroups. In this article, we use the representations of the symmetries of the dihedral groups as permutations of the vertices and determine concretely all its subgroups of index 2. Under this representation or embedding, the article determines the intersection of the dihedral group with the corresponding alternating groups when they are …

Sugar: A Sequence Unfolding Based Transformer Model For Group Activity Recognition, Yash U. Gondkar

Graduate Masters Theses

Large Language Models have improved significantly in the past couple of years due to the adoption of transformers. However, transformers still find it challenging to process videos due to limited context size caused by their quadratic computing cost. Therefore, we studied a booming field in machine learning which powers applications like social scene analysis and video surveillance systems called Group Activity Recognition (GAR). We found that recent models were able to achieve more than 90% accuracy on popular datasets like the Volleyball dataset, however, it turned out that even they relied on transformers.

Therefore, in this work, we developed a …

Universal Centralizers, Morita Abelianization, And Wonderful Models In Lie Theory, Peter Crooks

Funded Research Records

No abstract provided.

Career: Algebra And Representation Theory For Non-Semisimple Topological Field Theory, Matthew Young

Funded Research Records

No abstract provided.

An Exploration Of The Structure Of The Linear Algebraic Model Of Color, Isaac Martin

University Honors Theses

This thesis surveys the mathematical grounding of linear algebraic models of color. It aims to build from the ground up the framework by which additive color is broadly understood in the digital age. Primarily building on the work of Jozef Cohen, Eric Dubois, David H. Krantz, and Günter Wyszecki, it aims to chart the construction of a model of color that underpins most modern understandings of color. While the construction is certainly established in colorimetric circles, the construction is, in the thesis author's opinion, either obtuse or non-rigorous. Ideally, this thesis serves to make the construction accessible to an audience …

Constructing Code-Based Zero-Knowledge Proofs Leveraging Generic Errors And Bounded Vectors, Freeman Slaughter

All Dissertations

Quantum computing is developing at an expeditious rate, and once fully scalable quantum computers become realized, classical cryptographic systems face obsolescence. This approaching peril has prompted a paradigm shift away from pre-quantum cryptography and towards post-quantum primitives, such as those that arise from the field of coding theory. Among these, zero-knowledge proofs have emerged as a dynamic tool instrumental in constructing quantum-resilient digital signature schemes.

We being by introducing HammR, a pre-quantum zero-knowledge proof protocol designed to verify Hamming weight and entry constraints of error vectors, and comprehensively establish its security. Subsequently, we extend HammR to the multi-party computation setting, …

A Theory Of Fundamental Strata For Twisted Formal Connections, Sorawit Viwanthananut

LSU Doctoral Dissertations

Let $G$ be a complex reductive group. A fundamental stratum for $G$ is a triple $(x,r,\beta)$ where $x$ is a point in the Bruhat-Tits building of $G$, $r$ is a nonnegative real number called depth of the stratum, and $\beta$ is a semistable functional on the Moy-Prasad filtration of $\fg$ associated to $x$ at level $r$. Fundamental strata were first introduced to classify admissible representations of a $p$-adic reductive group. More recently, Bremer and Sage have shown that fundamental strata play an important role in the geometric Langlands program and developed a theory of fundamental strata for $G$-connections. In this …

Probabilities With Values In Scaled Hyperbolic Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a notion of a probabilistic measure which takes values in t-scaled hyperbolic numbers for t ∈ R, with a system of axioms generalizing directly Kolmogorov’s axioms. i.e., we establish a suitable measure theory in the set Dt of all t-scaled hyperbolic numbers for arbitrarily fixed t ∈ R.

Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses

Master's Theses

This thesis explores the evolution of reciprocity laws in number theory in order to provide a conceptual bridge between the classical ideas of quadratic reciprocity and the modern framework of the Langlands program. We develop the necessary algebraic background to understand how the splitting behavior of primes in number fields reflects deep arithmetic structure in ℚ. Starting with quadratic fields and cyclotomic extensions, we motivate the development of the Kronecker–Weber theorem and the characterization of abelian extensions of ℚ. We then introduce Artin reciprocity and show how it generalizes quadratic reciprocity through the formalism of Frobenius elements and Artin L-functions. …

Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie

Master's Theses

A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …

The Mckay-Navarro Conjecture For The Prime 2, L. Ruhstorfer, A. A. Schaeffer Fry

Mathematics: Faculty Scholarship

We complete the proof of the McKay-Navarro conjecture (also known as the Galois-McKay conjecture) for the prime 2, by completing the proof of the inductive McKay-Navarro conditions introduced by Navarro-Späth-Vallejo for this prime.

Math 115: College Algebra Instructor Guide, Seth Lehman

Open Educational Resources

OER instructor guide for Math 115, College Algebra, Queens College

Crystallization Of The Quantized Function Algebras Of Suq(N + 1), Manabendra Giri

Doctoral Theses

The $q$-deformation of a connected, simply connected Lie group $G$ is typically studied through two Hopf algebras associated with it: the quantized universal enveloping algebra $\mathcal{U}_q(\mathfrak{g})$ and the quantized function algebra $\mathcal{O}(G_q)$. If $G$ has a compact real form $K$, one can use the Cartan involution to give a $*$-structure on $\mathcal{O}(G_q)$. The QFA $\mathcal{O}(G_q)$ with this $*$ structure is denoted by $\mathcal{O}(K_q)$ and its $C^*$-completion by $C(K_q)$. Here we study the crystal limits of $\mathcal{O}(SU_q(n+1))$ and $C(SU_q(n+1))$ and classify all irreducible representations of the crystallized algebras. We also prove that the crystallized algebra carries a natural bialgebra structure.

Understanding Student And Teacher Perceptions Of A Checking For Accuracy Method In Algebra Class, Abigail A. Jameson

Masters of Education in Teaching and Learning

Mathematics encompasses learning from one’s mistakes and developing accuracy with problem-solving. This action research study analyzed the classroom teacher’s and students' perceptions of a checking for accuracy math method that was implemented in an eighth-grade algebra classroom. Additionally, the researcher wanted to understand how the participants felt about the accuracy method and its influence on students’ attitudes toward math and mastery of math concepts. Student surveys, individual teacher (with student artifacts) and student interviews and focus groups (with individual artifacts) were the qualitative data collected using the constant comparative method. Descriptive statistics was used to collect quantitative data with calculated …

A Preliminary Study Of Hilbert–Kunz Functions: Coefficient Behavior In A Normal Affine Semigroup Ring, Jesus A. Mendiola Herrera

Theses and Dissertations

In 1890, David Hilbert published a set of notes on what now constitutes one of the bases of Commutative Algebra; his work would eventually influence the efforts of mathematicians like Ernst Kunz. In 1969, Ernst Kunz introduced a particular mapping regarding modules of regular local rings. His goal was to characterize Noetherian local rings of prime characteristic by computing the length of the composition series under Frobenius power transformations. In this thesis, the focus will be on stating the initial steps on finding the coefficients of the Hilbert-Kunz function of the normal affine semigroup ring of the form R = …

A Zariski-Nagata Theorem For Smooth Toric Surfaces, Jordan Vincent Barrett

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

The Affine Zariski-Nagata theorem is a classical result in commutative algebra that gives an expression for the n^th symbolic power of a radical ideal I in a polynomial ring over a field in terms of the n^th ordinary powers of the maximal ideals in [affine variety]_max(I). In this thesis we discuss a well-known projective analog of Zariski-Nagata and provide the necessary background on toric varieties to present a generalization of this result to toric surfaces. We conclude with a brief discussion about work toward characterizing which abstract toric varieties have smooth point fibers with …

Some Interpolation Problems In The Projective Plane, Lilah Estes

Mathematical Sciences Undergraduate Honors Theses

Given some set of r general points in the projective plane, we want to better understand: what is the smallest degree of any polynomial passing through the points m times? How many linearly independent equations of this degree pass through the points m times? The investigation of these questions, particularly for the case of m=3 and r< 16, motivates the development of several results. We translate Terracini's inductive argument, a tool for evaluating the expectedness of certain sets of double points, into a version which can be used for triple points, and prove that the argument holds. We compute the minimal graded free resolutions for the ideals corresponding to up to 15 points, for m up to 6, and we conjecture a connection between the expectedness of these ideals and what their resolutions look like. Further, we prove that this conjecture holds when m=1, and we either fully or partially prove that these ideals are …

Geometric Algebra For Field Theory In Curved Spacetime, Kevin Rhine

All Graduate Reports and Creative Projects, Fall 2023 to Present

Physics seeks to understand the universe by uncovering the fundamental laws that govern matter, energy, space, and time. At its heart lies the challenge of unification: finding a mathematical framework that consistently describes these interactions across all scales, from the subatomic to the cosmological.

This thesis explores geometric algebra, a mathematical language that unifies algebra and geometry, as a tool for advancing this understanding. By extending this framework to curved spacetimes, where gravity influences the structure of space and time, we investigate its ability to describe physical phenomena such as electromagnetism and general relativity. A notable contribution includes the geometric …

Modules Of Finite Projective Dimension And Singularities, Nawaj Kc

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

In the first part of this thesis, we study liftings of modules of finite projective dimension. We introduce a notion of “Serre liftable” modules and deduce applications to multiplicity conjectures in local algebra. In the second part, we introduce and study a module of mixed K\"ahler differentials for finite algebras over ramified discrete valuation rings of mixed characteristic. We state and prove a Jacobian criterion for computing the singular loci of such algebras.

Advisors: Jack Jeffries and Mark Walker

Car Price Prediction Using Machine Learning: Analyzing The Dvm-Car Dataset, Yaman Abu Ghareebaih

Electronic Theses and Dissertations

The objective of this study is to predict car prices using machine learning models and the DVM-CAR dataset, which includes over 1.4 million images and car specifi- cations from 899 car models. Key factors such as mileage, engine power, and year of registration were analyzed for their correlation with car prices. Extensive data cleaning was performed, including filling missing values, identifying outliers, and normalizing numerical variables. Discrete variables like car make and body type were encoded using one-hot encoding. Linear relationships were analyzed with Multiple Logistic Regression, and Random Forest models were used for nonlinear patterns. Model performance was evaluated …

Quotients, Equivalence Relations, And Normality In Non-Associative Algebra With Regards To Loops And Quasigroups, Matthew L. Mulholland

All NMU Master's Theses

This thesis will contain a detailed overview of relations, quotients, normality, loops, quasigroups, and related theorems and varieties. Nonassociative algebra is a relatively new area of mathematics, it came about in the past hundred years, and has started making progress in the past 60 years. In nonassociative algebra, varieties do not necessarily satisfy associativity. Several interesting problems with relations, quotients, and normality arise from the setting of nonassociative algebra. In the language of equivalence relations, quotients, and subsets what are the conditions of normality, or existence of a subalgebra in quasigroups and loops? A quasigroup, Q, is defined to be …