Algebra Commons^™

The Subconstituent Algebra Of A Hypercube, Jared B. Billet

UNF Graduate Theses and Dissertations

We study the hypercube and the associated subconstituent algebra. Let Q_D denote the hypercube with dimension D and let X denote the vertex set of Q_D. Fix a vertex x in X. We denote by A the adjacency matrix of Q_D and by A* = A*(x) the diagonal matrix with yy-entry equal to D − 2i, where i is the distance between x and y. The subconstitutent algebra T = T(x) of Q_D with respect to x is generated by A and A* . We show that A 2A* − 2AA*A + A*A 2 = 4A* A*2A − 2A*AA* + …

Isolated Point Theorems For Uniform Algebras On Smooth Manifolds, Swarup Ghosh

Faculty Articles & Research

In 1957, Andrew Gleason conjectured that if A is a uniform algebra on its maximal ideal space X and every point of X is a one-point Gleason part for A, then A must contain all continuous functions on X. Gleason’s conjecture was disproved by Brian Cole in 1968. In this paper, we establish a strengthened form of Gleason’s conjecture for uniform algebras generated by real-analytic functions on compact subsets of real-analytic three-dimensional manifolds-with-boundary.

An Admm-Factorization Algorithm For Low Rank Matrix Completion, Rahman Taleghani, Maziar Salahi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we propose an Alternating Direction Method of Multipliers (ADMM) based algorithm that is taking advantage of factorization for the fixed rank matrix completion problem. The convergence of the proposed algorithm to the KKT point is discussed. Finally, on several classes of test problems, its efficiency is compared with several efficient algorithms from the literature.

Invariant Sum Defined In Terms Of Complex Multivariate Polynomial Given Degree, Matthew Niemiro '20

Exemplary Student Work

We use a generalized version of arithmetic progressions to obtain a non- trivial everywhere-zero sum in terms of a complex univariate polynomial and its degree. We then remark on its generalization to multivariate polynomials.

On Ordered (P; Q)-Lateral Ideals In Ordered Ternary Semigroups, Mohammad Y. Abbasi, Sabahat A. Khan, Akbar Ali

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study some useful results of ordered (p; q)-lateral ideals in ordered ternary semigroups. Also, some properties of (p; q)-lateral simple ordered ternary semigroup have been examined. Further, we characterize the relationship between minimal (resp., maximal) ordered (p; q)- lateral ideals and (p; q)-lateral simple ordered ternary semigroups.

Fibonacci And Lucas Identities From Toeplitz–Hessenberg Matrices, Taras Goy, Mark Shattuck

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we consider determinants for some families of Toeplitz–Hessenberg matrices having various translates of the Fibonacci and Lucas numbers for the nonzero entries. These determinant formulas may also be rewritten as identities involving sums of products of Fibonacci and Lucas numbers and multinomial coefficients. Combinatorial proofs are provided of several of the determinants which make use of sign-changing involutions and the definition of the determinant as a signed sum over the symmetric group. This leads to a common generalization of the Fibonacci and Lucas determinant formulas in terms of the so-called Gibonacci numbers.

Hankel Rhotrices And Constructions Of Maximum Distance Separable Rhotrices Over Finite Fields, P. L. Sharma, Arun Kumar, Shalini Gupta

Applications and Applied Mathematics: An International Journal (AAM)

Many block ciphers in cryptography use Maximum Distance Separable (MDS) matrices to strengthen the diffusion layer. Rhotrices are represented by coupled matrices. Therefore, use of rhotrices in the cryptographic ciphers doubled the security of the cryptosystem. We define Hankel rhotrix and further construct the maximum distance separable rhotrices over finite fields.

Rank Reduction Of String C-Group Representations, Peter A. Brooksbank, Dimitri Leemans

Faculty Journal Articles

We show that a rank reduction technique for string C-group representations first used in [Adv. Math. 228 (2018), pp. 3207–3222] for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on d-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group Alt(11)—the only known group having “rank gaps”—is perhaps more unusual …

Testing Isomorphism Of Graded Algebras, Peter A. Brooksbank, James B. Wilson, Eamonn A. O'Brien

Faculty Journal Articles

We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that often dramatically improve the performance of the algorithm and report on an implementation in Magma.

Groups Satisfying The Converse To Lagrange's Theorem, Jonah N. Henry

Graduate Theses/Dissertations

Lagrange’s theorem, which is taught early on in group theory courses, states that the order of a subgroup must divide the order of the group which contains it. In this thesis, we consider the converse to this statement. A group satisfying the converse to Lagrange’s theorem is called a CLT group. We begin with results that help show that a group is CLT, and explore basic CLT groups with examples. We then give the conditions to guarantee either CLT is satisfied or a non-CLT group exists for more advanced cases. Additionally, we show that CLT groups are properly contained between …

Distributive Laws In Residuated Binars, Wesley Fussner, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

In residuated binars there are six non-obvious distributivity identities of ⋅,/,∖ over ∧,∨. We show that in residuated binars with distributive lattice reducts there are some dependencies among these identities; specifically, there are six pairs of identities that imply another one of these identities, and we provide counterexamples to show that no other dependencies exist among these.

Supporting The Algebra I Curriculum With An Introduction To Computational Thinking Course, Michelle M. Laskowski

LSU Master's Theses

The Louisiana Workforce Commission predicts a 33.6% increase in computer science and mathematical occupations by 2022 and the Bureau of Labor Statistics foresees a 16% increase in computer scientists from 2018-2028. Despite these opportunities for job and financial security, the number of Louisiana students enrolled in a nationally accredited computing course is less than 1%, compared to national leaders California and Texas which have 3% and 3.8% of students respectively. Furthermore, the international assessments of mathematical literacy, PISA and TIMMS, both report American students continue to fall further behind their international peers in mathematics achievement.

This thesis rejects these statistics …

Alpha Capture Reaction Rates For Nucleosynthesis Within An Ab Initio Framework, Alison Constance Dreyfuss

LSU Doctoral Dissertations

Clustering in nuclear systems has broad impacts on all phases of stellar burning, and plays a significant role in our understanding of nucleosynthesis, or how and where nuclei are produced in the universe. The role of alpha particles in particular is extremely important for nuclear astrophysics: ⁴He was one of the earliest elements produced in the Big Bang, it is one of the most abundant elements in the universe, and helium burning -- in particular, the triple-alpha process -- is one of the most important ``engines'' in stars. To better understand nucleosynthesis and stellar burning, then, it is important …

A Concise Workbook For College Algebra 2nd Edition, Fei Ye

Open Educational Resources

This is the second edition of the book "A Concise Workbook for College Algebra". In this new edition, some tips and notes, more exercises and examples were added.

Beauty, Bees, And God: The Fibonacci Sequence As A Theological Springboard In Secondary Mathematics, John D. Brahier

Journal of Catholic Education

Catholic schools primarily should be in the business of making saints. This article identifies and explores a meaningful, engaging point of contact between mathematics and theology for high school math classes, the Fibonacci Sequence. This sequence serves as an engaging introduction to sequences and series; more importantly, the topic can be used as a springboard to theological discussions. The paper will provide a brief historical background to the Fibonacci Sequence, an explanation of how it can be used in a high school math classroom, and an exploration of three different theological touchpoints that the Fibonacci Sequence offers.

College Algebra (Atlm), Shreyas Desai, Anthonia Ekwuocha, Noel Whelchel

Mathematics Grants Collections

This Grants Collection for College Algebra was created under a Round Twelve ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Zeta Functions Of Classical Groups And Class Two Nilpotent Groups, Fikreab Solomon Admasu

Dissertations, Theses, and Capstone Projects

This thesis is concerned with zeta functions and generating series associated with two families of groups that are intimately connected with each other: classical groups and class two nilpotent groups. Indeed, the zeta functions of classical groups count some special subgroups in class two nilpotent groups.

In the first chapter, we provide new expressions for the zeta functions of symplectic groups and even orthogonal groups in terms of the cotype zeta function of the integer lattice. In his paper on universal $p$-adic zeta functions, J. Igusa computed explicit formulae for the zeta functions of classical algebraic groups. These zeta functions …

The Derived Category Of A Locally Complete Intersection Ring, Joshua Pollitz

Department of Mathematics: Dissertations, Theses, and Student Research

Let R be a commutative noetherian ring. A well-known theorem in commutative algebra states that R is regular if and only if every complex with finitely generated homology is a perfect complex. This homological and derived category characterization of a regular ring yields important ring theoretic information; for example, this characterization solved the well-known ``localization problem" for regular local rings. The main result of this thesis is establishing an analogous characterization for when R is locally a complete intersection. Namely, R is locally a complete intersection if and only if each nontrivial complex with finitely generated homology can build a …

Mathematics And Programming Exercises For Educational Robot Navigation, Ronald I. Greenberg

Computer Science: Faculty Publications and Other Works

This paper points students towards ideas they can use towards developing a convenient library for robot navigation, with examples based on Botball primitives, and points educators towards mathematics and programming exercises they can suggest to students, especially advanced high school students.

Tracing Cyclic Homology Pairings Under Twisting Of Graded Algebras, Sayan Chakraborty, Makoto Yamashita

Journal Articles

We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss–Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.