Algebra Commons^™

Algebraic Number Theory And Simplest Cubic Fields, Jianing Yang

Honors Theses

The motivation behind this paper lies in understanding the meaning of integrality in general number fields. I present some important definitions and results in algebraic number theory, as well as theorems and their proofs on cyclic cubic fields. In particular, I discuss my understanding of Daniel Shanks' paper on the simplest cubic fields and their class numbers.

Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, William Atkins

Murray State Theses and Dissertations

We present two families of diamond-colored distributive lattices – one known and one new – that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as filling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is …

I’M Being Framed: Phase Retrieval And Frame Dilation In Finite-Dimensional Real Hilbert Spaces, Jason L. Greuling

Honors Undergraduate Theses

Research has shown that a frame for an n-dimensional real Hilbert space offers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the Han-Larson-Naimark Dilation Theorem, which gives us the necessary and sufficient conditions required to dilate a frame for an n-dimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an n-dimensional real Hilbert space does not ensure that its dilation will offer phase retrieval. In this thesis, we will explore and provide what necessary and sufficient …

Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell

Honors Theses

Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example "...ababababab...". The Morse-Hedlund theorem says that a bi-infinite word f repeats itself, in at most n letters, if and only if the number of distinct subwords of length n is at most n. Using the example, "...ababababab...", there are 2 subwords of length 3, namely "aba" and "bab". Since 2 is less than 3, we must have that "...ababababab..." repeats itself after at most 3 letters. In fact it does repeat itself every two letters. …

Ancient Cultures + High School Algebra = A Diverse Mathematical Approach, Laryssa Byndas

Masters Essays

No abstract provided.

College Algebra Through Problem Solving (2018 Edition), Danielle Cifone, Karan Puri, Debra Masklanko, Ewa Dabkowska

Open Educational Resources

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

Logic -> Proof -> Rest, Maxwell Taylor

Senior Independent Study Theses

REST is a common architecture for networked applications. Applications that adhere to the REST constraints enjoy significant scaling advantages over other architectures. But REST is not a panacea for the task of building correct software. Algebraic models of computation, particularly CSP, prove useful to describe the composition of applications using REST. CSP enables us to describe and verify the behavior of RESTful systems. The descriptions of each component can be used independently to verify that a system behaves as expected. This thesis demonstrates and develops CSP methodology to verify the behavior of RESTful applications.

Special Subset Vertex Subgraphs For Social Networks, Florentin Smarandache, W. B. Vasantha Kandasamy, Ilanthenral K

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the new notion of special subset vertex subgraph of subset vertex graphs introduced recently in [46]. These subset vertex graphs takes the vertex set values from the power set P(X) of any set X. The main speciality of these subset vertex graphs is that once a set of subsets from P(X) is given, the edges of the graph are fixed in a unique way, so for a given collection of subset vertices the graph is always unique [46]. The special subset vertex subgraphs of G are the ones, which have the …

New Trends In Neutrosophic Theory And Applications Volume Ii, Florentin Smarandache, Surapati Pramanik

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory was proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications, …

Subset Vertex Graphs For Social Networks, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of subset vertex graph using the vertex set as the subset of the power set P(S), S is assumed in this book to be finite; however it can be finite or infinite. We have defined two types of subset vertex graphs, one is directed and the other one is not directed. The most important fact which must be kept in record is that for a given set of vertices there exists one and only one subset vertex graph be it of type I or type II. Several important and …

Neutrosophic Logic: The Revolutionary Logic In Science And Philosophy -- Proceedings Of The National Symposium, Florentin Smarandache, Huda E. Khalid, Ahmed K. Essa

Branch Mathematics and Statistics Faculty and Staff Publications

The first part of this book is an introduction to the activities of the National Symposium, as well as a presentation of Neutrosophic Scientific International Association (NSIA), based in New Mexico, USA, also explaining the role and scope of NSIA - Iraqi branch. The NSIA Iraqi branch presents a suggestion for the international instructions in attempting to organize NSIA's work. In the second chapter, the pivots of the Symposium are presented, including a history of neutrosophic theory and its applications, the most important books and papers in the advancement of neutrosophics, a biographical note of Prof. Florentin Smarandache in Arabic …

Mod Rectangular Natural Neutrosophic Numbers, Florentin Smarandache, K. Ilanthenral, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the new notion of MOD rectangular planes. The functions on them behave very differently when compared to MOD planes (square). These are different from the usual MOD planes. Algebraic structures on these MOD rectangular planes are defined and developed. However we have built only MOD interval natural neutrosophic products

Group Rings, Christopher Wrenn

Masters Essays

No abstract provided.

Dimensional Analysis: Physical Insight Gained Through Algebra, John A. Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Linear Algebra Applications In 3d Computer Graphics, Albert A. Antero, Chris Choung, Chris Goff

Math 365 Class Projects

Linear Transformations, Homogenous Coordinates, World Matrix, Project Matrices, Normalized Device Coordinates, View Matrix

On The Density Of The Odd Values Of The Partition Function, Samuel Judge

Dissertations, Master's Theses and Master's Reports

The purpose of this dissertation is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities is that, under suitable …

A Journey To The Adic World, Fayadh Kadhem

College of Graduate Studies: Theses & Dissertations

The first idea of this research was to study a topic that is related to both Algebra and Topology and explore a tool that connects them together. That was the entrance for me to the “adic world”. What was needed were some important concepts from Algebra and Topology, and so they are treated in the first two chapters.

The reader is assumed to be familiar with Abstract Algebra and Topology, especially with Ring theory and basics of Point-set Topology.

The thesis consists of a motivation and four chapters, the third and the fourth being the main ones. In the third …

Abelian Subalgebras Of Maximal Dimension In Euclidean Lie Algebras, Mark Curro

Theses and Dissertations (Comprehensive)

In this paper we define, discuss and prove the uniqueness of the abelian subalgebra of maximal dimension of the Euclidean Lie algebra. We also construct a family of maximal abelian subalgebras and prove that they are maximal.

On Representations Of The Jacobi Group And Differential Equations, Benjamin Webster

UNF Graduate Theses and Dissertations

In PDEs with nontrivial Lie symmetry algebras, the Lie symmetry naturally yield Fourier and Laplace transforms of fundamental solutions. Applying this fact we discuss the semidirect product of the metaplectic group and the Heisenberg group, then induce a representation our group and use it to investigate the invariant solutions of a general differential equation of the form .

Noncommutative Reality-Based Algebras Of Rank 6, Allen Herman, Mikhael Muzychuk, Bangteng Xu

EKU Faculty and Staff Scholarship

We show that noncommutative standard reality-based algebras (RBAs) of dimension 6 are determined up to exact isomorphism by their character tables. We show that the possible character tables of these RBAs are determined by seven real numbers, the first four of which are positive and the remaining three real numbers can be arbitrarily chosen up to a single exception. We show how to obtain a concrete matrix realization of the elements of the RBA-basis from the character table. Using a computer implementation, we give a list of all noncommutative integral table algebras of rank 6 with orders up to 150. …