Algebra Commons^™

The Spectral Theorem For Unitary Operators Based On The S-Spectrum, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem.

In this paper we prove the quaternionic spectral theorem for unitary operators using the S-spectrum. In the case of quaternionic matrices, the S-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of S-spectrum is relatively new, see [17], and has been used for …

A New Realization Of Rational Functions, With Applications To Linear Combination Interpolation, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the following linear combination interpolation problem (LCI): Given N distinct numbers w1,…wN and N+1 complex numbers a1,…,aN and c, find all functions f(z) analytic in a simply connected set (depending on f) containing the points w1,…,wN such that ∑u=1Nauf(wu)=c. To this end we prove a representation theorem for such functions f in terms of an associated polynomial p(z). We first introduce the following two operations, (i) substitution of p, and (ii) multiplication by monomials zj,0≤j

Higher Order Z-Ideals In Commutative Rings, Themba Dube, Oghenetega Ighedo

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study ideals that resemble z-ideals in commutative rings with identity. For each positive integer n, we say an ideal of a commutative ring A is a zⁿ-ideal in case it has the property that if a and b belong to the same maximal ideals of A, and aⁿ ϵ I , then bⁿ is also in I. The set of all zⁿ-ideals of A is denoted by A --> ʒⁿ (A). This gives an ascending chain ʒ(A) < ʒ²(A) < ʒ³(A) <… of collections of ideals, starting with the collection of z-ideals. …

The Spectral Theorem For Quaternionic Unbounded Normal Operators Based On The S-Spectrum, Daniel Alpay, Fabrizio Colombo, David P. Kimsey

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The fact that the correct notion of …

Concurrent Kleene Algebra With Tests And Branching Automata, Peter Jipsen, M. Andrew Moshier

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce concurrent Kleene algebra with tests (CKAT) as a combination of Kleene algebra with tests (KAT) of Kozen and Smith with concurrent Kleene algebras (CKA), introduced by Hoare, Möller, Struth and Wehrman. CKAT provides a relatively simple algebraic model for reasoning about semantics of concurrent programs. We generalize guarded strings to guarded series-parallel strings , or gsp-strings, to give a concrete language model for CKAT. Combining nondeterministic guarded automata of Kozen with branching automata of Lodaya and Weil one obtains a model for processing gsp-strings in parallel. To ensure that the model satisfies the weak exchange law (x‖y)(z‖w)≤(xz)‖(yw) of …

On The Construction Of Simply Connected Solvable Lie Groups, Mark E. Fels

Research Vignettes

This worksheet contains the implementation of Theorems 4.2, 5.4 and 5.7 in the paper On the Construction of Solvable Lie Groups. All the examples in the paper are demonstrated here, along with one in Section 6 that was too long to include in the article.

Constructions And Isomorphism Types Of Images, Jessica Luna Ramirez

Electronic Theses, Projects, and Dissertations

In this thesis, we have presented our discovery of true finite homomorphic images of various permutation and monomial progenitors, such as 2^*7: D₁₄, 2^*7 : (7 : 2), 2^*6 : S₃ x 2, 2^*8: S₄, 2*⁷²: (3²:(2S₄)), and 11^*2 :_m D₁₀. We have given delightful symmetric presentations and very nice permutation representations of these images which include, the Mathieu groups M₁₁, M₁₂, the 4-fold cover of the Mathieu group M₂₂, 2 x …

Non-Commutative Holomorphic Functions On Operator Domains, Jim Agler, John E. Mccarthy

Mathematics Faculty Research

We characterize functions of d-tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets.

Review: The Classical Hom-Yang-Baxter Equation And Hom-Lie Bialgebras, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Generating All Finite Modular Lattices Of A Given Size, Peter Jipsen, Nathan Lawless

Mathematics, Physics, and Computer Science Faculty Articles and Research

Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold [8] developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18. Here we adapt and improve this algorithm to construct and count modular lattices up to size 24, semimodular lattices up to size 22, and lattices of size 19. We also show that 2 ⁿ⁻³ is a lower bound for the number of nonisomorphic modular lattices of size n.

College Algebra (University Of North Georgia), Michael Goodroe, Berhanu Kidane, Julian Allagan, John Williams

Mathematics Grants Collections

This Grants Collection Open Textbook for College Algebra was created under a Round Two 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.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Foundations For College Algebra, Michael Goodroe, Berhanu Kidane, Julian Allagan, John Williams

Mathematics Grants Collections

This Grants Collection for Foundations for College Algebra was created under a Round Two 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.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Support For College Algebra, Michael Goodroe, Berhanu Kidane, Julian Allagan, John Williams

Mathematics Grants Collections

This Grants Collection for Support for College Algebra was created under a Round Two 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.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Intermediate Algebra, Michael Goodroe, Berhanu Kidane, Julian Allagan, John Williams

Mathematics Grants Collections

This Grants Collection for Intermediate Algebra was created under a Round Two 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.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Candy Crush Combinatorics, Dana Rowland

Mathematics Faculty Publications

In the popular game Candy Crush, differently colored candies are arranged in a grid and a player swaps adjacent candies in order to crush them by lining up three or more of the same color. At the beginning of each game, the grid cannot have three consecutive candies of the same color in a row or column, but it must be possible to swap two adjacent candies in order to get at least three consecutive candies of the same color. How many starting configurations are there? We derive recurrence relations to answer this question for a single line of candy, …

Geometric Constructions From An Algebraic Perspective, Betzabe Bojorquez

Electronic Theses, Projects, and Dissertations

Many topics that mathematicians study at times seem so unrelated such as Geometry and Abstract Algebra. These two branches of math would seem unrelated at first glance. I will try to bridge Geometry and Abstract Algebra just a bit with the following topics. We can be sure that after we construct our basic parallel and perpendicular lines, bisected angles, regular polygons, and other basic geometric figures, we are actually constructing what in geometry is simply stated and accepted, because it will be proven using abstract algebra. Also we will look at many classic problems in Geometry that are not possible …

The Design And Validation Of A Group Theory Concept Inventory, Kathleen Mary Melhuish

Dissertations and Theses

Within undergraduate mathematics education, there are few validated instruments designed for large-scale usage. The Group Concept Inventory (GCI) was created as an instrument to evaluate student conceptions related to introductory group theory topics. The inventory was created in three phases: domain analysis, question creation, and field-testing. The domain analysis phase included using an expert consensus protocol to arrive at the topics to be assessed, analyzing curriculum, and reviewing literature. From this analysis, items were created, evaluated, and field-tested. First, 383 students answered open-ended versions of the question set. The questions were converted to multiple-choice format from these responses and disseminated …

Systems Of Parameters And The Cohen–Macaulay Property, Katharine Shultis

Department of Mathematics: Dissertations, Theses, and Student Research

Let R be a commutative, Noetherian, local ring and M a finitely generated R-module. Consider the module of homomorphisms Hom_R(R/a,M/bM) where b [subset of] a are parameter ideals of M. When M = R and R is Cohen-Macaulay, Rees showed that this module of homomorphisms is isomorphic to R/a, and in particular, a free module over R/a of rank one. In this work, we study the structure of such modules of homomorphisms for a not necessarily Cohen-Macaulay R-module M.

The Impact Of A Quantitative Reasoning Instructional Approach To Linear Equations In Two Variables On Student Achievement And Student Thinking About Linearity, Paul Thomas Belue

Boise State University Theses and Dissertations

A control group and an experimental group of college students at a community college in the Pacific Northwest were taught a unit on linear equations in two variables. The control group was taught using a traditional instructional approach that focused on learning procedures and the experimental group was taught using a quantitative reasoning instructional approach that focused on learning proportional and functional reasoning. Both groups were then given the same unit assessment that had 10 procedural understanding items and 10 conceptual understanding items related to linear equations in two variables. The assessment was given to determine the impact of the …

A System Of Equations: Mathematics Lessons In Classical Literature, Valery F. Ochkov, Andreas Look

Journal of Humanistic Mathematics

The aim of this paper is to showcase a handful of mathematical challenges found in classical literature and to offer possible ways of integrating classical literature in mathematics lessons. We analyze works from a range of authors such as Jules Verne, Anton Chekhov, and others. We also propose ideas for further tasks. Most of the problems can be restated in terms of simple mathematical equations, and they can often be solved without a computer. Nevertheless, we use the computer program Mathcad to solve the problems and to illustrate the solutions to enhance the reader’s mathematical experience.