Algebra Commons^™

Free Resolutions Of Orbit Closures For Representations With Finitely Many Orbits, Federico Galetto

Mathematics Dissertations

The irreducible representations of reductive groups with finitely many orbits are parametrized by graded simple Lie algebras. For the exceptional Lie algebras, Kraśkiewicz and Weyman exhibit the Hilbert polynomials and the expected minimal free resolutions of the normalization of the orbit closures. We present an interactive method to construct explicitly these and related resolutions in Macaulay2. The method is then used in the cases of the Lie algebras of type E₆, F₄, G₂, and select cases of type E₇ to confirm the shape of the expected resolutions as well as some geometric properties of the orbit ...

Decreasing Math Anxiety Through A Quadratics Unit, Lee C. Hooker

Education and Human Development Master's Theses

Anxiety related to the learning of mathematics is referred to as math anxiety and has been shown to have a negative influence on student performance. Research reveals that math anxiety is something that can be unlearned and informs about the potential causes and treatments of math anxiety in the mathematics classroom. A majority of math anxiety experienced by students has been caused by teachers’ repetitious teaching styles. Research presents various teaching strategies that have helped teachers when working with students who have math anxiety. These strategies include writing, class discussions, cooperative groups, kinesthetic activities, use of manipulatives, and various, frequent ...

Apolarity For The Determinant And Permanent, Masoumeh Shafiei

Mathematics Dissertations

We show that the apolar ideals to the determinant and permanent of a generic matrix, the Pfaffian of a generic skew symmetric matrix and the determinant and the hafnian of a generic symmetric matrix are each generated in degree two. We also show that unlike the previous polynomials, the apolar ideal to the permanent of a generic symmetric matrix is generated in degrees two and three. In each case we specify the generators and give a Gröbner basis of the apolar ideal. As a consequence, using a result of K. Ranestad and F.-O. Schreyer we give lower bounds to ...

Discrete Structures In Finite Type Cluster Algebras, Salvatore Stella

Mathematics Dissertations

Due to their recursive definition, manipulating cluster algebras in an efficient way can be hard. Several combinatorial models have been developed in order to overcome this difficulty; here we investigate some of them in the finite type case.

In the first part of this thesis, using the parametrization of cluster variables by their g-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we extend the original construction of generalized associahedra by F. Chapoton, S. Fomin and A. Zelevinsky to any choice of acyclic initial cluster, and compare it to the one given by C. Hohlweg, C. Lange, and H ...

An Analysis Of Differences In Approaches To Systems Of Linear Equations Problems Given Multiple Choice Answers, Amber Lagasse

Honors Theses

This descriptive study focuses on the approaches college students (ages 20 -24) use when solving systems of linear equations problems that have multiple choice answers. Participants were from a midsize public university in the northeast. Four approaches were considered – three forwards approaches: 1) substitution, 2) elimination, and 3) graphing, and one backwards approach: plugging in the x and y values from each multiple choice option. Participants solved systems of linear equations problems and answered questions based on their methods in a structured clinical interview. Each participant also filled out a questionnaire. It was shown from the results of this study ...

Symbolic Powers Of Ideals In K[PN], Michael Janssen

Dissertations, Theses, and Student Research Papers in Mathematics

Let I ⊆ k[P^N] be a homogeneous ideal and k an algebraically closed field. Of particular interest over the last several years are ideal containments of symbolic powers of I in ordinary powers of I of the form I^(m) ⊆ I^r, and which ratios m/r guarantee such containment. A result of Ein-Lazarsfeld-Smith and Hochster-Huneke states that, if I ⊆ k[P^N], where k is an algebraically closed field, then the symbolic power I^(Ne) is contained in the ordinary power I^e, and thus, whenever m/r ≥ N we have the containment I^(m) ⊆ I^r. Therefore ...

Periodic Modules Over Gorenstein Local Rings, Amanda Croll

Dissertations, Theses, and Student Research Papers in Mathematics

It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}] associated to R. This module, denoted (R), is the free Z[t,t^{-1}]-module on the isomorphism classes of finitely generated R-modules modulo relations reminiscent of those defining the Grothendieck group of R. The main result is a structure theorem for J(R) when R is a complete Gorenstein local ring; the link between periodicity and torsion stated above is a ...

Propeller, Joel Kahn

The STEAM Journal

This image is based on several different algorithms interconnected within a single program in the language BASIC-256. The fundamental structure involves a tightly wound spiral working outwards from the center of the image. As the spiral is drawn, different values of red, green and blue are modified through separate but related processes, producing the changing appearance. Algebra, trigonometry, geometry, and analytic geometry are all utilized in overlapping ways within the program. As with many works of algorithmic art, small changes in the program can produce dramatic alterations of the visual output, which makes lots of variations possible.

Polynomial Functions Over Finite Fields, John J. Hull

Georgia State Undergraduate Research Conference

No abstract provided.

How To Create A Jordan Algebra, Ian M. Anderson, Thomas J. Apedaile

How to... in 10 minutes or less

We show how to create a Jordan algebra in Maple using the commands AlgebraLibraryData and AlgebraData.

How To Create The Quaternion & Octonion Algebras, Ian M. Anderson, Thomas J. Apedaile

How to... in 10 minutes or less

We show how to create the quaternion and octonion algebras with the DifferentialGeometry software. For each algebra, there is a split-form also available.

On Contemplation In Mathematics, Frank Lucas Wolcott

Journal of Humanistic Mathematics

In a section about research, we make the case that intentional, structured reflection on the mathematical research process, by mathematical researchers themselves, would result in better mathematicians doing better mathematics. As supporting evidence, we describe the Flavors and Seasons project. In a section about teaching, we describe the contemplative education movement and share personal experiences using meditation in the math classroom. We conclude with an explicit proposal for elucidating the experiential context of mathematics, in both research and teaching environments.

Multi-Agent Flocking With Random Communication Radius, Samuel Martin, Arastoo Fazeli, Ali Jadbabaie, Antoine Girard

Departmental Papers (ESE)

In this paper, we consider a multi-agent system consisting of mobile agents with second-order dynamics. The communication network is determined by a metric rule based on a random interaction range. The goal of this paper is to determine a bound on the probability that the agents asymptotically agree on a common velocity (i.e. a flocking behavior is achieved). This bound should depend on practical conditions (on the initial positions and velocities of agents) only. For this purpose, we exhibit an i.i.d. process bounding the original system’s dynamics. We build upon previous work on multi-agent systems with ...

The (Nested) Word Problem: Formal Languages, Group Theory, And Languages Of Nested Words, Christopher S. Henry

Open Access Dissertations and Theses

This thesis concerns itself with drawing out some interesting connections between the fields of group theory and formal language theory. Given a group with a finite set of generators, it is natural to consider the set of generators and their inverses as an alphabet. We can then consider formal languages such that every group element has at least one representative in the language. We examine what the structure of the language can tell us about group theoretic properties, focusing on the word problem, automatic structures on groups, and generalizations of automatic structures. Finally we prove new results concerning applications of ...

Prime Ideals In Two-Dimensional Noetherian Domains And Fiber Products And Connected Sums, Ela Celikbas

Dissertations, Theses, and Student Research Papers in Mathematics

This thesis concerns three topics in commutative algebra:

1) The projective line over the integers (Chapter 2),

2) Prime ideals in two-dimensional quotients of mixed power series-polynomial rings (Chapter 3),

3) Fiber products and connected sums of local rings (Chapter 4),

In the first chapter we introduce basic terminology used in this thesis for all three topics.

In the second chapter we consider the partially ordered set (poset) of prime ideals of the projective line Proj(Z[h,k]) over the integers Z, and we interpret this poset as Spec(Z[x]) U Spec(Z[1/x]) with an appropriate ...

Commutative Rings Graded By Abelian Groups, Brian P. Johnson

Dissertations, Theses, and Student Research Papers in Mathematics

Rings graded by Z and Z^d play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the ...

How To Create A Lie Algebra, Ian M. Anderson

How to... in 10 minutes or less

We show how to create a Lie algebra in Maple using three of the most common approaches: matrices, vector fields and structure equations. PDF and Maple worksheets can be downloaded from the links below.