Algebra Commons^™

Cartan Subalgebras, Compact Roots And The Satake Diagram For Su(2, 2), Ian M. Anderson

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In this worksheet we use the 15-dimensional real Lie algebra su(2, 2) to illustrate some important points regarding the general structure theory and classification of real semi-simple Lie algebras.

1. Recall that a real semi-simple Lie algebra g is called a compact Lie algebra if the Killing form is negative definite. The Lie algebra g is compact if and only if all the root vectors for any Cartan subalgebra are purely imaginary. However, if the root vectors are purely imaginary for some choice of Cartan subalgebra it is not necessarily true that the Lie algebra is compact.

2. A real …

Jordan Algebras And The Exceptional Lie Algebra F4, Ian M. Anderson

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This worksheet analyzes the structure of the Jordan algebra J(3, O) and its split and exceptional versions. The algebra of derivations is related to the exceptional Lie algebra f4.

Cohomology Of Absolute Galois Groups, Claudio Quadrelli

Electronic Thesis and Dissertation Repository

The main problem this thesis deals with is the characterization of profinite groups which are realizable as absolute Galois groups of fields: this is currently one of the major problems in Galois theory. Usually one reduces the problem to the pro-p case, i.e., one would like to know which pro-p groups occur as maximal pro-p Galois groups, i.e., maximal pro-p quotients of absolute Galois groups. Indeed, pro-p groups are easier to deal with than general profinite groups, yet they carry a lot of information on the whole absolute Galois group.

We define a new class of pro-p groups, called Bloch-Kato …

Long Wavelength Analysis Of A Model For The Geographic Spread Of A Disease, Layachi Hadji

Applications and Applied Mathematics: An International Journal (AAM)

We investigate the temporal and spatial evolution of the spread of an infectious disease by performing a long-wavelength analysis of a classical model for the geographic spread of a rabies epidemic in a population of foxes subject to idealized boundary conditions. We consider twodimensional and three-dimensional landscapes consisting of an infinite horizontal strip bounded by two walls a finite distance apart and a horizontal region bounded above and below by horizontal walls, respectively. A nonlinear partial differential evolution Equation for the leading order of infectives is derived. The Equation captures the space and time variations of the spread of the …

The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle

Mathematics Faculty Publications

Since the 1970’s, great interest has been taken in the study of pure O-sequences, which are in bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these by their shape. It has been shown that all monomial complete intersections, Artinian algebras in two variables and Artinian level monomial algebras with type two in both three and four variables have unimodal Hilbert functions. This paper proves that Artinian level monomial algebras of type three in three variables have unimodal Hilbert functions. We will also discuss the licciness of these algebras.

Continuous Dependence Of Solutions Of Equations On Parameters, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

It is shown under very general conditions that the solutions of equations depend continuously on the coefficients or parameters of the equations. The standard examples are solutions of monic polynomial equations and the eigenvalues of a matrix. However, the proof methods apply to any finite map T : Cⁿ -> Cⁿ.

Review: Crystal Bases Of Q-Deformed Kac Modules Over The Quantum Superalgebras Uq(Gl(Mln)), Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Kazhdan-Lusztig Cells In Planar Hyperbolic Coxeter Groups And Automata, Mikhail V. Belolipetsky, Paul E. Gunnells, Richard A. Scott

Paul Gunnells

Let C be a one- or two-sided Kazhdan–Lusztig cell in a Coxeter group (W, S), and let Red(C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Red(C) is regular. In this paper, we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.

On Sign-Solvable Linear Systems And Their Applications In Economics, Eric Hanson

Journal of Undergraduate Research at Minnesota State University, Mankato

Sign-solvable linear systems are part of a branch of mathematics called qualitative matrix theory. Qualitative matrix theory is a development of matrix theory based on the sign (¡; 0; +) of the entries of a matrix. Sign-solvable linear systems are useful in analyzing situations in which quantitative data is unknown or had to measure, but qualitative information is known. These situations arise frequently in a variety of disciplines outside of mathematics, including economics and biology. The applications of sign-solvable linear systems in economics are documented and the development of new examples is formalized mathematically. Additionally, recent mathematical developments about sign-solvable …

Calculation Of The Killing Form Of A Simple Lie Group, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

The Killing form of a simple Lie Algebra is determined from invariants of the extended root diagrams of the Lie algebra.

Polynomial Identities On Algebras With Actions, Chris Plyley

Electronic Thesis and Dissertation Repository

When an algebra is endowed with the additional structure of an action or a grading, one can often make striking conclusions about the algebra based on the properties of the structure-induced subspaces. For example, if A is an associative G-graded algebra such that the homogeneous component A₁ satisfies an identity of degree d, then Bergen and Cohen showed that A is itself a PI-algebra. Bahturin, Giambruno and Riley later used combinatorial methods to show that the degree of the identity satisfied by A is bounded above by a function of d and |G|. Utilizing a …

Algebraic Properties Of Ext-Modules Over Complete Intersections, Jason Hardin

Department of Mathematics: Dissertations, Theses, and Student Research

We investigate two algebraic properties of Ext-modules over a complete intersection R of codimension c. Given an R-module M, Ext(M,k) can be viewed as a graded module over a polynomial ring in c variables with an action given by the Eisenbud operators. We provide an upper bound on the degrees of the generators of this graded module in terms of the regularities of two associated coherent sheaves. In the codimension two case, our bound recovers a bound of Avramov and Buchweitz in terms of the Betti numbers of M. We also provide a description of the differential graded (DG) R-module …

Ghost Number Of Group Algebras, Gaohong Wang

Electronic Thesis and Dissertation Repository

The generating hypothesis for the stable module category of a finite group is the statement that if a map in the thick subcategory generated by the trivial representation induces the zero map in Tate cohomology, then it is stably trivial. It is known that the generating hypothesis fails for most groups. Generalizing work done for p-groups, we define the ghost number of a group algebra, which is a natural number that measures the degree to which the generating hypothesis fails. We describe a close relationship between ghost numbers and Auslander-Reiten triangles, with many results stated for a general projective class …

Randomized Detection Of Extraneous Factors, Manfred Minimair

Manfred Minimair

A projection operator of a system of parametric polynomials is a polynomial in the coefficients of the system that vanishes if the system has a common root. The projection operator is a multiple of the resultant of the system, and the factors of the projection operator that are not contained in the resultant are called extraneous factors. The main contribution of this work is to provide a randomized algorithm to check whether a factor is extraneous, which is an important task in applications. A lower bound for the success probability is determined which can be set arbitrarily close to one. …

Isotopic Form Of M-Rings, M. R. Molaei, A. Keyhaninejad

Applications and Applied Mathematics: An International Journal (AAM)

The aim of this work is to generalize the notion of isofields by presenting the notion of Misorings. A method for constructing new M-isorings is presented. It is proved that an M-isoring for which its isounit is the fixed point of its identity function is an M-ring. Two methods for constructing new M-rings are presented.

Homormophic Images And Their Isomorphism Types, Diana Herrera

Electronic Theses, Projects, and Dissertations

In this thesis we have presented original homomorphic images of permutations and monomial progenitors. In some cases we have used the double coset enumeration tech- nique to construct the images and for all of the homomorphic images that we have discovered, the isomorphism type of each group is given. The homomorphic images discovered include Linear groups, Alternating groups, and two sporadic simple groups J₁ and J₂X2 where J₁ is the smallest Janko group and J₂ is the second Janko sporadic group.

Monoid Rings And Strongly Two-Generated Ideals, Brittney M. Salt

Electronic Theses, Projects, and Dissertations

This paper determines whether monoid rings with the two-generator property have the strong two-generator property. Dedekind domains have both the two-generator and strong two-generator properties. How common is this? Two cases are considered here: the zero-dimensional case and the one-dimensional case for monoid rings. Each case is looked at to determine if monoid rings that are not PIRs but are two-generated have the strong two-generator property. Full results are given in the zero-dimensional case, however only partial results have been found for the one-dimensional case.

The Tame-Wild Principle For Discriminant Relations For Number Fields, John W. Jones, David P. Roberts

Mathematics Publications

Consider tuples ( K₁ , … , K_r ) of separable algebras over a common local or global number field F₁, with the K_i related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants of K_i ∕ F . We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.

Permutation Groups And Puzzle Tile Configurations Of Instant Insanity Ii, Amanda N. Justus

Electronic Theses and Dissertations

The manufacturer claims that there is only one solution to the puzzle Instant Insanity II. However, a recent paper shows that there are two solutions. Our goal is to find ways in which we only have one solution. We examine the permutation groups of the puzzle and use modern algebra to attempt to fix the puzzle. First, we find the permutation group for the case when there is only one empty slot at the top. We then examine the scenario when we add an extra column or an extra row to make the game a 4 × 5 puzzle or …

Convexity Properties Of The Diestel-Leader Group Γ_3(2), Peter J. Davids

Honors Projects

The Diestel-Leader groups are a family of groups first introduced in 2001 by Diestel and Leader in [7]. In this paper, we demonstrate that the Diestel-Leader group Γ₃(2) is not almost convex with respect to a particular generating set S. Almost convexity is a geometric property that has been shown by Cannon [3] to guarantee a solvable word problem (that is, in any almost convex group there is a finite-step algorithm to determine if two strings of generators, or “words”, represent the same group element). Our proof relies on the word length formula given by Stein and Taback …

Calculator Usage In Secondary Level Classrooms: The Ongoing Debate, Nicole Plummer

Honors College Theses

With technology becoming more prevalent every day, it is imperative that students gain enough experience with different technological tools in order to be successful in the “real-world”. This thesis will discuss the debate and overall support for an increased usage of calculators as tools in the secondary level classroom. When the idea of calculators in the classroom first came to life, many educators were very apprehensive and quite hesitant of this change. Unfortunately, more than 40 years later, there is still hesitation for their usage; and rightfully so. While there are plenty of advantages of calculator use in the classroom, …

Polynomial Factoring Algorithms And Their Computational Complexity, Nicholas Cavanna

Honors Scholar Theses

Finite fields, and the polynomial rings over them, have many neat algebraic properties and identities that are very convenient to work with. In this paper we will start by exploring said properties with the goal in mind of being able to use said properties to efficiently irreducibly factorize polynomials over these fields, an important action in the fields of discrete mathematics and computer science. Necessarily, we must also introduce the concept of an algorithm’s speed as well as particularly speeds of basic modular and integral arithmetic opera- tions. Outlining these concepts will have laid the groundwork for us to introduce …

Properties Of Ideal-Based Zero-Divisor Graphs Of Commutative Rings, Jesse Gerald Smith Jr.

Doctoral Dissertations

Let R be a commutative ring with nonzero identity and I an ideal of R. The focus of this research is on a generalization of the zero-divisor graph called the ideal-based zero-divisor graph for commutative rings with nonzero identity. We consider such a graph to be nontrivial when it is nonempty and distinct from the zero-divisor graph of R. We begin by classifying all rings which have nontrivial ideal-based zero-divisor graph complete on fewer than 5 vertices. We also classify when such graphs are complete on a prime number of vertices. In addition we classify all rings which …

Fan-Linear Maps And Fan Algebras, John Hull

Georgia State Undergraduate Research Conference

No abstract provided.

Construction Algorithms For Expander Graphs, Vlad S. Burca

Senior Theses and Projects

Graphs are mathematical objects that are comprised of nodes and edges that connect them. In computer science they are used to model concepts that exhibit network behaviors, such as social networks, communication paths or computer networks. In practice, it is desired that these graphs retain two main properties: sparseness and high connectivity. This is equivalent to having relatively short distances between two nodes but with an overall small number of edges. These graphs are called expander graphs and the main motivation behind studying them is the efficient network structure that they can produce due to their properties. We are specifically …

Fast Monte Carlo Algorithms For Computing A Low-Rank Approximation To A Matrix, Vlad S. Burca

Senior Theses and Projects

Many of today's applications deal with big quantities of data; from DNA analysis algorithms, to image processing and movie recommendation algorithms. Most of these systems store the data in very large matrices. In order to perform analysis on the collected data, these big matrices have to be stored in the RAM (random-access memory) of the computing system. But this is a very expensive process since RAM is a scarce computational resource. Ideally, one would like to be able to store most of the data matrices on the memory disk (hard disk drive) while loading only the necessary parts of the …

Review: The Relationships Among Multiplicities Of A J-Self-Adjoint Differential Operator's Eigenvalue, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Classification Of W-Groups Of Pythagorean Formally Real Fields, Fatemeh Bagherzadeh Golmakani

Electronic Thesis and Dissertation Repository

In this work we consider the Galois point of view in determining the structure of
a space of orderings of fields via considering small Galois quotients of absolute Galois
groups G _F of Pythagorean formally real fields. Galois theoretic, group theoretic and
combinatorial arguments are used to reduce the structure of W-groups.

The Irreducible Representations Of D2n, Melissa Soto

Electronic Theses, Projects, and Dissertations

Irreducible representations of a finite group over a field are important because all representations of a group are direct sums of irreducible representations. Maschke tells us that if φ is a representation of the finite group G of order n on the m-dimensional space V over the field K of complex numbers and if U is an invariant subspace of φ, then U has a complementary reducing subspace W .

The objective of this thesis is to find all irreducible representations of the dihedral group D2n. The reason we will work with the dihedral group is because it is one …

Revisiting Fibonacci And Related Sequences, Arthur Benjamin, Jennifer Quinn

Jennifer J. Quinn

This issue focuses on proving several interesting facts about the Fibonacci Sequence using a combinatorial proof. The aim of Delving Deeper is for teachers to pose and solve novel math problems, expand on mathematical connections, or offer new insights into familiar math concepts. Delving Deeper focuses on mathematics content appealing to secondary school teachers. It provides a forum that allows classroom teachers to share their mathematics from their work with students, their classroom investigations and products, and their other experiences. Delving Deeper is a regular department of Mathematics Teacher.