Algebra Commons^™

Cyclicizers, Centralizers, And Normalizers, David Patrick, Eric Wepsic

Mathematical Sciences Technical Reports (MSTR)

Our goal is to define the cyclicizer, which is analogous to the centralizer and normalizer, and to examine groups in which these subsets have certain special properties.

Counting Centralizers In Finite Groups, Sarah Marie Belcastro, Gary J. Sherman

Mathematical Sciences Technical Reports (MSTR)

We discuss various results on the number of commuting pairs and the sizes of the centralizers of a group.

An Upper Bound For 3-Rewriteability In Finite Groups, Jordan Ellenberg

Mathematical Sciences Technical Reports (MSTR)

An ordered triple of group elements (x,y,z) is said to be rewriteable if the product xyz is equal to one of the products xzy, yxz, yzx, zxy, zyx. In the present paper, we shall ask the following question: how rewriteable can a finite group be if its derived group has order greater than 2?

Finite Abelian Groups In Which The Probability Of An Automorphism Fixing An Element Is Large, Gary J. Sherman

Mathematical Sciences Technical Reports (MSTR)

Let G be a finite group and let A be its automorphism group. We obtain various results on the probability that a random element of A fixes a random element of G.

Moore Cohomology, Principal Bundles, And Actions Of Groups On C*-Algebras, Ian Raeburn, Dana P. Williams

Dartmouth Scholarship

In recent years both topological and algebraic invariants have been associated to group actions on C*-algebras. Principal bundles have been used to describe the topological structure of the spectrum of crossed products [18, 19], and as a result we now know that crossed products of even the very nicest non-commutative algebras can be substantially more complicated than those of commutative algebras [19, 5]. The algebraic approach involves group cohomological invariants, and exploits the associated machinery to classify group actions on C*-algebras; this originated in [2], and has been particularly successful for actions of R and tori ([19; Section 4], [21]). …

Modern Algebra And Discrete Structures, R. F. Lax

E-Textbooks

This text is intended for either an applied algebra course or a modern algebra course that includes more applications than has been traditional. It is at an advanced undergraduate (junior-senior) level and is suitable for a one-semester or two-quarter course. We assume that students have already had a course in linear algebra (although we briefly review concepts from linear algebra when they are needed in the sections on fields and linear codes).

Our treatment is fairly rigorous, with almost every proof supplied. However, we have tried to concentrate on examples and applications. In an applied algebra course, which usually consists …

Analytic Continuation In Bergman Spaces And The Compression Of Certain Toeplitz Operators, William T. Ross

Department of Math & Statistics Faculty Publications

Let G be a Jordan domain and K C G be relatively closed with Area(K) = 0. Let A² (G\K) and A²(G) be the Bergman spaces on G\K, respectively G and define N = A²(G\K) Ɵ A² (G). In this paper we show that with a mild restriction on K, every function in N has an analytic continuation across the analytic arcs of aG that do not intersect K. This result will be used to discuss the Fredholm theory of the operator C_f = P_NT_f│N, where f ϵ C(G) …

Some Remarks On Reproducing Kernel Krein Spaces, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

The one-to-one correspondence between positive functions and reproducing kernel Hilbert spaces was extended by L. Schwartz to a (onto, but not one-to-one) correspondence between difference of positive functions and reproducing kernel Krein spaces. After discussing this result, we prove that matrix value function K(z,ω) symmetric and jointly analytic in z and ω in a neighborhood of the origin is the reproducing kernel of a reproducing kernel Krein space. We conclude with an example showing that such a function can be the reproducing kernel of two different Krein spaces.

Dihedral Rewriteability, Cheryl P. Grood

Mathematical Sciences Technical Reports (MSTR)

In this paper we compute the probability that an n-tuple for a group G is S-rewritable for a given set S of permutations for several classes of groups.

Some Facts About Cwat-Sets, Martin Wattenburg

Mathematical Sciences Technical Reports (MSTR)

In [1] and [2], Sherman and Atkins, in connection with a problem in statistics, introduced a generalization of the concept of a subgroup of Z₂. These generalized subgroups, which we call CWAT-sets (where "CWAT" is an acronym for "Closed With A Twist" ), have a rich algebraic structure. In this paper we establish some simple combinatorial facts about CWAT-sets, as well as provide two construction methods, prove a divisibility theorem, and make a classification conjecture.

A Numerical Approach To Rewriteability In Finite Groups, J.L. Leavitt, G.J. Sherman, M.E. Walker

Mathematical Sciences Technical Reports (MSTR)

In this paper we compute the probability that an n-tuple for a group G is S-rewritable for a given set S of permutations for several classes of groups.

Maximal Order Three-Rewriteable Subgroups Of Symmetric Groups, John T. O'Bryan

Mathematical Sciences Technical Reports (MSTR)

Recently, Burns and Goldsmith [2] characterized the maximal order Abelian subgroups of the symmetric groups using elementary techniques and the results of Hoffman [5]. This classification could also be directly inferred from the results of Kovacs and Praeger [7]. A natural extension would be to consider the weaker, more general form of commutativity, three-rewriteability. The purpose of this paper is to completely characterize the maximal order three-rewriteable subgroups of the symmetric groups.

How Hamiltonian Can A Finite Group Be?, G.J. Sherman, T.J. Tucker, M.E. Walker

Mathematical Sciences Technical Reports (MSTR)

No abstract provided.

Fibonacci Sequences In Finite Groups, Steven W. Knox

Mathematical Sciences Technical Reports (MSTR)

This paper extend the notion of Fibonacci sequence mod m to Fibonacci sequences in finite groups.

Sets Of Typical Subsamples, Joel Atkins, G.J Sherman

Mathematical Sciences Technical Reports (MSTR)

A group theoretic condition on a set of subsamples of a random sample from a continuous random variable symmetric about 0 is shown to be sufficient to provide typical values for 0.

An Artificial Neural Approach To The Decomposition Problem, Chandrashekar L. Masti

Electrical & Computer Engineering Theses & Dissertations

The goal of this thesis is to develop an artificial neural approach toward addressing the intractability involved with the decomposition problem. The search for the lattice of substitution property (s. p.) partitions essential to decompositions is cast into the framework of constraint satisfaction. An artificial neural network is developed to provide solutions by performing optimization of a mathematically derived objective function over the problem space. The issue of transitivity is verified to belong to a class of problems beyond the scope of solvability for conventional quadratic-order constraint satisfaction neural networks. A theorem is stated and proved establishing that third-order correlations …

Mat 751 Algebraic Topology I - Fall '89, David Handel

Mathematics Faculty Research Publications

A collection of notes for the course Mat 751, Algebraic Topology I, prepared by Professor David Handel of the Wayne State University Mathematics Department. The notes include examples, exercises, and additional lecture notes on related concepts.

Dilatations Des Commutants D'Opérateurs Pour Des Espaces De Krein De Fonctions Analytiques, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

Let K1 and K2 be two Krein spaces of functions analytic in the unit disk and invariant for the left shift operator R0(R0f(z)=(f(z)−f(0))/z), and let A be a linear continuous operator from K1 into K2 whose adjoint commutes with R0. We study dilations of A which preserve this commuting property and such that the Hermitian forms defined by I−AA∗ and I−BB∗ have the same number of negative squares. We thus obtain a version of the commutant lifting theorem in the framework of Krein spaces of analytic functions. To prove this result we suppose that the graph of the operator A∗, …

The Space Of Minimal Prime Ideals Of C(X) Need Not Be Basically Disconnected, Alan Dow, Melvin Henriksen, Ralph Kopperman, J. Vermeer

All HMC Faculty Publications and Research

Problems posed twenty and twenty-five years ago by M. Henriksen and M. Jerison are solved by showing that the space of minimal prime ideals of the ring C(X) of continuous real-valued functions on a compact (Hausdorff) space need not be basically disconnected-or even an F-space.

Locally Finite Families, Completely Separated Sets And Remote Points, Melvin Henriksen, Thomas J. Peters

All HMC Faculty Publications and Research

It is shown that if X is a nonpseudocompact space with a σ-locally finite π-base, then X has remote points. Within the class of spaces possessing a σ-locally finite π-base, this result extends the work of Chae and Smith, because their work utilized normality to achieve complete separation. It provides spaces which have remote points, where the spaces do not satisfy the conditions required in the previous works by Dow, by van Douwen, by van Mill, or by Peters.

The lemma: "Let X be a space and let {Cε: € < α} be a locally finite family of cozero sets of X. Let {Zε: € < α } be a family of zero sets of X such that for each € < α, Zε с Cε. Then ∪_ε<α Zε is completely separated from X/∪εCε", is a fundamental …

The Persistence Of Universal Formulae In Free Algebras, Anthony M. Gaglione, Dennis Spellman

Mathematics Faculty Publications

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements ...

Reproducing Kernel Krein Spaces Of Analytic Functions And Inverse Scattering, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

The purpose of this thesis is to study certain reproducing kernel Krein spaces of analytic functions, the relationships between these spaces and an inverse scattering problem associated with matrix valued functions of bounded type, and an operator model.

Roughly speaking, these results correspond to a generalization of earlier investigations on the applications of de Branges' theory of reproducing kernel Hilbert spaces of analytic functions to the inverse scattering problem for a matrix valued function of the Schur class.

The present work considers first a generalization of a portion of de Branges' theory to Krein spaces. We then formulate a general …

Commutator Identities Obtained By The Magnus Algebra, Anthony M. Gaglione, Dennis Spellman

Mathematics Faculty Publications

In this paper, two related commutator identities are established through the use of the Magnus Algebra (the algebra of noncommutative formal power series with integral coefficients).

Math 752 Algebraic Topology Ii - Winter '84, David Handel

Mathematics Faculty Research Publications

A collection of notes for the course MAT 752, Algebraic Topology II, prepared by Professor David Handel of the Wayne State University Mathematics Department. This course builds on MAT 751, Algebraic Topology I, and the notes include examples, exercises, and suggestions for further reading.

Some Properties Of Positive Derivations On F-Rings, Melvin Henriksen, Frank A. Smith

All HMC Faculty Publications and Research

Throughout A denotes an f-ring; that is, a lattice-ordered ring that is a subdirect union of totally ordered rings. We let D(A) denote the set of derivations D: A --> A such that a ≥ 0 implies Da ≥ 0, and we call such derivations positive. In [CDK], P. Coleville, G. Davis, and K. Keimel initiated a study of positive derivations on f-rings. Their main results are (i) D ε D(A) and A archimedean imply D = 0, and (ii) if A has an identity element 1 and a is the supremum of a set …

The Classification Of Extensions Of C*-Algebras, Jonathan Rosenberg, Claude Schochet

Mathematics Faculty Research Publications

Our objective in this note is to outline a number of results concerning the Kasparov groups Ext(A,B) which are analogous to known information about the Brown, Douglass, and Fillmore (BDF) theory regarding groups which classify extensions of the form 0 → B ⊗ K → E → A → 0. These results enable one to compute the groups with relatively mild restrictions on the C*-algebras A and B. This in turn should make it possible to analyze the way in which a wide variety of C*-algebra extensions are put together, at least stably.

A Note On The Degree Of Approximation With An Optimal, Discrete Polynomial, J. J. Swetits, B. Wood

Mathematics & Statistics Faculty Publications

A saturation theorem and an asymptotic theorem are proved for an optimal, discrete, positive algebraic polynomial operator. The operator is based on the Gauss-Legendre quadrature formula.

A Summary Of Results On Order-Cauchy Completions Of Rings And Vector Lattices Of Continuous Functions, Melvin Henriksen

All HMC Faculty Publications and Research

This paper is a summary of joint research by F. Dashiell, A. Hager and the present author. Proofs are largely omitted. A complete version will appear in the Canadian Journal of Mathematics. It is devoted to a study of sequential order-Cauchy convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified for compact spaces X in …

Approximation By Discrete Operators, J. J. Swetits, B. Wood

Mathematics & Statistics Faculty Publications

A discrete, positive, weighted algebraic polynomial operator which is based on Gaussian quadrature is constructed. The operator is shown to satisfy the Jackson estimate and an optimal version is obtained.

An Algebraic Characterization Of The Freudenthal Compactification For A Class Of Rimcompact Spaces, Melvin Henriksen

All HMC Faculty Publications and Research

Throughout C(X) will denote the ring of all continuous real-valued functions on a Tychonoff space X, and C*(X) will denote the subring of bounded elements of C(X). The real line is denoted by R, and N denotes the (discrete) subspace of positive integers. A subset S of X such that the map f → f|_s is an epimorphism of C(X) (resp. C*(X)) is said to be C-embedded (resp. C*-embedded) in X. As is well-known, every f Є C*(X) has a unique continuous extension βf over its Stone-Čech compactification βX [GJ, Chapter 6]. That is, X is …