Algebra Commons^™

Writing Assignments In An Abstract Algebra Course, Krystina K. Leganza

Humanistic Mathematics Network Journal

No abstract provided.

Introduction To Linear Algebra: Models, Methods, And Theory, Alan Tucker

Department of Applied Mathematics & Statistics Faculty Books

This book develops linear algebra around matrices. Vector spaces in the abstract are not considered, only vector spaces associated with matrices. This book puts problem solving and an intuitive treatment of theory first, with a proof-oriented approach intended to come in a second course, the same way that calculus is taught. The book's organization is straightforward: Chapter 1 has introductory linear models; Chapter 2 has the basics of matrix algebra; Chapter 3 develops different ways to solve a system of equations; Chapter 4 has applications, and Chapter 5 has vector-space theory associated with matrices and related topics such as pseudoinverses …

Conjugacy Classes Of Triple Products In Finite Groups, Kevin Hutson, Emily Salvo

Mathematical Sciences Technical Reports (MSTR)

For an underlying finite group G, we establish estimates on the number of triples that bind a certain set to one conjugacy class, or else breaks it into two conjugacy classes.

Metode De Calcul În Analiza Matematică, Florentin Smarandache, C. Dumitrescu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Square Roots Of Finite Groups - Ii, Matthew Devos, David Mcadams, Rebecca Rapoport

Mathematical Sciences Technical Reports (MSTR)

A subset R of a finite group G is a square root of G if R² = G. If R is a square root of G for which |R|² = G, then R is referred to as a perfect square root of G. It can be shown using character theory that perfect square roots do not exist. The purpose of this paper is to work toward an elementary proof of this result.

Lattice-Ordered Algebras That Are Subdirect Products Of Valuation Domains, Melvin Henriksen, Suzanne Larson, Jorge Martinez, R. G. Woods

All HMC Faculty Publications and Research

An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if A/P is a valuation domain for every prime ideal P of A. If M is a maximal ℓ-ideal of A , then the rank of A at M is the number of minimal prime ideals of A contained in M, rank of A is the sup of the ranks of A at each of its maximal ℓ-ideals. If the latter is a positive integer, then A is said to have finite rank, and if A …

Square Roots Of Finite Groups, Kashi Abhyankar, Daniel Grossman

Mathematical Sciences Technical Reports (MSTR)

Let G be a finite group of order n². A perfect square root of G is a subset X of G such that |X| = n and X² = G. Neither generalized dihedral groups nor groups of nilpotency class two have perfect square roots.

Finite Groups Can Be Arbitrarily Hamiltonian, Stephen Ahearn, Mark Huber

Mathematical Sciences Technical Reports (MSTR)

Let r be a rational in (0,1]. There exists a finite group G which is the direct product of at most four metacyclic groups and whose proportion of normal subgroups is r. An analogous result holds for three other measures of "Hamiltonianess".

Counting Order Classes Of Triple Products In Finite Groups, Scott Annin, Jennifer Ziebarth

Mathematical Sciences Technical Reports (MSTR)

Let G be a finite group and let W_G denote the proportion of triples, (.x:, y , z) , i n G3 for which x yz , x zy, y x z, zx y , yzx , and z y x have the same order. The following results are established.

i) G is abelian if, and only if, W_G = 1.

ii) W_G can be arbitrarily close to 1.

iii) Additional estimates on W_G

Hyperinvariant Subspaces Of The Harmonic Dirichlet Space, William T. Ross, Stefan Richter, Carl Sundberg

Department of Math & Statistics Faculty Publications

No abstract provided.

Invariant Subspaces Of Bergman Spaces On Slit Domains, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper, we characterize the z-invariant subspaces that lie between the Bergman spaces A^p(G) and A^p(G\K), where 1 < p < ∞, G is a bounded region in C, and K is a closed subset of a simple, compact, C¹ arc.

Analytic Besov Spaces And Invariant Subspaces Of Bergman Spaces, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper, we examine the invariant subspaces (under the operator f -->z f) M of the Bergman space ^p_a (G\T) (where 1 < p < 2, G is a bounded region in C containing D, T is the unit circle, and D is the unit disk) which contain the characteristic functions x_D and x_G, i.e. the constant functions on the components of G\T. We will show that such M are in one-to-one correspondence with the invariant subspaces of the analytic Besov space AB_q (q is the conjugate index to p) and …

Bounds On Squares Of Two-Sets, Dan Slilaty, Jeff Vanderkam

Mathematical Sciences Technical Reports (MSTR)

For a finite group G, let p_i(G) denote the proportion of (x,y) in GxG for which the set {x²,xy,yx,y²} has cardinality i. In this paper we develop estimates on the p_i(G) for various i.

Hypergraph Representations And Orders Of Cwatsets, Julie Kerr

Mathematical Sciences Technical Reports (MSTR)

We determine upper bounds on the order of cwatsets of odd order.

When Is The Number Of P-Subgroups Of A Group Satisfying A Property Congruent To 1 (Mod P)?, Jason Fulman, Jeff Vanderkam

Mathematical Sciences Technical Reports (MSTR)

Let T be a property which holds for a group independent of whether or not this group is embedded in a group G or in a p-Sylow subgroup of G. Using a generalization of Sylow's second Theorem, we prove that if for any p-group P the number of subgroups of P satisfying T is congruent to 1 (mod p), then for any group G, the number of p-subgroups satisfying T is also congruent to 1 (mod p). As an application, we give simple proofs of several theorems, including the well-known Frobenius theorem.

The Kaplansky Test Problems - An Approach Via Radicals, R. Gobel, Brendan Goldsmith

Articles

The existence of non-free, K-free Abelian groups and modules (over some non-left perfect rings R) having prescribed endomorphism algebra is established within ZFC + 0 set theory. The principal technique used exploits free resolutions of non-free R-modules X and is similar to that used previously by Griffith and Eklof; much stronger results than have been obtained heretofore are obtained by coding additional information into the module X. As a consequence we can show, inter alia, that the Kaplansky Test Problems have negative answers for strongly K,-free Abelian groups of cardinality K1 in ZFC and assuming the weak Continuum Hypothesis.

The Commutant Of A Certain Compression, William T. Ross

Department of Math & Statistics Faculty Publications

Let G be any bounded region in the complex plane and K Ϲ G be a simple compact arc of class C¹. Let A²(G\K) (resp. A²(G)) be the Bergman space on G\K (resp. G). Let S be the operator multiplication by z on A²(G\K) and C = P_N S│_N be the compression of S to the semi-invariant subspace N = A²(G\K) Ɵ A²(G). We show that the commutant of C* is the set of all operators …

The Sharp Lipschitz-Constants For Feasible And Optimal-Solutions Of A Perturbed Linear Program, Wu Li

Mathematics & Statistics Faculty Publications

The purpose of this paper is to derive the sharp Lipschitz constants for the feasible solutions and optimal solutions of a linear program with respect to right-hand-side perturbations. The Lipschitz constants are given in terms of pseudoinverses of submatrices of the matrices involved and are proven to be sharp.

Divisibility By |G| For Powers Of Ordered K-Sets, Jeffery Vanderkam

Mathematical Sciences Technical Reports (MSTR)

It is shown that the number of ordered k-sets of a group G whose nth power contains exactly i elements is always a multiple of IGI. An elementary proof of the fact that the number of ordered pairs ( x , y ) such that x² = y² is equal to k_r lGI is also given.

Cubing Ordered 2-Sets, Jeffery Vanderkam

Mathematical Sciences Technical Reports (MSTR)

Given a group G, we define p_i as the probability that, given an ordered pair X = (x,y), there are exactly i elements in X³ = {x₁x₂x₃ l x_i in X}. We show that P₂( G) = 0 if, and only if, IGI is odd, and that p₃(G) = 0 if, and only if, IGI is not divisible by three. The groups for which p₄ ( G) = 0 and p₅ ( G) = 0 are also determined.

Counting Nilpotent Pairs, Jason Fulman, Michael Galloy, Jeffery Vanderkam

Mathematical Sciences Technical Reports (MSTR)

In this paper , we consider the probability that two elements chosen at random from a finite group G generate a subgroup of a given nilpotency class. It is shown that in solvable non­-nilpotent groups, the probability that two elements generate a nilpotent subgroup is <= l/p,, where p, is the smallest prime dividing the order of the group, and it is also shown that there exist groups such that the probability of two elements generating a subgroup of class i approaches one (and other groups for which it approaches zero) for all i =>2. It is also shown …

The 2/3 Bound For Rewritable N-Tuples, Lawren Smithline, Catherine Sugar

Mathematical Sciences Technical Reports (MSTR)

We study the number of rewritings of an n-tuple in a finite a group, for certain classes of groups.

Some Upper Bounds For Commutativity And Cyclicity Measures In Finite Groups, David Patrick, Catherine Sugar, Eric Wepsic

Mathematical Sciences Technical Reports (MSTR)

The number of commuting and cyclic n-tuples in a group are enumerated for certain classes of groups.

Counting Nilpotent Pairs In Finite Groups: Some Conjectures, H Dubose-Schmidt, Michael D. Galloy, D,L, Wilson

Mathematical Sciences Technical Reports (MSTR)

The number of nilpotent pairs is determined for a number of small groups.

Some Facts About Cycels And Tidy Groups, Kevin O'Bryant, D. Patrick, Lawren Smithline, Eric Wepsic

Mathematical Sciences Technical Reports (MSTR)

No abstract provided.

A4 Rewriteability, Eric Wepsic, Kevin O'Bryant, Lawren Smithline

Mathematical Sciences Technical Reports (MSTR)

The concept of 4-rewriteability with permutation coming form the alternating group A₄ is explored.

Rewriteability, Commutators, And Fundamental N-Rewritings, Lawren Smithline

Mathematical Sciences Technical Reports (MSTR)

We consider the relationship between commutators and rewriteability.

Splitting Of The Identity Component In Locally Compact Abelian Groups, Peter Loth

Mathematics Faculty Publications

In this paper we are concerned with the splitting of the identity component G₀ in an LCA group G. As Pontrjagin duality shows, this splitting is to the splitting of the torsion part tA in a discrete abelian group A, if G is assumed to be compact.

A Theorem On Reproducing Kernel Hilbert Spaces Of Pairs, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we study reproducing kernel Hilbert and Banach spaces of pairs. These are a generalization of reproducing kernel Krein spaces and, roughly speaking, consist of pairs of Hilbert (or Banach) spaces of functions in duality with respect to a sesquilinear form and admitting a left and right reproducing kernel. We first investigate some properties of these spaces of pairs. It is then proved that to every function K(z, ω) analytic in z and ω* there is a neighborhood of the origin that can be associated with a reproducing kernel Hilbert space of pairs with left reproducing kernel K(z, …

More Upper Bounds On The 3-Rewriteability Of Non-3-Rewriteable Groups, Eric Wepsic

Mathematical Sciences Technical Reports (MSTR)

We find an upper bound on the probability that a randomly selected triple in a group is 3-rewriteable, and a bound for the core set rewriteability.