Mathematics Commons^™

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 …

An Invariant Subspace Problem For P = 1 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 A¹(G) and A¹(G/K), where G is a bounded region in the complex plane and K is a compact subset of a simple arc of class C¹.

New Constructions Of Menon Difference Sets, K. T. Arasu, James A. Davis, Jonathan Jedwab, Surinder K. Sehgal

Department of Math & Statistics Faculty Publications

Menon difference sets have parameters (4N², 2N² − N, N² − N). These have been constructed for N = 2^a3^b, 0 ⩽ a,b, but the only known constructions in abelian groups require that the Sylow 3-subgroup be elementary abelian (there are some nonabelian examples). This paper provides a construction of difference sets in higher exponent groups, and this provides new examples of perfect binary arrays.

A Note On New Semi-Regular Divisible Difference Sets, James A. Davis, Jonathan Jedwab

Department of Math & Statistics Faculty Publications

We give a construction for new families of semi-regular divisible difference sets. The construction is a variation of McFarland's scheme [5] tor noncyclic difference sets.

Rely To "Comment On 'Nonexistence Of Certain Perfect Binary Arrays' And 'Nonexistence Of Perfect Binary Arrays'", Jonathan Jedwab, James A. Davis

Department of Math & Statistics Faculty Publications

Yang's comment [C] is based on a lemma which claims to construct an s₀ x s₁ x s₂ x ... x s, perfect binary array (PBA) from an s₀s₁ x s₂ x ... x s_r PBA.

Symmetric Designs, Difference Sets, And A New Way To Look At Macfarland Difference Sets, John Bowen Polhill Jr.

Honors Theses

In this paper, the topics of symmetric designs and difference sets are discussed both separately and in relation to each other. Then an approach to MacFarland Difference Sets using the theory behind homomorphisms from groups into the complex numbers is introduced. This method is contrasted with the method of finding this type of difference set used by E.S. Launder in his book Symmetric Designs: An Algebraic Approach.

Topics In Cyclotomic And Quadratic Fields, Pam Mellinger

Honors Theses

This paper introduces cyclotomic and quadratic fields and explores some of their properties and applications to problems in number theory. After some preliminary definitions and theorems, the paper looks at the relationship between quadratic and cyclotomic fields, at Kummer's lemma on units in the pth cyclotomic field, and at the Quadratic Reciprocity Law and its applications to diophantine equations.

Nonexistence Of Certain Perfect Binary Arrays, Jonathan Jedwab, James A. Davis

Department of Math & Statistics Faculty Publications

A perfect binary array (PBA) is an r-dimensional matrix with elements ±I such that all out-of-phase periodic autocorrelation coefficients are zero. The two smallest sizes for which the existence of a PBA is undecided, 2 x 2 x 3 x 3 x 9 and 4 x 3 x 3 x 9, are ruled out using computer search and a combinatorial argument.

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 …

A Summary Of Menon Difference Sets, James A. Davis, Jonathan Jedwab

Department of Math & Statistics Faculty Publications

A (v, k, λ) difference set is a k-element subset D of a group G of order v for which the multiset {d₁d₂^-1 : d_1,d₂ ∈ D, d₁ ≠ d₂} contains each nonidentity element of G exactly λ times. A difference set is called abelian, nonabelian or cyclic if the underlying group is. Difference sets a.re important in design theory because they a.re equivalent to symmetric (v, k, λ) designs with a regular automorphism group. Abelian difference sets arise naturally in …

Weak-Star Limits On Polynomials And Their Derivatives, William T. Ross, Joseph A. Ball

Department of Math & Statistics Faculty Publications

Let μ and v be regular finite Borel measures with compact support in the real line ℝ and define the differential operator D :L ∞(μ) → L ∞(v) with domain equal to the polynomials P by Dp = p′. In this paper we will characterize the weak-star closure of the graph of D in ∞(μ) ⊕ ∞(y). As a consequence we will characterize when D is closable (i.e. the weak-star closure of G contains no non-zero elements of the form o ⊕ g) and when g is weak-star dense in L∞(μ) ⊕ …

Computational Problems With Binomial Failure Rate Model And Incomplete Common Cause Failure Reliability Data, Paul H. Kvam

Department of Math & Statistics Faculty Publications

In estimating the reliability of a system of components, it is ordinarily assumed that the component lifetimes are independently distributed. This assumption usually alleviates the difficulty of analyzing complex systems, but it is seldom true that the failure of one component in an interactive system has no effect on the lifetimes of the other components. Often, two or more components will fail simultaneously due to a common cause event. Such an incident is called a common cause failure (CCF), and is now recognized as an important contribution to system failure in various applications of reliability. We examine current methods for …

Almost Difference Sets And Reversible Divisible Difference Sets, James A. Davis

Department of Math & Statistics Faculty Publications

Let G be a group of order mn and N a subgroup of G of order n. If D is a k-subset of G, then D is called a (m, n, k, λ₁, λ₂) divisible difference set (DDS) provided that the differences dd^'-1 for d, d' ∈ D, d ≠ d' contain every nonidentity element of N exactly λ₁ times and every element of G - N exactly λ₂ times. Difference sets are used to generate designs, as described by [4] and [9]. D will be …

Construction Of Relative Difference Sets In P-Groups, James A. Davis

Department of Math & Statistics Faculty Publications

Jungnickel (1982) and Elliot and Butson (1966) have shown that (p^j+1,p,p^j+1,p^j) relative difference sets exist in the elementary abelian p-group case (p an odd prime) and many 2-groups for the case p = 2. This paper provides two new constructions of relative difference sets with these parameters; the first handles any p-group (including non-abelian) with a special subgroup if j is odd, and any 2-group with that subgroup if j is even. The second construction shows that if j is odd, every abelian group …

A Generalization Of Kraemer's Result On Difference Sets, James A. Davis

Department of Math & Statistics Faculty Publications

Kraemer has shown that every abelian group of order 2^{2d+ 2} with exponent less than 2^{2d+ 3} has a difference set. Generalizing this result, we show that any nonabelian group with a central subgroup of size 2^{d+ 1} together with an exponent-like condition will have a difference set.

An Exponent Bound For Relative Difference Sets In P-Groups, James A. Davis

Department of Math & Statistics Faculty Publications

An exponent bound is presented for abelian (p^i+j, pⁱ, p^i+j, pⁱ) relative difference sets: this bound can be met for i≤j.

[Introduction To] The Vax Book: An Introduction, John R. Hubbard

Bookshelf

This book is an expansion of the book, A Gentle Introduction to the Vax System. The purpose of the book is to guide the novice, step-by-step, through the initial stages of learning to use the Digital Equipment Corporation's Vax computers, running under the VMS operating system (Version 5.0 or later). As a tutorial for beginners, this book assumes no previous experience with computers.

Difference Sets In Abelian 2-Groups, James A. Davis

Department of Math & Statistics Faculty Publications

Examples of difference sets are given for large classes of abelian groups of order 2^{2d + 2}. This fills in the gap of knowledge between Turyn's exponent condition and Dillon's rank condition. Specifically, it is shown thatℤ/(2^d)×ℤ/(2^d+2) andℤ/(2^d+1)×Z/(2^d+1) both admit difference sets, and these have many implications.

A Result On Dillon's Conjecture In Difference Sets, James A. Davis

Department of Math & Statistics Faculty Publications

Dillon has conjectured that any group of order 2^2d+2 with a normal subgroup isomorphic to Z₂^d+1 will have a difference set. He was able to show that this is true if the subgroup is central: this paper extends that idea to noncentral subgroups.

Some Non-Existence Results On Divisible Difference Sets, K. T. Arasu, James A. Davis, Dieter Jungnickel, Alexander Pott

Department of Math & Statistics Faculty Publications

In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches:

(i) character sum arguments similar to the work of Turyn [25] for ordinary difference sets,

(ii) involution arguments, and

(iii) multipliers in conjunction with results on ordinary difference sets.

Among other results, we show that an abelian affine difference set of odd order s (s not a perfect square) in G can exist only if the Sylow 2-subgroup of G is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd …

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) …

A Note On Nonabelian (64, 28, 12) Difference Sets, James A. Davis

Department of Math & Statistics Faculty Publications

The existence of difference sets in abelian 2-groups is a recently settled problem [5]; this note extends the abelian constructs of difference sets to nonabelian groups of order 64.

A Note On Intersection Numbers Of Difference Sets, K. T. Arasu, James A. Davis, Dieter Jungnickel, Alexander Pott

Department of Math & Statistics Faculty Publications

We present a condition on the intersection numbers of difference sets which follows from a result of Jungnickel and Pott [3]. We apply this condition to rule out several putative (non-abelian) difference sets and to correct erroneous proofs of Lander [4] for the nonexistence of (352, 27, 2)- and (122, 37, 12)-difference sets.

On Uniform And Relative Distribution In The Brauer Group, Gary R. Greenfield

Department of Math & Statistics Technical Report Series

In this progress/technical report our objective is twofold. First, to formalize and expand upon remarks appearing in [7] concerning the relativization of the fundamental identity in the setting of the Brauer group of a ring, and second to exhibit a construction which shows how to interpret uniform distribution as a homological phenomenon.

Partially Confluent Maps And N-Ods, Van C. Nall

Department of Math & Statistics Faculty Publications

Let f : X-->Y be a map between topological spaces. A Wf-set in Y is a continuum in Y which is the image under f of a continuum in X. The map f is partially confluent if each continuum in Y is the union of a finite number of Wf-sets, and n-partially confluent if each continuum in Y is the union of n Wf-sets. In this paper, it is shown that every partially confluent map onto an n-cell is weakly confluent. Also, the relationship between partially confluent maps and continua which …

Maps Which Preserve Graphs, Van C. Nall

Department of Math & Statistics Faculty Publications

In 1976 Eberhart, Fúgate, and Gordh proved that the weakly confluent image of a graph is a graph. A much weaker condition on the map is introduced called partial confluence, and it is shown that the image of a graph is a graph if and only if the map is partially confluent.

In addition, it is shown that certain properties of one-dimensional continua are preserved by partially confluent maps, generalizing theorems of Cook and Lelek, Tymchatyn and Lelek, and Grace and Vought. Also, some continua in addition to graphs are shown to be the images of partially confluent maps only.

[Introduction To] A Gentle Introduction To The Vax System, John R. Hubbard

Bookshelf

This book was written originally for students enrolled in computer science courses at the University of Richmond. Very few had worked on a large time-sharing system like the VAX.

The purpose of this book is to help the novice become comfortable using any of the Digital Equipment Corporations VAX computers, from the Micro-VAX to the powerful VAX 8000 system. The book is meant to be used as a tutorial.

Weak Confluence And W-Sets, Van C. Nall

Department of Math & Statistics Faculty Publications

A mapping between continua is weakly confluent if for each subcontinuum K of the range some component of the preimage of K maps onto K. Class [W] is the class of all continua which are the images of weakly confluent maps only. The notion of Class [W] was introduced by Andrej Lelek in 1972. Since then it has been widely explored and some characterizations of these continua have been given. J. Grispolakis and E. D. Tymchatyn have given a characterization in terms of hyperspaces [4]. J. Davis has shown that acyclic atriodic continua are in Class [W]i therefore, atriodic tree-like …

Approximation Of Compact Homogeneous Maps, John R. Hubbard

Department of Math & Statistics Faculty Publications

Within the clasp of continuous homogeneous maps between Banach spaces, it is proved that every compact map can be uniformly approximated by finite-rank maps. This result is obtained by means of the classical metric projection on Banach spaces.

Air Force Rotc At The University Of Richmond, 1951-1957, Robert J. Dandridge

Honors Theses

In the summer of 1950, after the seriousness of the Korean conflict had become evident, a special committee appointed three years earlier by the Executive Committee of the Board of Trustees was reactivated for the purpose of considering "the feasibility of applying for a Reserve Office Training Corps unit" for the University of Richmond. This special committee, called the ROTC Committee, after thorough study, rendered a unanimous recommendation to the Board that a unit of Army ROTC be applied for. The application was completed and mailed to the Department of the Army. Approval of the application was not long in …