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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.

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.

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.