Physical Sciences and Mathematics Commons^™

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.

Simulated Annealing And Optimal Codes, Gary R. Greenfield

Department of Math & Statistics Technical Report Series

Following standard notation, an (n, m, d) code C denotes a binary code C which has length n, size m, and Hamming distance d. According to Hill [6] the “main coding theory problem” is to optimize one of these three parameters when the other two are held fixed. The usual version of this optimization problem is to find the largest code for a given length and given minimum distance. This is the problem we shall consider, thus making it clear what we mean by an “optimal code.”

Graphical Evolution Experiments In Artificial Life, Gary R. Greenfield

Department of Math & Statistics Technical Report Series

Larry Yaeger's alife simulation running on a Silicon Graphics Iris Workstation is called Poly World. Our description of PolyWorld is based on notes taken during an oral presentation and video demonstration given in the Artificial Life Panel Session of SIGGRAPH '92: In PolyWorld the visual organisms roam on a bounded two dimensional grid. The organisms "brains" are small neural nets enabling the organisms to control their external visual appearance and to perceive the external world by processing pixmaps. The simulation controls for total energy while striving to explore competition and self-organization. Genes present are for size, strength, maximum speed, mutation …

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 …

On The Construction Of A Potential From Cauchy Data, Lester Caudill, Bruce D. Lowe

Department of Math & Statistics Faculty Publications

We investigate the problem of recovering a potential q(y)in the differential equation:

-∆u+q(y)u = 0, (x, y) ∈ (0, 1) x (0, 1),
u(0, y) = u(1, y) = u(x, 0) = 0,
u(x, 1) = f(x), u_y(x, 1) = g(x).

The method of separation of variables reduces the recovery of q(y) to a nonstandard inverse Sturm-Liouville problem. An asymptotic formula is developed that suggests that under appropriate conditions on the Cauchy pair (f, g), q(y) is uniquely determined up to the mean. Moreover, the recovery of …

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 …

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 …

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) ⊕ …

Pairwise And Many-Body Contributions To Interaction Potentials In He(N) Clusters, Carol A. Parish, Clifford E. Dykstra

Chemistry Faculty Publications

High level ab initio calculations have been carried out to assess the pairwise additivity of potentials in the attractive or well regions of the potential surfaces of clusters of helium atoms. A large basis set was employed and calculations were done at the Brueckner orbital coupled cluster level. Differences between calculated potentials for several interacting atoms and the corresponding summed pair potentials reveal the three‐body and certain higher order contributions to the interaction strengths. Attraction between rare gas atoms develops from dispersion, and so helium clusters provide the most workable systems for analyzing nonadditivity of dispersion. The results indicate that …