Physical Sciences and Mathematics Commons^™

Orthogonal Designs V: Orders Divisible By Eight, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Constructions are given for orthogonal designs in orders divisible by eight. These are then used to show all two variable orthogonal designs exist in orders 24, 32 and 48. The existence of two variable designs in order 40 and three variable designs in order 24 is discussed.

The conjectures on the existence of all orthogonal designs (1, k) and skew-symmetric weighing matrices for weights k = 1, 2, ..., 2.^t9-1 are resolved in the affirmative for orders 2.^t9, t > 3 a positive integer.

On The Existence Of Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Given any natural number q > 3 we show there exists an integer t ≤ [2 log2 (q – 3)] such that an Hadamard matrix exists for every order 2sq where s > t. The Hadamard conjecture is that s = 2. This means that for each q there is a finite number of orders 2vq for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q. In particular it is already known that an Hadamard matrix exists for each 2sq where if q = 2m – 1 then s ≥ …

Reduction Of Angular Momentum Expressions By Matrix Arithmetic, D. J. Newman, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

In perturbation calculations using basis states defined in terms of spherically symmetric potentials it is often necessary to simplify complicated expressions involving n-j symbols. A well known graphical technique can be used to aid in this process. We represent the graphs by their incidence matrices, so that the algebraic manipulations can be carried out by matrix arithmetic. It is shown that the sequence of operations required to simplify a given graph can be determined from structural considerations based on the properties of certain polynomials in the adjacency matrix. This provides a method of performing complete perturbation calculations of this type …

A Note On Orthogonal Designs In Order Eighty, Joan Cooper, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

This is a short note showing the existence of all twovariable designs in order 80 except possibly (13, 64) and (15, 62) which have not yet been construced. The designs are constructed using designs in order 8, 16, 20, and 40 and applying lemmas and theorems concerning orthogonal designs. Three-variable designs (a, b, n-a-b), which are useful in constructing Hadamard matrices, are also considered for n = 40 and 80.

“George Szekeres”, J R. Giles, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

George Szekcres was born in Budapest on 29th May, 1911 the second of three sons to wealthy Jewish paren ts. As a youlh he was shy and retiring, but early it became clear that his gifts lay in the direction of science and mathematics. At high school George was greatly influenced by his teacher in mathematics and physics, K. (Charles) Novobatzky, who worked actively in the theory of relativity and was in 1945 to become a professor of theoretical physics at the University of Budapest. Small wonder that George's first great mathematical interest was relativity. The other major formative influence …

Some New Constructions For Orthogonal Designs, Anthony V. Geramita, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We give three new constructions for orthogonal designs using amicable orthogonal designs. These are then used to show (i) all possible n-tuples, n ~ 5 , are the types of orthogonal designs in order 16 and (ii) all possible n-tuples, n ~ 3 are the types of orthogonal designs in order 32 , (iii) all 4-tuples, (e, f, g, 32-e-f-g) , o ~ e T f T g ~ 32 are the types of orthogonal designs in order 32. These resultg are used in a paper by Peter J. Robinson, "Orthogonal designs of order sixteen", in this same volume, to …

Designs From Cyclotomy, Elizabeth J. Morgan, Anne P. Street, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

In this note we use the theory of cyclotomy to help us construct initial blccks from which we can develop balanced and partially balanced incomplete block designs. Our main construction method, using unions of cyclotomic classes, gives us upper bounds on m, the number of associate classes of the design, but not exact values for m; we discuss the possible values of m and the circumstances under which m = 1, so that the design is in fact balanced.

A Note On Using Sequences To Construct Orthogonal Designs, Peter J. Robinson, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Several constructions are given which show how to construct orthogonal designs from sequences of commuting variables with zero non-periodic auto-correlation function.

Some Asymptotic Results For Orthogonal Designs: Ii, Peter Eades, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

In a recent manuscript « Some asymptotic results for orthogonal designs » Peter Eades showed that for many types of orthogonal designs existence is established once the order is large enough. This paper examines 4-tuples (S1 S2, S3, S4) where Sl + S2 + S3 + S4 ~ 28 and establishes lower bounds for the existence of orthogonal designs of that type.

An Infinite Family Of Skew Weighing Matrices, Peter Eades, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We verify the skew weighing matrix conjecture for orders 2t.7, t ~ 3 a positive integer, by showing that orthogonal (1, k) exist for all t k = 0, 1, .... , 2.7 - 1 in order 2t.7 We discuss the construction of orthogonal designs using circulant matrices. In particular we construct designs in orders 20 and 28. The weighing matrix conjecture is verified for order 60.

Using Cyclotomy To Construct Orthogonal Designs, Joan Cooper, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

An orthogonal design of order n and type (s₁, S₂) on the commuting variables x_1, X₂ is a matrix of order n with entries from {O, ± x_1, ± x₂ } whose row vectors are formally orthogonal.

This note uses cyclotomy to construct orthogonal designs and finds several infinite families of new designs.