Physical Sciences and Mathematics Commons^™

On The Structure And Existence Of Some Amicable Orthogonal Designs, Peter J. Robinson, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

The structure is determined for the existence of some amicable weighing matrices. This is then used to prove the existence and non-existence of some amicable orthogonal designs in powers of two.

A Remark On The Excess Of Hadamard Matrices And Orthogonal Designs, J Hammer, R Levingston, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Some improved upper and lower bounds are given for the excess of Hadamard matrices. The excess of orthogonal designs is defined and discussed.

A Computer Listing Of Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

A computer has been used to list all known Hadamard matrices of order less than 40,000. If an Hadamard matrix is not known of order 4q (q odd) then the smallest t so that there is an Hadamard matrix of order 2tq is given. Hadamard matrices are not yet known for orders 268, 412, 428.

A Class Of Codes Generated By Circulant Weighing Matrices, K Wehrhahn, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Some properties of a new class of codes constructed using circulant matrices over GF(3) will be discussed. In particular we determine the weight distributions of the (14, 7) and two inequivalent (26, 13)-codes arising from the incidence matrices of projective planes of orders 2 and 3.

Optimal Design Of A Quasi-Redundant Protective System For Nuclear Reactors, J. M. Kontoleon

Faculty of Informatics - Papers (Archive)

In many instances protective systems used in nuclear reactors are quasi-redundant systems; each of a number of safety channels feeds a number of independent protective units. A reactor shutdown is initiated if more than a specified number of units are in favour of shut down. The objective is to achieve a very high reliability at a reasonable cost. An analysis is presented to obtain the reliability, failsafe and fail-danger probabilities of a quasi-redundant system. Three algorithms are given for: (a) the design of a quasi-redundant system having the maximum reliability subject to a cost constraint, (b) the optimal design satisfying …

A Class Of Group Divisible Designs, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

No abstract provided.

On Skew Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Recently I have proved that for every odd integer q there exists integers t and s (dependent on q) so that there is an Hadamard matrix of order 2tq and a symmetric Hadamard matrix with constant diagonal order 2s q2. We conjecture that "for every odd integer q there exists an integer t (dependent on q) so that there is a skew-Hadamard matrix of order 2tq”. This paper makes progress toward proving this conjecture. In particular we prove the result when q = 5 (mod 8) = s2 + 4r2 is a prime power and all orthogonal designs of type …

Generation Of A Frequency Square Orthogonal To A 10 X 10 Latin Square, H C. Kirton, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

In general it is a difficult if not impossible task to find a latin square orthogonal to a given latin square. Because of a practical problem it was required to find a frequency square orthogonal to a given latin square. We describe a computer approach which was successful in finding a (4,23) frequency square orthogonal to a given 10 x 10 latin square.

An Algorithm For The Permanent Of Circulant Matrices, Larry J. Cummings, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

The permanent of an n x n matrix A = (a;j) is the matrix function ( 1) per A = ∑ al1r(1)••• a .. ",( .. )".C~" where the summation is over all permutations in the symmetric group, S ... An n x n matrix A is a circulant if there are scalars ab ... ,a,. such that (2) A= ∑ a;pi-l where P is the n x n permutation matrix corresponding to the cycle (12• .. n) in s". In general the computation of the permanent function is quite difficult chiefly because it is not invariant under addition of …

Orthogonal Designs In Powers Of Two, Peter J. Robinson, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Repeat designs are introduced and it is shown how they may be used to give very powerful constructions for orthogonal designs in powers of two. These results are used to show all full four variable and all three variable designs exist in 2t , t ≤ 9. We believe these constructions demonstrate the existence of all possible four variable designs with no zeros in every power of two but we have not been able to prove this.

A Note On Asymptotic Existence Results For Orthogonal Designs, Peter Eades, Jennifer Seberry, Nicholas Wormald

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 uses sequences with zero non-periodic and periodic autocorrelation function to establish the asymptotic existence of many orthogonal designs with four variables. Bounds are also established for orthogonal designs of type (1, k) where k ≤ 63 and (t) where I ≤ 52. It is shown that any 4 sequences with zero non-periodic auto-correlation function and 8k - 1 entries +1 or -1 must have length at least 2k + …

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.

Beach Erosion, May-June, 1974, Central And South Coast, Nsw, Edward A. Bryant, R. Kidd

Faculty of Science - Papers (Archive)

Between May 24th and June 18th 1974, three periods of erosive wave conditions dramatically changed the character of many beaches along the central and southern New South Wales Coast. This paper documents and evaluates regional variations in the responses of beaches to these erosional events for selected portions of this coast (Figure 1).

Some Results On Weighing Matrices, Jennifer Seberry, Albert Leon Whiteman

Faculty of Informatics - Papers (Archive)

It is shown that if q is a prime power then there exists a circulant weighing matrix of order q² + q + 1 with q² non-zero elements per row and column.

This result allows the bound N to be lowered in the theorem of Geramita and Wallis that " given a square integer k there exists an integer N dependent on k such that weighing matrices of weight k and order n and orthogonal designs (1, k) of order 2n exist for every n > N".

On Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and four (1, -1) matrices A, B, C, D of order m which are of Williamson type, that is pairwise satisfy

(i) MN^T = NM^T and

(ii) AA^T + BB^T + CC^T + DD^T = 4ml_m.

If (i) is replaced by (i')MN = NM we have Goethals-Seidel matrices. These matrices are very important to the determination of the Hadamard conjecture: that there exists an Hadamard matrix of order 4t for all natural numbers t. This paper …

Orthogonal Designs: Ii, Anthony V. Geramita, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1, k) and order n exist for every k < n when n = 2^t+2. 3 and n = 2^t+2.5 (where t is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n.

Coupled with some results of earlier work, this means that the weighing matrix conjecture 'For every order n = 0 (mod4) there is, for each …

Orthogonal Designs Iv: Existence Questions, Anthony V. Geramita, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

In [5] Raghavarao showed that if n = 2 (mod 4) and A is a {O, 1, -1} matrix satisfying AA^t = (n - 1) I_n. then n - 1 = a² - b² for a, b integers. In [4] van Lint and Seidel giving a proof modeled on a proof of the Witt cancellation theorem, proved more generally that if n is as above and A is a rational matrix satisfying AA^t = kI_n then k = q_1² + q₂² (q₁, q₂ E Q, the …

Construction Of Williamson Type Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and four (1, - 1) matrices A, B, C, D of order m which are of Williamson type, that is they pair-wise satisfy

i) MN^T = NM^T, M, N E {A, B, C, D} and

ii) AA^T + BB^T + CC^T + DD^T = 4mI_m.

It is shown that Williamson type matrices exist for the orders m = s(4s - 1), m = s(4s + 3) for s E {1, 3, 5, ... ,25} and …

Orthogonal Designs, Anthony V. Geramita, Joan Murphy Geramita, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Orthogonal designs of special type have been extensively studied, and it is the existence of these special types that has motivated our study of the general problem of the existence of orthogonal designs.

This paper is organized in the following way. In the first section we give some easily obtainable necessary conditions for the existence of orthogonal designs of various order and type. In Section 2 we briefly survey the examples of such designs that we have found in the literature. In the third section we describe several methods for constructing orthogonal designs. In the fourth section we obtain some …

Construction Of Amicable Orthogonal Designs, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Infinite families of amicable orthogonal designs are constructed with

(i) both of type (1, q) in order q + 1 when q = 3, (mod 4 ) is a prime power,

(ii) both of type (1, q) in order 2(q+1) where q = 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix,

(iii) both of type (2, 2q) in order 2(q+l) when q = 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix.