The purpose of this thesis is to attempt to evaluate the permanent function of a n×n complex matrix with entries aij = θij being a primitive n root of unity. If this matrix is denoted by An then its permanent function is given by per An = [formula omitted] In this thesis the following results are proved. Per An is always an integer; with per An ≡ 0 mod n. If n is even per An = 0. For n odd however, the problem is in general not resolved. It is shown that if n=p² with p a prime, that per An = 0 mod p⁴ and that for any prime n, per An can be narrowed down to be one of a restricted class of numbers. / Science, Faculty of / Mathematics, Department of / Graduate

"The present paper comprises a survey of investigations undertaken to determine possible values of n for which a Hadamard matrix of order n may be constructed"--Introduction. / "August, 1960." / Typescript. / "Submitted to the Graduate School of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: M. F. Tinsley, Professor Directing Paper. / Includes bibliographical references (leaf 33).

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Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. The most desirable class of matrices for our purpose are orthogonal designs constructible from 16 circulant matrices. In studying these matrices, we show that the OD (12;1,1,1,9) is the only orthogonal design constructible from 16 circulant matrices of type OD (4n;1,1,1,4n-3), whenever n > 1 is an odd integer. We then use an exhaustive search to show that the only orthogonal design constructible from 16 circulant matrices of order 12 on 4 variables is the OD (12;1,1,1,9). It is known that by using of T-matrices and orthogonal designs constructible from 16 circulant matrices one can produce an infinite family of orthogonal designs. To complement our studies we reproduce and important recent construction of T-matrices by Xia and Xia. We then turn our attention to the applications of orthogonal matices. In some recent works productive regular Hadamard matrices are used to construct many new infinite families of symmetric designs. We show that for each integer n for which 4n is the order of a Hadamard matrix and 8n2 - 1 is a prime, there is a productive regular Hadamard matrix of order 16n2(82-1)2. As a corollary, we get many new infinite classes of symmetric designs whenever either of 4n(8n2-1)-1,4n(82-1) +1 is a prime power. We also review some other constructions of productive regular Hadamard matrices which are related to our work. / iv, 64 leaves : ill., map ; 29 cm.

Let A be an m x n, (0, 1)-matrix. A submatrix of A is odd if the sum of its entries is an odd integer and even otherwise. The maximum number of 2 x 2 odd submatrices in a (0, 1)-matrix is related to the existence of Hadamard matrices and bounds on Turán numbers. Pinelis [On the minimal number of even submatrices of 0-1 matrices, Designs, Codes and Cryptography, 9:85-93, 1994] exhibits an asymptotic formula for the minimum possible number of p x q even submatrices of an m x n (0, 1)-matrix. Assuming the Hadamard conjecture, specific techniques are provided on how to assign the 0's and 1's, in order to yield the maximum number of 2 x 2 odd submatrices in an m x n (0, 1)-matrix. Moreover, formulas are determined that yield the exact maximum counts with one exception, in which case upper and lower bounds are given. These results extend and refine those of Pinelis.

published_or_final_version / Mathematics / Master / Master of Philosophy