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Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and Williamson-type matrices. These latter are four (1,-1) matrices A,B,C,D, of order m, which pairwise satisfy (i) MNT = NMT, M,N ε {A,B,C,D}, and (ii) AAT+BBT+CCT+DDT = 4mIm, where I is the identity matrix. Currently Williamson matrices are known to exist for all orders less than 100 except: 35,39,47,53,59,65,67,70,71,73,76,77,83,89,94. This paper gives two constructions for Williamson matrices of even order, 2n. This is most significant when no Williamson matrices of order n are known. In particular we give matrices for the new orders 2.39,2.203,2.303,2.333,2.689,2.915, 2.1603. [ABSTRACT FROM AUTHOR]

Several authors in recent years investigated the properties of the Moore-Penrose inverse of an arbitrary Boolean relation matrix. The concept of a Moore-Penrose inverse for Boolean relation matrices was discussed first by Rutherford [11] and then independently discovered by Markowsky [8], Plemmons [10], and the author ([3] and [4]). It is natural to inquire whether or not the Moore-Penrose inverse is unique, if it exists. In this paper, the properties of unique Moore-Penrose inverse of an arbitrary Boolean relation matrix are examined in connection with partial order relation and three computational methods for the unique Moore-Penrose inverse for an arbitrary Boolean relation matrix is developed. [ABSTRACT FROM AUTHOR]

Given an n × n grid of n2 points we must select as many as possible so that no three are in a straight line. This paper reviews results concerning the problem and provides a few minor proofs, additions and generalisations. [ABSTRACT FROM AUTHOR]