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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.
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...
If G is a graph with vertex set V(G) and (vertex) automorphism group γ(G), then a sequence s={vπ(i)}i=1k of distinct vertices of G is a partial stabilising sequence for G if $$\Gamma \left( {G_{S_n } } \right) = \Gamma \left( G \right)_{S_n } $$ for n = 1,...,k. Here S is the set $$\bigcup\limits_{i = 1}^n {V_{\pi (i)} ,G_{S_n } } $$ is the subgraph of G induced by the subset V(G) — Sn...
A subset S of an additive group G is said to be a sum-free set if S ∩ (S + S) = Ø. A sum-free set S is said to be locally maximal if for every sum-free set T such that S⊑T⊑G, we have S = T. Here we determine some sum-free cyclotomic classes in finite fields and from them, we construct new supplementary difference sets, association schemes and block designs. We also continue our study of locally...
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:...
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