Orthogonal diagonal latin squares of order n, ODLS(n), are orthogonal latin squares of order n with transversals on both the main diagonal and the back diagonal of each square. It has been proven that ODLS(n) exist for all n except n = 2, 3, 6, 10, 14, 15, 18 and 26, in which the first three are impossible. In this note an example of ODLS(14) is given.

Let all the zeros of a polynomial P(z) of degree n lie in |z|≤ 1 and a be a given complex number. In this paper we study the location of the zeros of higher derivatives of the polynomial (z – z) P(z) and obtain certain generalizations of some results of Rahman and Rubinstein. We shall also extend a result of Goodman, Rahman and Ratti for the zeros of the polar derivative of the polynomial P(z) given P(l) = 0.

We define and investigate the notion of a decomposable hypergraph, showing that such a hypergraph always is conformal, that is, can be viewed as the class of maximal cliques of a graph. We further show that the clique hypergraph of a graph is decomposable if and only if the graph is triangulated and characterise such graphs in terms of a combinatorial identity.

The notion of a recursive causal graph is introduced, hopefully capturing the essential aspects of the path diagrams usually associated with recursive causal models. We describe the conditional independence constraints which such graphs are meant to embody and prove a theorem relating the fulfilment of these constraints by a probability distribution to a particular sort of factorisation. The relation of our results to the usual linear structural equations on the one hand, and to log-linear models, on the other, is also explained

Vertices u and v of a graph G are pseudo-similar if G – u ≅ G – v, but no automorphisms of G maps u to v. Let H be a graph with a distinguished vertex a. Denote by G(u. H) and G(v. H) the graphs obtained from G and H by identifying vertex a of H with pseudo-similar vertices u and v, respectively, of G. Is it possible for G(u.H) and G(v.H) to be isomorphic graphs? We answer this question in the affirmative by constructing graphs G for which G(u. H)≅ G(v. H).

We provide an algorithm for calculating the Alexander polynomial of a two-bridge link by putting every two-bridge link in a special type of Conway diagram. Using this algorithm, some necessary conditions for a polynomial to be the Alexander polynomial of a two-bridge link are given, in particular, certain alternating and monotonicity conditions on the coefficients, analogous to corresponding known properties of the reduced Alexander polynomial.

Let G be transitive permutation group of degree n and let K be a nontrivial pronormal subgroup of G (that is, for all g in G, K and Kg are conjugate in (K, Kg)). It is shown that K can fix at most ½(n – 1) points. Moreover if K fixes exactly ½(n – 1) points then G is either An or Sn, or GL(d, 2) in its natural representation where n = 2d-1 ≥ 7. Connections with a result of Michael O'Nan are dicussed, and an application to the Sylow subgroups of a one point stabilizer is given.

Let ℒ be a lattice of subsets of a set X. Let MR (ℒ) denote the set of all ℒ-regular (finitely additive) measures on the algebra generated by ℒ. Under the assumption that ℒ is disjunctive, in the first part of the paper, criteria are obtained for the σ-smoothness, τ-smoothness, and tightness of elements of MR(ℒ) in terms of the general Wallman remander. in the second part of the paper, various applications are given, and, in particular, extensions and refinements of the Yoside-Hewitt Decom position Theorem are obtained.

We study the existence of zeros of a perturbed nonlinear operator near a zero of the unperturbed operator in the case where both operators are invariant under a symmetry group. To do this, we first correct some work of Rubinsztein on the G-homotopy groups of spheres.

In this paper we give a sufficient condition of quasi-pseudometrization for bitopological spaces. From this condition we obtain, as immediate corollaries, some known results.

We show that a bounded linear operator on a uniformly convex space may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius.

This paper is concerned with the numerical range and some related properties of the operator Δ/ S: T → AT – TB(T∈S), where A, B are (bounded linear) operators on the normed linear spaces X and Y. respectively, and S is a linear subspace of the space ℒ (Y, X) of all operators from Y to X. S is assumed to contain all finite operators, to be invariant under Δ, and to be suitably normed (not necessarily with the operator norm). Then the algebra numerical range of Δ/ S is equal to the difference of the algebra numerical ranges of A and B. When X = Y and S = ℒ (X), Δ is Hermitian (resp. normal) in ℒ (ℒ(X)) if and only if A–λ and B–λ are Hermitian (resp. normal) in ℒ(X)for some scalar λ;if X: = H is a Hilbert space and if S is a C *-algebra or a minimal norm ideal in ℒ(H)then any Hermitian (resp. normal) operator in S is of the form Δ/ S for some Hermitian (resp. normal) operators A and B. AT = TB implies A*T = TB* are hyponormal operators on the Hilbert spaces H1 and H2, respectively, and T is a Hilbert-Schmidt operator from H2 to H1.