The generalized Hadamard product S * T and the Jordan-Hadamard product S ∘ T of two operator-matrices S and T are introduced. They coincide with the usual Hadamard product of two complex matrices when the underlying Hilbert spaces are one-dimensional. Some inequalities which hold true for the usual Hadamard product of positive definite complex matrices are shown to be true for these two new products of positive invertible operator-matrices.

The conjugacy classes in the finite-dimensional projective full linear, special linear and projective special linear groups over an arbitrary commutative field are determined. The results over a finite field are applied to certain enumerative problems.

The present paper deals with some oscillating and asymptotic properties of the functional differential equations of the form

where λ is an arbitrary positive constant and τ > 0 is a constant delay.

The object of this paper is to establish limitation theorems for the ordinary and also absolute generalized Nörlund methods which include some known results as special cases. We shall give a different proof of the recent result of S. Narang (Proc. Indian Acad. Sci. Sect. A 88 (1979), 115–123), and we get a generalization of the result of G. Das (J. London Math. Soc. 41 (1966), 685–692) which states the summability factors of the absolute Nörlund methods.

The Fibonacci varieties introduced by Ann Chi Kim, Bull. Austral. Math. Soc. 19 (1978), 191–196 (1979), are here generalised in an obvious way suggested by the work of D.L. Johnson, J.W. Wamsley, and D. Wright, Proc. London Math. Soc. (3) 29 (1974), 577–592. The underlying groups of algebras in these varieties are studied, and are shown to be abelian when the algebras are generated by a single element, and in general are found to be extensions of central subgroups by groups of finite exponent.

A characterization of the class of operator semistable probability measures on a real separable Hilbert space is given.

The classification of strongly subdirectly irreducible rings, which was begun in a previous paper, is completed by showing that a mixed group G is strongly subdirectly irreducible if and only if G ⋍ Z(p∞) ⊕ H, H a rank one, p-divisible, torsion free nil group.

Iséki's BCK-algebras form a quasivariety of groupoids and a finite BCK-algebra must satisfy the identity (En): xyn = xyn+1, for a suitable positive integer n. The class of BCK-algebras which satisfy (En) is a variety which has strongly equationally definable principal congruences, congruence-3-distributivity, and congruence-3-permutability. Thus, a finite BCK-algebra generates a 3-based variety of BCK-algebras. The variety of bounded commutative BCK-algebras which satisfy (En) is generated by n finite algebras, each of which is semiprimal.

It is shown that a simple trick involving Legendre polynomials readily yields the irrationality of ea, , π2, and of the zeros of Bessel functions of integer order. Generalisation of this idea yields the irrationality of ζ(3).

Let k be a finite field of characteristic 2, and let G be the three dimensional special linear group over k. The principal indecomposable modules of G over k are constructed from tensor products of the irreducible modules, and formulae for their dimensions are given.

The Cauchy problem x′ = f(t, x), x(0) = x0, is considered in a non-reflexive Banach space E, where f is weakly continuous. A local existence theorem is proved using the measure of weak noncompactness.

A faulty typescript of [2] was, regrettably, submitted. The following changes should be made:

Page 398, line 7: replace A є G with A є L.

Page 398, line 12: replace A є G with A є L.