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.

Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” TkA → A on A of one or more endofunctors of A, subjected to equational axioms. Such problems include those of free monads and free monoids, of cocompleteness in categories of monads and of monoids, of orthogonal subcategories (= generalized sheaf-categories), of categories of continuous functors, and so on; apart from problems involving the algebras for their own sake.

Desirable properties of the category of algebras - existence of free ones, cocompleteness, existence of adjoints to algebraic functors - all follow if this category can be proved reflective in some well-behaved category: for which we choose a certain comma-category T/A

We show that the reflexion exists and is given as the colimit of a simple transfinite sequence, if A is cocomplete and the Tk preserve either colimits or unions of suitably-long chains of subobjects.

The article draws heavily on the work of earlier authors, unifies and simplifies this, and extends it to new problems. Moreover the reflectivity in T/A is stronger than any earlier result, and will be applied in forthcoming articles, in an enriched version, to the study of categories with structure.

This expository article on Shape Theory contains the main concepts of this theory with a formulation of the most important results of this theory and also with some open problems. The proofs are omitted, however the article gives references to the hooks and papers, in which the reader can find the proofs.

For simplicity, we formulate several theorems only in the form which clearly gives their geometric sense, even if there are known more general results with rather complicated formulations. The reader more interested in details, can find them in the original papers quoted in the list of references.

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.

This note shows that the variety generated by Murskii's 3-element algebra contains an infinite descending chain of subvarieties, thus falsifying a conjecture made previously by the author and M.R. Vaughan-Lee.

In this note we study certain properties of quasi-similarity orbit of a subclass of compact operators defined on a separable Hilbert space. This class and its quasi-similarity orbit were introduced and studied by Fialkow in Pacific J. Math. 70 (1977), 151–161.

Let E be a barrelled space with dual F ≠ E*. It is shown that F has uncountable codimension in E*. If M is a vector subspace of E* of countable dimension with M ∩ F = {o}, the topology τ(E, F+M) is called a countable enlargement of τ(E, F). The results of the two previous papers are extended: it is proved that a non-barrelled countable enlargement always exists, and sufficient conditions for the existence of a barrelled countable enlargement are established, to include cases where the bounded sets may all be finite dimensional. An example of this case is given, derived from Amemiya and Kōmura; some specific and general classes of spaces containing a dense barrelled vector subspace of codimension greater than or equal to c are discussed.

It is shown that the functional fails to possess a variational derivative, contrary to what is claimed by Gelfand and Fomin. A modified definition is given with respect to which the functional does possess a variational derivative.

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.

In various papers on some of the theoretical aspects of computing, the reader is confronted with a mathematical entity called a “multiset”, and is told that a multiset is “an unordered collection of elements that may have repeated occurrences of identical elements” (see, for example, Nachum Dershowitz and Zohar Manna, “Proving termination with multiset orderings”, Lecture Notes in Computer Science, 71). These entities are then manipulated in accordance with the classical laws of set algebra. The purpose of this note is to present a formal foundation for this concept of multiset, and to show by means of examples that if this foundation is adopted then, although certain sections of classical set algebra can be applied to multisets, it certainly cannot be applied in total.

Let n be an odd integer greater than 9. It is proved that the alternating group An has spread 3 in the sense that for any non-trivial elements x1, x2, x3 of An, there is an element y in An such that 〈xi, y〉 = An for i = 1, 2, 3.

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.

In this paper, smoothness properties of solutions of the initial-Dirichlet problem for parabolic equations in regions with edges are considered. We obtain bounds for solutions and derivatives, and prove the Hölder continuity of the first derivatives and of the second derivatives multiplied by a suitable power of the distance from the edges.

Kennison's concept of an ordered sheaf is used to show that any member of the variety of subresiduated lattices is canonically isomorphic to the algebra of all ordered sections in a certain ordered sheaf, whose base is the Priestley space of the residuating sublattice.

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.

An invariant measure which is absolutely continuous with respect to Lebesgue measure is constructed for a particular first order difference equation that has an extensive biological pedigree. In a biological context this invariant measure gives the density of the population whose growth is governed by the difference equation. Further asymptotically universal results are obtained for a class of difference equations.

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

The object of this paper is to establish some inclusion relations between two generalized Nörlund methods of summability. Our results are generalizations of Das's Theorem and one of them is also a generalization of a theorem of the author. They include the well-known fact that the Cesàro method (C, α) is weaker than (C, β) for β > α > 0.

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.

Let S denote the class of functions f analytic and univalent in the open disc {z: |z| < 1} and normalized by f(0) = 0 = f′(0) − 1, and S*(α) denote the set of starlike functions of order α (0 ≤ α ≤ 1) in S. In this paper, the results of William M. Causey and William L. White [J. Math. Anal. Appl. 64 (1978), 458–466] are generalized by the following:

THEOREM 1. Let f, g and h belong to S*(α), S*(λ) and S*(γ) , respectively and let F be defined by

where a, c ∈ N. Then F belongs to S*(β) for |z| less than a suitably chosen number.

THEOREM 2. Let F, g and h belong to S*(α), S*(λ) and S*(γ) , respectively and f be given by **. Then f belongs to S*(β) , for |z| less than a suitably chosen number.