The generalised Riesz-Fischer theorem states that if

is convergent, with 1 < p ≤ 2, then

is the Fourier series of a function of class . When p > 2 the series (2) is not necessarily a Fourier series; neither is it necessarily a Fourier D-series. It will be shown below that it must however be what may be called a “Fourier Stieltjes” series. That is to say, the condition (1) with (p > 1) implies that there is a continuous function F (x) such that

Symmetric inverse monoids of objects in arbitrary categories are studied. Necessary and sufficient conditions are given for such monoids to be E-unitary or else form (complete) inverse algebras. Particular attention is given to symmetric inverse monoids of objects in free categories.

In this paper the “hypercore” of a semigroup S is defined to be the subsemigroup generated by the union of all the subsemigroups of S without non-universal cancellative congruences, provided that at least one such subsemigroup exists: otherwise it is taken to be the empty set. It is shown first that if the hypercore of S is nonempty (which holds, for example, when S contains an idempotent) then it is the largest subsemigroup of S with no non-universal cancellative congruence, is full and unitary in S, and is contained in the identity class of every group congruence on S (Theorem 1).

Let A be a quasi-accretive operator defined in a uniformly smooth Banach space. We present a necessary and sufficient condition for the strong convergence of the semigroups generated by – A and of the steepest descent methods to a zero of A.

In a recent generalization of the Bernstein polynomials, the approximated function f is evaluated at points spaced at intervals which are in geometric progression on [0, 1], instead of at equally spaced points. For each positive integer n, this replaces the single polynomial Bnf by a one-parameter family of polynomials , where 0 < q ≤ 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning when f is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if f is increasing then is increasing, and if f is convex then is convex, generalizing well known results when q = 1. It is also shown that if f is convex then, for any positive integer n This supplements the well known classical result that when f is convex.

In this paper two major questions concerning generalised quaternion groups and distributively generated (d.g.) near-rings are investigated. The d.g. near-rings generated, respectively, by the inner automorphisms, automorphisms, and endomorphisms of the group are described. It is also shown that these morphism near-rings are local near-rings and contain no non-trivial idempotents. Finally, it is demonstrated that exactly 16 d.g. near-rings can be defined on a given generalised quaternion group.

Let G be a finite group generated by n elements and defined by m relations, then G has a presentation, G = {x1, …, xn | R1, …, Rm} = F/R where F is free on generators x1, …, xn and R is the normal closure in F of R1, …, Rm. The deficiency of this presentation is n − m. Since G is finite the deficiency is non-positive and the deficiency of G is the maximal over the deficiencies of all presentations for G.

The notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded operators were further investigated by Ringrose (6, 7), Sills (8) and Berkson and Dowson (2). Berkson and Dowson have developed a more complete theory for the type (A) and type (B) well-bounded operators than is possible for the general well-bounded operator. Their work relies heavily on Sills' treatment of the Banach algebra structure of the second dual of the Banach algebra of absolutely continuous functions on a compact interval.

If by any change of axes ax2 + 2hxy + by2 by changed into a′x′2 + 2h′x′y′ + b′y′2, then will

It is known that if in a Banach*-algebra with unit the following holds:

then it is a C*-algebra (see [3]).

We shall show that the above theorem can be sharpened in the following way: we replace the submultiplicativity of the norm by the weaker assumption

Observe that under this assumption even the existence of exp(ih) is not at all obvious, but it will be proved to be true below. Our main result is Theorem 2 which depends on Theorem 1. Our last remark is the equivalent-norm-version of the statement.

It is a fundamental fact in the theory of radicals of associative rings that if S is a radical and I is a two-sided ideal of R then S(I)⊆S(R). In view of this result it seems to be interesting to investigate radicals satisfying such or similar connections for other type of subrings. There are many works devoted to similar problems (2, 8, 8, 10). In this paper we try to get a uniform description of some facts in this area.

In the present paper two problems on approximation by rational functions will be treated. The one concerns rational functions whose poles are of any order but lie at two preassigned points. The other problem relates to rational functions which have simple poles only.

It is well-known that a general net of quadric surfaces cannot be obtained as the net of polar quadrics of the points of a plane in regard to a cubic surface; in order that it may be so obtained it must have various properties that a general net of quadrics does not have. The locus of the vertices of the cones which belong to the net of quadrics is a curve ϑ –the Jacobian curve of the net of quadrics, and the trisecants of ϑ generate a scroll. Any plane which contains two trisecants of ϑ is a bitangent plane of the scroll and, for a general net of quadrics, there are eighteen of these bitangent planes passing through an arbitrary point. When however the net of quadrics is a net of polar quadrics it is found that any plane which contains two trisecants of ϑ contains two other trisecants also; it thus contains four trisecants in all and counts six times as a bitangent plane of the scroll. The bitangent developable of the scroll, which is, for a general net of quadrics, of class eighteen, degenerates, in the special case when the net of quadrics is a net of polar quadrics, into a developable of class three counted six times; the planes of the developable are therefore the osculating planes of a twisted cubic γ. The plane which, together with a cubic surface, gives rise to the net of polar quadrics must be one of the osculating planes of γ. It is also found, further, that the osculating planes of γ are grouped into pentahedra, the vertices of all these pentahedra lying on ϑ.