Consider uniform spaces X and Y and a separately uniformly continuous real-valued function f on X × Y. The following question arises in the theory of games: under what conditions can f be extended to a separately continuous function on × Ŷ, where , Ŷ are the completions of X and Y respectively? Firstly observe that such an extension is not always possible. If X = Y = (0, 1] with the usual uniform structure and f(x, y) = xy then f is separately uniformly continuous but has no separately continuous extension to × Ŷ = [0, 1]2 since such an extension would satisfy f(0, .) = 0 on Y and f(., 0) = 1 on X and so would necessarily have a discontinuity in one argument at the origin.

We are concerned with the stability properties of uniformly closed wedges in C(E) (resp. C+(E)), the real-valued (resp. non-negative) continuous functionson a compact space E, and solve the following problems in this area:

(a) Let A be a closed semi-algebra in C+(E) such that

The concept of a pseudo-ring was introduced by Patterson (1). Briefly, a pseudo-ring is an algebraic system consisting of an additive abelian group A, a distinguished subgroup A*, and a multiplication operation A* × A→A under which A* is a ring and A a left A*-module. For convenience, we denote the pseudo-ring by = (A*, A). For the definitions of the various types of ideal, we refer the reader to (1).

In an earlier paper (5) a description was given in set-theoretic terms of the semigroup generated by the idempotents of a full transformation semigroup , one of the results being that if X is finite then every element of that is not bijective is expressible as a product of idempotents. In view of this it was natural to ask whether by analogy every singular square matrix is expressibleas a product of idempotent matrices. This is indeed the case, as was shown by J. A. Erdos (2). Magill (6) has considered products of idempotents in thesemigroup of all continuous self-maps of a topological space X, but a comparable characterization of products of idempotents in this case appears to be extremelydifficult, and no solution is available yet.

Let X be a complex normed space, with dual space X′. Let T be a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x ∈ X, f ∈ X′, ‖ x ‖ = ‖ f ‖ = f(x) = 1}, and the numerical radius v(T) of T is defined as sup {|z|: z ∈ V(T)}. For a unital Banach algebra A, the numerical range V(a) of a ∈ A is defined as V(Ta), where Ta is the operator on A defined by Tab = ab. It is shown in (2, Chapter 1.2, Lemma 2) that V(a) = {f(a): f ∈ D(1)}, where D(1) = {f ∈ A′: ‖f‖ = f(1) = 1}.

Kilmister (1) has discussed the existence of linear integrals of a dynamical system specified by generalized coordinates qα(α = 1, 2, …, n) and a Lagrangian

repeated indices being summed from 1 to n. He derived covariant conditions for the existence of such an integral, conditions which do not imply the existence of an ignorable coordinate. Boyer (2) discussed the conditions and found the most general Lagrangian satisfying the conditions for the case of two degrees of freedom (n = 2).

The main theorem of this paper is a little involved (though the proof is straightforward using a well-known idea) but the immediate corollaries are interesting. For example, take a complex normed vector space A which is also a normed algebra with identity under each of two multiplications * and ∘. Then these multiplications coincide if and only if there exists α such that ‖a ∘ b ‖ ≦ α ‖ a * b ‖ for a, b in A. This is a condition for the two Arens multiplications on the second dual of a Banach algebra to be identical. By taking * to be the multiplication of a Banach algebra and ∘ to be its opposite, we obtain the condition for commutativity given in (3). Other applications are concerned with conditions under which a bilinear mapping between two algebras is a homomorphism, when an element lies in the centre of an algebra, and a one-dimensional subspace of an algebra is a right ideal. An example shows that the theorem is false for algebras over the real field, but Theorem 2 gives the parallel result in this case.

This paper owes its origin to the following question posed by A. M. Sinclair, “If a linear algebra with identity has two equivalent unital algebra norms, |.|1 and |.|2, whose corresponding numerical radii, v1 and v2, are equal on the whole algebra, are |.|1 and |.|2 related? Are they, for example, necessarily equal?” We do not give a complete answer to this question but are able to give sufficient conditions on algebras of operators for v1 = v2 to imply |.|1 = |.|2 That this implication does not hold for an arbitrary algebra with identity is demonstrated by means of a counter-example. The result for operator algebras is used to deduce some essentially non numerical range results for equivalent operator norms.

Let M be a positive integer, let a1, a2, …, aM be non-negative reals, and put aM+i = ai for i = 1, 2, 3. Further let each of v1, v2, v3 and δ1, δ2, δ3 be 0 or 1, giving 26 possibilities. This note is concerned with the problem of finding bounds for each of the non-trivial cases out of the 26 cyclic sums

In 1964 Green and Rivlin (1) introduced a theory of simple force and stress multipoles founded on conventional kinematics. Using a work formula, the force and stress multipoles were defined with the help of the velocity field and its spatial derivatives. More recently, within the framework of this general study, Bleustein and Green (2) examined the theory of the simplest multipolar fluid, the dipolar fluid, and formulated constitutive equations for a homogeneous incompressible linear dipolar fluid.

In this note we show that the variety of near-rings generated by d.g. near-rings is the class of all near-rings R satisfying

This extends a result of J. J. Malone (3) on the embedding of near-rings.

In (1; p. 38), A. P. Guinand discusses the plane partition function q(n). He observes that q(3), q(6), q(9), q(15), q(18), q(21), and q(24) are respectively 6, 48, 282, 1479, 6879, 29601, 118794, and 451194. As all these are multiples of 3 he suggests the conjecture that q(3n) ≡ 0 (mod 3) for all positive integers n.

The equation of the osculating plane at a point on the complete irreducible curve of intersection of two algebraic surfaces in [3] was found by Hesse (5, p. 283); the plane, having to contain the tangent of the curve, belongs to the pencil spanned by the tangent planes of the two surfaces, and it is a question of determining which plane of the pencil to choose. The equation also appears in the books of Salmon (6, p. 378) and Baker (1, p. 206). The analogous problem for the osculating solid at a point on the complete irreducible curve of intersection of three algebraic primals, or threefolds, in [4] does not appear to have been considered. The simplest instance is the octavic curve C of intersection of three quadrics, and this has the special interest of being a canonical curve; moreover the quadrics are of the same order, and so can be replaced by any three linearly independent members of the net which they determine, a replacement of which it may be prudent to take advantage with a view to simplifying the algebra. It is a question of determining which solid to choose among the tangent solids to the quadrics of the net at a point on C, but while Hesse's methods serve to carry one a certain distance there seems no obvious way of pushing them to a conclusion. It is then natural, with a view to reaching a conclusion, to choose a net of quadrics that, through having some particular property, is more amenable.

We refer to (1) for the definitions of Un and Hn. Our object is to find asymptotic expansions for Un and Hn for large n. This enables us to improve the approximations to Un and Hn for large n found in the last two pages of (1).