Minkowski proved the following (for a proof see (4)): if A and B are n × n positive semi-definite hermitian matrices then

It is known (4) that if both A and B are non-singular, then the equality holds in (1) if and only if B = cA where c is a positive number.

Some Hilbert spaces of continuous functions satisfying a mean value property are studied in which the generalised eigenfunctions of any selfadjoint operator again satisfy the same mean value property. Applications are made to nullspaces of some differential operators. The classes of functions involved in these applications are less general than those studied by K. Maurin (6); however, the Hilbert space norms may be arbitrary, while Maurin only considered L2-norms.

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).

A sequence x = {si} has been defined to be summable-(Z, p) to s if

Where p is a positive integer and

A bounded monotonic sequence is convergent. Dr J. M. Whittaker recently suggested to me a generalisation of this result, that, if a bounded sequence {an} of real numbers satisfies the inequality

then it is convergent. This I was able to prove by considering the corresponding difference equation

This note is concerned with “twisted“ analogues of the LP-structures (i.e.local-product structures) and grids of (4). To obtain these twisted structures, we modify the concepts of LP-structure and grid by removing the ordering of the local foliations involved in the definitions. The effect of this change is that global foliations need no longer exist, since the locally denned foils may now fit together to form self-intersecting immersed manifolds.

Patterson (4) introduced the concept of a pseudo-ring and considered the pseudo-ring of infinite matrices over a ring. In this paper we shall generalize and improve the work of Patterson, using certain additions to the general theory of pseudo-rings which have recently been introduced (1). We shall follow the conventions and notations used in (1) and (4).

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.

In his recent paper on partitions (1), Jakub Intrator proved that the number p(n, k) of partitions of n into exactly k summands, 1 < k ≦ n, is given by a polynomial of degree exactly k − 1 in n, the first [(k+1)/2] coefficients of which (starting with the coefficient of the highest degree term), are independent of n and the rest depend on the residue of n modulo the least common multiple of the integers 1, 2, 3, …, k. He even showed (ignoring the case k = 3) that the [(k+3)/2]-th coefficient in the polynomial depends only on the parity of n and is not the same for n even and n odd.

Let G be a group and let K be an algebraically closed field of characteristic p>0. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be a vector space over K with basis elements , …; a multiplication is defined on this basis of Kt(G) and extended by linearity to Kt(G) by letting

where α(x, y) is a non-zero element of K, subject to the condition that

which is both necessary and sufficient for associativity. If, for all x, y ∈ G, α{x, y) is the identity of K then Kt(G) is the usual group algebra K(G) of G over K. We denote the Jacobson radical of Kt(G) by JKt(G). We are interested in the relationship between JKt(G) and JKt(H) where H is a normal subgroup of G. In § 2 we show, among other results, that if certain centralising conditions are satisfied and if JK(H) is locally nilpotent then JK(H)K(G) is also locally nilpotent and thus contained in JK(G). It is observed that in the absence of some centralising conditions these conclusions are false. We show, in particular, that if H and G/C(H) are locally finite, C(H) being the centraliser of H, and if G/H has no non-trivial elements of order p, then JK(G) coincides with the locally nilpotent ideal JK(H)K(G). The latter, and probably more significant, part of this paper is concerned with particular types of groups. We introduce the notion of a restricted SN-group and show that if G is such a group and if G has no non-trivial elements of order p then JKt(G) = {0}. It is also shown that if G is polycyclic then JKt(G) is nilpotent.

Let ∑an be a given infinite series and {λn} a non-negative, strictly increasing, monotonic sequence, tending to infinity with n. We write, for w > λ0,

and, for r>0, we write

is known as the Riesz sum of “ type ” λn and “ order ” r, and

is called the Riesz mean of type λn and order r.

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.

The study of near-rings is motivated by consideration of the system generated by the endomorphisms of a (not necessarily commutative) group. Such endomorphism near-rings also furnish the motivation for the concept of a distributively generated (d.g.) near-ring. Although d.g. near-rings have been extensively studied, little is known about the structure of endomorphism near-rings. In this paper results are presented which enable one to give the elements of the endomorphism near-ring of a given group. Also, some results relating to the right ideal structure of an endomorphism near-ring are presented. These concepts are applied to present a detailed picture of the properties of the endomorphism near-ring of (S3, +).

In this paper we prove that the states of a unital Banach algebra generate the dual Banach space as a linear space (Theorem 2). This is a result of R. T. Moore (4, Theorem 1(a)) who uses a decomposition of measures in his proof. In the proof given here the measure theory is replaced by a Hahn-Banach separation argument. We shall let A denote a unital Banach algebra over the complex field, and D(1) denote {f ∈ A′: ‖f‖ = f(1) = 1} where A′ is the dual of A. The motivation of Moore's results is the theorem that in a C*-algebra every continuous linear functional is a linear combination of four states (the states are the elements of D(1)) (see (2, 2.6.4, 2.1.9, 1.1.10)).

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

We shall use results of Palmer (10, 11) and of Edwards and Ionescu Tulcea (6) to show that a commutative V*-algebra (with identity) of operators on a weakly complete Banach space is isomorphic to such an algebra on a Hilbert space, the isomorphism extending to the weak closures of the algebras. This result leads to an extension of Stone's theorem on unitary groups (a similar extension is proved by different methods in (2, p. 350) and of Nagy's theorems on semigroups of normal operators. The same technique yields an easy proof of Dunford's theorem on the existence of a σ-complete extension of a bounded Boolean algebra of projections on a weakly complete Banach space. We are indebted to H. R. Dowson for suggesting this topic and for help and guidance in pursuing it.

In this paper we study closed sets having a neighbourhood with compact closure which are positively asymptotically stable under a flow on a metric space X. For an understanding of this and the rest of the introduction it is sufficient for the reader to have in mind as an example of a flow a system of first order, autonomous ordinary differential equations describing mathematically a time-independent physical system; in short a dynamical system. In a flow a set M is positively stable if the trajectories through all points sufficiently close to M remain in the future in a given neighbourhood of M. The set M is positively asymptotically stable if it is positively stable and, in addition, trajectories through all points of some neighbourhood of M approach M in the future.

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.

A tournamentTn consists of a finite set of nodes1, 2, …, n such that each pair of distinct nodes i and j is joined by exactly one of the arcsij or ji. If the arc ij is in Tn we say that i beats j or j loses to i and write i→j. If each node of a subtournament A beats each node of a subtournament B we write A→B and let A + B denote the tournament determined by the nodes of A and B.

Let S be a completely 0-simple semigroup and let Λ(S) be the lattice of congruences on S. G. Lallement (2) has described necessary and sufficient conditions on S for Λ(S) to be modular, and has shown that Λ(S) is always semimodular . This result may be stated: If S is 0-bisimple and contains a primitive idempotent, then Λ(S) is semimodular.