The object of this note is to generalize the notion of quasi-monotony for sequences of real numbers and to prove corresponding generalizations of certain known theorems. First, we recall the definition of quasi-monotony.

Precise asymptotic estimates for the eigenvalues of a uniformly right definite two parameter system of Sturm–Liouville problems are developed. The work extends recent results of B. P. Rynne.

In their paper N. Divinsky and A. Sulinski [6] have introduced the notion of mutagenic radical property—that is, a radical property which is far removed from hereditariness—and constructed two such examples. The first is the lower radical property determined by a ring Swo (N. Divinsky [5]) and is an almost subidempotent radical property in the sense of F. Szász [9], and the second is a weakly supernilpotent radical property, that is the lower radical property determined by Swo and all nilpotent rings.

In a recent paper, J. L. Synge gives an interesting derivation of the conservation equations Tij,j = 0 satisfied by the energy tensor Tij of a continuous medium. Previous to the appearance of this paper, these equations were generally obtained by assuming the classical equations of motion and continuity, after which it was necessary to appeal to the Principle of Equivalence. It then follows that the path of a free particle is a geodesic. Synge however starts with the hypothesis that the path of a particle between collisions is a geodesic and that the proper mass is constant. The conservation equations are then deduced exactly from the law of conservation of momentum for collisions.

While engaged in a study of the Methodus Differentialis of Jas. Stirling (1730) I have been struck by the fact that Nicole's Papers on the same subject, printed in the Memoires de l'Academie Royale des Sciences (Paris), appear to form a fitting prelude to the work published by Stirling. The dates of Nicole's Papers are 1717, 1723, 1724, 1727, and it is almost certain that Stirling was well acquainted with their contents, for he remarks on page 24 of the Methodus Differentialis:—“Hac de re primus quod sciarn egit D. Taylor in Methodo Incrementorum. Eadem etiam fusius et elegantissime traditur a D. Nicol in Actis Academiae Regiae Parisiensis.”

The closed wedges in C(X) (the space of real continuous functions on a compact Hausdorff space X) which are also inf-lattices have been characterized by Choquet and Deny (2); see also (5). The present note extends their result to certain wedges of affine continuous functions on a Choquet simplex, the generalization being in the same spirit as the generalization of the Kakutani- Stone theorem obtained by Edwards in (4).

I should like to thank my supervisor, Dr D. A. Edwards, for suggesting this problem and for his subsequent help. I am also grateful to the referee for correcting several slips.

If F is a (commutative) field let denote the class of all groups G such that every irreducible FG-module has finite dimension over F. The introduction to [7] contains motivation for considering these classes and surveys some of the results to date concerning them. In [7] for every field F we determined the finitely generated soluble groups in . Here, for fields F of characteristic zero, we determine, at least in principle, the soluble groups in . Our main result is the following.

Let E be a real Hausdorff locally convex space with topological dual E′, topologised by the strong topology. Let (x, x′) denote the bilinear mapping defining the duality between E and E′ (x∈E, x′∈E′). By a unitary representation of E′ we mean an operator valued function U(x′) = Ux′. defined on E′, whose values are unitary operators in a separable Hilbert space H such that

In (8) Stonehewer referred to the following open question due to Amitsur: If G is a torsion-free group and F any field, is the group algebra, FG, of G over F semi-simple? Stonehewer showed the answer was in the affirmative if G is a soluble group. In this paper we show the answer is again in the affirmative if G belongs to a class of generalised soluble groups

The equation

need not have a solution z in the complex plane, even when ƒ is entire. For example, let ƒ(z) = ez, z1 = z0+2kπi. Thus the classical mean value theorem does not extend to the complex plane. McLeod has shown (2) that if ƒ is analytic on the segment joining z1 and z0, then there are points w1 and w2 on the segment such that where

The purpose of this article is to give a local mean value theorem in the complex plane. We show that there is at least one point z satisfying (1), which we will call a mean value point, near z1 and z0 but not necessarily on the segment joining them, provided z1 and z0 are sufficiently close. The proof uses Rouché's Theorem (1).

In the present paper we determine the Laplace transforms of the modified Bessel function of the second kind Kn(t±mx), where m is any positive integer. The Laplace transforms are given in (2) and (4) below, p being the transform parameter and having positive real part.

We prove that if t is a compact linear operator that is not quasi-nilpotent and is appropriately normalised, then the closed semi-algebra A(t) generated by t is locally compact. The theory of locally compact semi-algebras (2) is therefore applicable to A(t), and we show that it can be used to obtain spectral properties of t.

Definition. The isogonals of the medians of a triangle are called the symmedians

If the internal medians be taken, their isogonals are called the internal symmedians or the insymmedians; if the external medians be taken, their isogonals are called the external symmedians, or the exsymmedians

The purpose of this paper is to provide a vector version of the characterisations of the multipliers for L1(G) (see (9), p. 2). This problem was considered by Akinyele (1). However, the Banach algebras involved in that paper are commutative semi-simple Banach algebras and the proof of the main theorem seems to be incomplete. Indeed we give at the end of this paper an example which shows that statements (i) and (ii) of Theorem 3.2 of Akinyele ((1), p. 490) are not equivalent in general.

M. L. Cartwright has given ((2), 180–181) the following theorem, together with a neat proof of it.

Theorem C.Suppose that f(z) is regular and

in the half-strip S

of the complex plane.

Suppose also that for some constant α in α<a<β

as y→∞. Then for every δ>0.

uniformly asy→∞ for α+δ≦x≦β–δ.