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

My object in the following notes is to call attention to some points in integration by successive reduction which may be of use in directing the choice of the particular form for the reduced integral in any given case. The remarks apply chiefly to the binomial differential x(α + bxn)pdx which has been discussed from a different point of view by Dr Muir in vol. iii., p. 100 of the Proceedings.

The main result of this paper is the following

Theorem A. Let G be a π-separable finite group with Hall πsubgroup H. Suppose θεIrr(H). Then there exists a unique subgroup M, maximal with the property that it contains H and θ can be extended to a character of M.

McAlister proved that every regular locally inverse semigroup can be covered by a regular Rees matrix semigroup over an inverse semigroup by means of a homomorphism which is locally an isomorphism. We generalize this result to the class of semigroups with local units whose local submonoids have commuting idempotents and possessing what we term a ‘McAlister sandwich function’.

AMS 2000 Mathematics subject classification: Primary 20M10. Secondary 20M17

1. In the accompanying figure DEF is the pedal triangle of ABC and P, Q, R are the orthocentres of AFE, BDF, CED.

AP, BQ, CR evidently meet in the circumcentre, O, of ABC, which is the orthocentre of PQR,

2. Now the circumradius of AFE(ρa) = RcosA,

hence the sides of PQR are equal and parallel to the sides of DEF, i.e., the triangles are congruent, and their centre of perspective, L, bisects DP, EQ, FR.

A bilateral Laplace multiplier theory, based on Rooney's class , is developed for certain operators defined on the Fréchet spaces Dp,μ. The theory is applied to Riesz fractional integrals associated with the one-dimensional wave operator.

Recent work has shown that the solutions of the second-kind integral equation arising from a difference kernel can be expressed in terms of two particular solutions of the equation. This paper establishes analogous results for a wider class of integral operators, which includes the special case of those arising from difference kernels, where the solution of the general case is generated by a finite number of particular cases. The generalisation is achieved by reducing the problem to one of finite rank. Certain non-compact operators, including those arising from Cauchy singular kernels, are amenable to this approach.

[The substance of this paper will appear in the second volume of Professor Chrystal's Algebra.]

In analogy with a construction from function theory, we herein define right, left, and two-sided Bourgain algebras associated with an operator algebra A. These algebras are defined initially in Banach space terms, using the weak-* topology on A, and our main result is to give a completely algebraic characterization of them in the case where A is a nest algebra. Specifically, if A = alg N is a nest algebra, we show that each of the Bourgain algebras defined has the form A + K ∩ B, where B is the nest algebra corresponding to a certain subnest of N. We also characterize algebraically the second-order (and higher) Bourgain algebras of a nest algebra, showing for instance that the second-order two-sided Bourgain algebra coincides with the two-sided Bourgain algebra itself in this case.

The method of treating tangents to confocal conicoids, of which I propose to give an account, is discussed in the number for December 1867 of the Nouvelles Annales de Mathématiques. The writer of the paper referred to is M. Ph. Gilbert, Professor at the University of Louvain. The results arrived at are in nearly every case already well known, but the method of reaching them is somewhat novel, and I have thought that it might interest the members of this society if I were to give a statement of the chief methods and results of Gilbert's paper. In one or two cases I have altered the proofs, and I have added two or three propositions that seemed to follow naturally from the equations dealt with. Gilbert deals only with central conicoids; but I have put in an equation for the paraboloids that corresponds to Gilbert's fundamental one.

It is proposed to establish, by elementary methods, a theorem for matrices analogous to Fermat's Theorem in the Theory of Numbers. In Jordan's Traité des Substitutions (Paris, 1870) pp. 127, 128, the order of any given linear substitution or matrix A with reference to any prime number p is determined, but the result given depends on the particular characteristic equation satisfied by the matrix A, and a general result applicable to all matrices of n rows and n columns does not seem to have been published hitherto.

We consider the one-dimensional operator,

on 0<x<∞ with . The coefficients p, V1 and V2 are assumed to be real, locally Lebesgue integrable functions; c1 and c2 are positive numbers. The operator L acts in the Hilbert space H of all equivalence classes of complex vector-value functions such that . L has domain D(L) consisting of ally∈H such that y is locally absolutely continuous and Ly∈H; thus in the language of differential operators L is a maximal operator. Associated with L is the minimal operator L0 defined as the closure of where is the restriction of L to the functions with compact support in (0,∞).