The radioactivity of potassium was discovered by Campbell and Wood (1907). Since the β-ray activity shown by all the potassium salts investigated was very small, the suspicion naturally arose that this activity was due to a trace of radioactive impurity and not to the potassium itself. All tests for the presence of an emanation gave, however, negative results, and it became necessary to conclude that the observed emission of the β-radiation was indeed a property of the potassium atom. The subsequent investigations of Campbell (1907) and of Hoffman (1924), whose attempts to alter the activity by chemical operations were unsuccessful, supported this conclusion, as did also the work of Biltz and Marcus, who prepared specimens of potassium sulphate from potassium minerals of different geological ages and found that the activity of the preparations per atom of potassium was always the same.

In this paper the authors investigate two classes of triple series equations of trigonometric type. Closed form solutions are given for eight triple series in each class.

Margulis has given conditions under which a lattice in a semisimple Lie group admits the structure of an arithmetic subgroup. We show that these arithmetic structures are unique. The result is not subject to the condition “rkR(G) ≧ 2” required by the Margulis result. In the lowest dimensions, the result has previously been observed by Takeuchi, Maclachlan and Reid.

In a previous publication we investigated certain idempotent residuated mappings and showed how these could be used to provide a solution to the problem of finding a Baer semigroup coordinatisation of bounded modular lattices. Here we use essentially the same idempotents to provide a coordinatisation of bounded distributive lattices. Specifically, we prove that a bounded lattice L is distributive if and only if it can be coordinatised by a Baer semigroup S such that if eS, fS, gS ∈ R(S) with eS ∩ fS = eS ∩ gS then there are idempotents ē, , ∈ S such that ēS = eS, S = fS, S = gS and ē commutes with both and .

Included in the area of distribution of Salmo salar are the western coasts of Europe as far south as the Franco-Spanish border as well as the British Isles and Iceland, and, in addition, the eastern coast of Canada and the United States down to the State of Maine. A very large number of investigations have been made in Great Britain and various European countries, both by marking the fish in order to trace their subsequent growth and movements, and by reading their age and history from the scales. Length calculations from scale measurements have also been made in Scotland, Norway, and Sweden.

The positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.

Complex and chaotic structures in certain dynamical systems in biology arise as a consequence of noncomplete integrability of two-degree-of-freedom Hamiltonian systems. A study of this problem is made using Ziglin theory and implemented with the aid of the Kovacic algorithm.

The preparation of 1-benzhydrylnaphthalene, 1, 2, 3, 4-tetrahydro-1-benzoylnaphthalene, and 1-diphenylmethylenetetralin is described.

Thanks are expressed to the Department of Scientific and Industrial Research for a maintenance grant (to D. K.); to the Royal Society of London for a grant (to A. J. P.); to Miss E. W. Robertson for help with the experimental work; and to the Education Committee of Stirlingshire for permission to use laboratory facilities.

Several properties of symmetric matrices and positive definite matrices are derived. These are used to improve an estimate given for regular solutions of the n-metaharmonic differential equation by Chow and Dunninger [1].

It is shown that eigenvalues of infinite multiplicity can exist for the Schrödinger equation holding in the whole N-dimensional space RN(N ≧ 2). In the example which is constructed, the potential is separable and bounded in RN, and the method is an application of inverse spectral theory.

The singular limit as one diffusion coefficient approaches zero is considered for travelling wave solutions to a pair f reaction diffusion equations. An explicit criterion determining the sign of the wave speed is obtained. The limit behaviour turns out to be of a different nature for positive and negative wave speed. Different techniques, which may be applicable to a range of examples, are needed in the two cases.

The behaviour of heat transfer and skin friction is analysed in a compressible laminar boundary layer with external velocity Ue(x)(l + α sin ωt The Mach number M is assumed small but finite so that high frequency flows (s ≫1) in which c =αγM2s/2 = O(1) are considered. Solutions, obtained by matching in the Stokes and Prandtl layers, involve summation of Fourier-like series to give the dominant terms in the heat transfer and skin friction. Results, for c =½, verify that a previous approximate method gives a reasonable description of unsteady heat transfer and skin friction; forc =1 there is a substantial increase in amplitude of heat transfer but little change of phase.

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterization of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well-known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

Stable homotopy decompositions of the classifying spaces of the gauge groups of principal SO(3) and U(2)-bundles over the sphere S2 are obtained using a fibrewise stable splitting theorem for the loop space of an unreduced suspension. The stable decomposition is related to a description of the integral cohomology ring.

Recently, Sarrión and the authors gave a sufficient condition on invertible Lamperti operators on Lp which guarantees the convergence in the Cesàro-α sense of the ergodic averages and the ergodic Hilbert transform for all f ∈ Lp with p > 1/(1 + α) and −1 < α ≤ 0. The result does not hold for the space L1/(1 + α). In this paper we give a positive result for the smaller Lorentz space L1/(1 + α),1.

Recently, Sun posed a series of conjectures on the log-concavity of the sequence , where is a familiar combinatorial sequence of positive integers. Luca and Stănică, Hou et al. and Chen et al. proved some of Sun's conjectures. In this paper, we present a criterion on the log-concavity of the sequence . The criterion is based on the existence of a function f(n) that satisfies some inequalities involving terms related to the sequence . Furthermore, we present a heuristic approach to compute f(n). As applications, we prove that, for the Zagier numbers , the sequences are strictly log-concave, which confirms a conjecture of Sun. We also prove the log-concavity of the sequence of Cohen–Rhin numbers.

Let W be a 2-dimensional Coxeter group, that is, one with 1/mst + 1/msr + 1/mtr ≤ 1 for all triples of distinct s, t, r ∈ S. We prove that W is biautomatic. We do it by showing that a natural geodesic language is regular (for arbitrary W), and satisfies the fellow traveller property. As a consequence, by the work of Jacek Świątkowski, groups acting properly and cocompactly on buildings of type W are also biautomatic. We also show that the fellow traveller property for the natural language fails for $W=\widetilde {A}_3$.