Using a criterion due to J. Tits [4] we shall show that it is easy to give a pair of conjugate matrices in GLn(ℂ) which freely generate a free group of rank two. The difficulty lies in producing two such matrices whose freeness does not follow directly from the known freeness of two 2 × 2 matrices. Finally we show that the well known problem as to whether the Burau representation of Artin's braid group B4 is faithful turns out to be arbitrarily close to being true in a large number of ways. Thanks are due to the referee for his helpful comments.

1. This paper is concerned with certain properties of the solutions of differential equations of the type

where f, g have continuous partial derivatives up to the second order satisfying Lipschitz conditions in some bounded domain and are periodic in t with period 2π, ω is a positive constant and k is a small parameter. If f = 0, (1·1) is equivalent to

an equation representing the forced vibrations of a quasi-linear system of one degree of freedom.

Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. For any homotopy given by a composite homotopy data triple we define a complete invariant of nanophrases. This invariant is used to show that equivalence of nanophrases under such a homotopy can be calculated just by using the homotopies given by its prime factors.

Introduction. In this paper, a study is made of the solutions of Lamé's differential equation as series of associated Legendre functions. The particular feature studied is the representation of the second solution corresponding to the case when the first solution is a Lamé polynomial.

1. The conductivity of liquid sulphur is found to vary with the temperature in a similar manner to that in which the viscosity varies.

2. The conduction is electrolytic in character.

3. There is a definite E.M.F. of polarisation.

The study of differential properties of functions of two real variables in the light of the modern theory of functions of real variables was started by H. Rademacher in 1919. Although much work has since been done on the subject, and a great many general results have been obtained, a number of questions have remained open, some of which will be discussed in the present work.

In this paper I find the critical lattices of the region Kt defined by the inequalities | xy | ≤ 1, x2 + y2 ≤ t (t > 0). I deduce in §9 the arithmetic minima of a pair of binary quadratic forms

where f1 is indefinite, f2 is positive definite, and f1, f2 are harmonically related, that is to say

and

Measurements have been made on the proportion of the effect induced in silver which is due to neutrons strongly absorbed by cadmium, using cylinders of water and boric acid of different sizes to slow down the neutrons. In addition to furnishing data for use in the foregoing paper, the results show that practically all the neutrons of groups A and B must have energies greater than any of those of group C.

The author desires to thank Lord Rutherford for his interest in the work and for the use of a Ra + Be source, and Dr Oliphant for discussion and advice. He is also indebted to the Department of Scientific and Industrial Research for a Senior Research Award.

Let $\mathcal{M}$ denote the mapping class group of Σ, a compact connected oriented surface with one boundary component. The action of $\mathcal{M}$ on the nilpotent quotients of π1(Σ) allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of $\mathcal{M}$, called the Lagrangian filtration. He also introduced a version of the Johnson homomorphisms for this new filtration. The first term of the Lagrangian filtration is the Lagrangian mapping class group, whose definition involves a handlebody bounded by Σ, and which contains the Torelli group. These constructions extend in a natural way to the monoid of homology cobordisms. Besides, D. Cheptea, K. Habiro and G. Massuyeau constructed a functorial extension of the LMO invariant, called the LMO functor, which takes values in a category of diagrams. In this paper we give a topological interpretation of the upper part of the tree reduction of the LMO functor in terms of the homomorphisms defined by J. Levine for the Lagrangian mapping class group. We also compare the Johnson filtration with the filtration introduced by J. Levine.

The relation between the reaction velocity of catalytic dehydrogenation and the rate of passage of alcohol vapour over the catalyst has been found by experiment. At low rates of flow the concentration of the aldehyde and hydrogen in the effluent gas, and therefore over the catalyst, is comparable with that of the reactant. The whole of the surface is thus not covered with the reactant, and the adsorbed reaction products cause marked decrease in the reaction velocity.

When the concentration of the reaction products becomes sufficiently small, that is, when the rate of passage of the alcohol vapour exceeds some limit, the velocity is approximately independent of the rate of flow of reactant. With very high speeds the reaction velocity tends to fall again, but this is probably due to the difficulty of keeping the temperature of the catalyst constant.

The formula previously deduced connecting the partial pressure of the reactant in a mixture with the fractional reaction velocity has been verified for mixtures of alcohol vapour with water, acetone and benzene; and the composition of the adsorbed gas film with the fractional partial pressure of the gases for mixtures of carbon monoxide with oxygen on platinum.

The effect of mixture of vapours on the temperature coefficient of reaction is considered, and equations deduced to show the alteration produced. If the heat of desorption of the reactant is greater than that of the diluent, then the temperature coefficient is diminished, if they are equal there is no alteration, and if less the temperature coefficient is increased. Experiments show that water present in alcohol has little effect on the temperature coefficient of the reaction.

Introduction. Let be the complex separable Hilbert space. We say that the closed linear operator T, with domain dense in. , is represented by the infinite matrix H if T is the operator T˜1(H) defined† by H (with respect to some complete orthonormal set). We define an (nJ)-matrix as a Hermitian matrix H = [hij]i, j ≥ 1 for which hij = 0 when i − j > n and hij ╪ 0 when i − j = n. (Thus a Jacobi matrix is a (1J)-matrix.) If, in addition, hij = 0 when 0 < i − j < n, we call H an (nJ ┴)-matrix.

The algebraic invariants of the Riemann tensor for a static axially symmetric gravitational field are examined. Contrary to common expectation they are shown to be well defined in a region of the field known to be physically singular.

Energy of the exterior and the interior fields of a particle, charged or otherwise, embedded in an empty space is calculated, using Møller's tetrad formulation of the energy-momentum complex in general relativity. It is found that the exterior field contains positive gravitational energy, and in addition, the electromagnetic energy if the particle is a charged one and that the sum total of the energies of the interior and the exterior fields is equivalent to the Newtonian mass of the particle.

1.Introduction. To every closed subalgebra S of a regular algebra R(D) of functions tending to zero at infinity on a discrete space D we define the Rudin spectrum of S to be the partition of D consisting of the equivalence classes of the equivalence relation .

It is well known that the operators mainly employed in quantum theory are hermitian; it is less well known amongst physicists that they are required, in addition, to be self-adjoint. This is essential for the validity of the result known in quantum theory as the representation theorem and in the mathematical theory as the resolution of the identity. The purpose of this paper is to show that the self-adjoint operators can be characterized by a condition which is nearer to having a physical significance than those given in the literature.

In a finite projective plane S2, q of order q, a (k, n)-arc is a set of k points, of which some n, but no n + 1 are collinear. Barlotti (1) proved that if (n, q) = 1 then

For α a positive irrational, let 𝛢α be the subalgebra of continuous functions on the two-torus whose Fourier transform vanishes at (m, n) if m+αn < 0. These algebras were studied by Wermer and others, who proved properties such as maximality and characterised the Gelfand space. One of the major themes of current work in operator algebras is classification, but none of the properties which were investigated earlier distinguished between 𝛢α and αβ, if β is another positive irrational. We address this question. We also determine the automorphism group of 𝛢α.

In this paper, we consider the energy behaviour of Rayleigh surface waves in a homogeneous isotropic 3-dimensional half space with free boundary condition. It is known that Rayleigh waves with finite energy are asymptotically concentrated along the boundary. We clarify the domain where the energy of the Rayleigh waves concentrates and obtain the decay order of the energy distributed in the complement of the domain.

In a recent paper Ragab and Simary (6) have deduced some integrals involving the products of generalized Whittaker functions. We feel that some of the integrals are not correctly evaluated. Indeed the integral (2) should be symmetrical in α and β, as we see by changing the variables by s = − t in the integral, while it is not symmetrical. A similar remark will follow in the case of the integrals (3) and (6).

Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$. Let $(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence $(a_n)$ is positive.