We show that under quite general circumstances a Banach algebra R need not be closed in its tilda algebra . In answer to a query of Katznelson we show that there is a very strongly homogeneous algebra B ≠ C0(R) on R with R as carrier space such that non-analytic functions operate on B.

Let ∥x∥ denote distance to a nearest integer. Heilbronn's theorem in Diophantine approximation [8] may be stated as follows.

We study the prime ideal structure of the Iwasawa algebra $\Lambda_G$ of an almost simple compact $p$-adic Lie group $G$. When the Lie algebra of $G$ contains a copy of the two-dimensional non-abelian Lie algebra, we show that the prime ideal structure of $\Lambda_G$ is somewhat restricted. We also provide a potential example of a prime c-ideal of $\Lambda_G$ in the case when the Lie algebra of $G$ is ${\frak s}{\frak l}_2(\mathbb{Q}p)$.

Soit $f$ une fonction arithmétique. Posons $$m(n;f) \,{\coleq}\, \min_{d|n}\!{\Vert}f(d){\Vert}\qquad(n\,{\geq}\,1),$$ oú ${\Vert} u {\Vert}$ désigne la distance du nombre réel $u$ à l'ensemble des entiers. Le comportement normal de $m$($n$;$f$) est une mesure de la nature probabiliste de l'ensemble des diviseurs d'un entier [Lt ]statistique[Gt ]. Une hypothèse standard d'équirépartition conduit à l'évaluation $$m(n;f) = 1/\tau(n)^{1+o(1)}\quad {\rm pp}$$, oú $\tau$($n$) désigne le nombre total des diviseurs d'un entier naturel $n$, et oú, ici et dans la suite, nous utilisons la mention pp (presque partout) pour indiquer qu'une propriété relative à un entier générique est valable sur une suite de densité naturelle unité.

1. Pólya's walk in the Euclidean plane is the motion of a particle, which takes successive independent random steps of unit length parallel to the coordinate axes, the probabilities of each of the four possible directions at each step being ¼. We imagine the particle starts at the origin, and that, in taking the ith step (i = 1,2,…) from (xi-1, yi-1) to (xi, yi), it leaves a straight track of unit length between these two points. Let Tn denote the track thus marked out in the first n steps; so that Tn is a connected chain of n straight segments each of unit length. Define the interior of Tn to be the set of points, which neither lie on Tn nor can be connected to the point at infinity without crossing Tn at least once; and let An denote the area of the interior of Tn. Associated with each walk ω, there is a sequence of random numbers {An(ω)}; what can be said about the behaviour of this sequence, especially for large n? We shall proveTheorem 1. For all ω,0 ≤ An(ω) ≤ 1/16 n2.

The aim of this note is to construct generators for II*(MU). Generators of a similar type were constructed by various authors; Hazewinkel (2), Kozma (3) and Liulevicius (4). The construction in (4) is in fact for II*(BP) at the prime 2.

In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$-type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.

In 1974, Erdős posed the following problem. Given an oriented graph H, determine or estimate the maximum possible number of H-free orientations of an n-vertex graph. When H is a tournament, the answer was determined precisely for sufficiently large n by Alon and Yuster. In general, when the underlying undirected graph of H contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of F-free graphs. As the main contribution of the paper, we resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to Erdős’s question. Moreover, we determine the answer exactly when H is an odd cycle and n is sufficiently large, answering a question of Araújo, Botler and Mota.

A point in Baire space is found for which the first derived $\omega$-limit set is not Borel, whilst the second is empty. A second point is found for which the sequence of derived $\omega$-limit sets does not stabilise until the first uncountable ordinal. The two points are recursive.

The most outstanding discrepancy between experiment and Dirac's theory of the electron is at present that shown by Dymond's experiments on the polarisation of beams of electrons by double scattering. It has been shown by Mott, using Dirac's relativistic wave equation, that one would expect a 15% asymmetry of beams of 140 k.v. electrons scattered twice at 90° by gold. The experiments of Dymond referred to above show that there cannot be more than 1% asymmetry. Several attempts have been made to explain this discrepancy, but without success. (For an account of these see Dymond's paper II.) One possibility which has not yet been investigated is that of exchange between the incident electron and the electrons of the struck atom. It is evident that only elastic exchange need be considered, since the probability of excitation of any form is known to be small. Only exchange with the K and L shells may be expected to give any appreciable effect.

A simple explanation of Pauli's principle was first given with the wave mechanics. Its interpretation in the new theory was that the wave functions of Schrödinger were antisymmetrical in all the electrons concerned. Thus when the interactions of the electrons may be neglected, the wave function (for a system of n electrons) can never be of the form

in nature, but only of the form

Hence σ, τ,…ω must be all distinct.

In paper(1) it has been proved that

where the Jacobi polynomial is denned as ((3), p. 255)

Suppose that X is any vector space on which it is possible to recognize a class of sets of such a nature that it is natural to call them ‘bounded’. (Precise conditions for such a class of sets are given in § 2.) Let L be any vector space of linear functionals on X which map each ‘bounded’ set into a bounded set. We say that a filter is boundedly-weakly convergent if it is convergent in the weak topology of the linear system [X, L] and contains a ‘bounded’ set. If M is a vector space of boundedly-weakly continuous linear functionals on X which includes L, we say that a subset S of M is limited if 〈 , f〉 converges uniformly for f ε S whenever is a boundedly-weakly convergent filter.

Necessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

Much attention has been focused on the excitation of atomic spectra by electrons whose velocities do not greatly exceed the critical value, and it is now possible to form a rough idea of the behaviour of the “excitation function,” or probability that an electron of given velocity shall excite on impact, when the energy of the electron is only slightly greater than that required for excitation. But the form of this excitation function for much higher velocities is still in considerable doubt.

The possibility is explored that the general static solution of Einstein's vacuum field equations is given in terms of a single harmonic function, i.e. solution of the flat space Laplace equation. It is shown that coordinate conditions can be chosen such that one of Einstein's equations reduces to Laplace's equation. To examine the extent to which the metric is determined by the resulting harmonic function ø, a power series expansion is considered in which in the first non-trivial order one gets a system of eight equations for six unknowns. This is reduced to a system of three equations for three unknowns. This is some indication that the coordinate conditions are consistent and shows the sense in which, to this order, the metric is determined by ø. This system of three equations is then solved explicitly for a particular choice of ø, thereby giving an approximate non-axisymmetric solution of Einstein's vacuum equations. This solution is asymptotically flat and could represent the approximate gravitational field of a non-axisymmetric bounded rigid body. The form of a class of time-dependent metric related to the general static metric is discussed.

The general solution of Chaplygin's equation for the motion of an inviscid compressible fluid is obtained for ‘simple wedge flows’. Some particular examples are given and a discrepancy in earlier Russian papers explained. After a discussion of the properties of the velocity potential and the expression for the physical coordinates in terms of the hodograph variables, a general theorem on sonic jets is proved, namely, that the physical changes due to the presence of solid boundaries in the flow are completed within a finite distance in those directions in which the flow becomes finally a sonic jet.

We give a precise description of combed trees in terms of Kelly–Mac Lane graphs. We show that any combed tree is uniquely expressed as an allowable Kelly–Mac Lane graph of a certain shape. Conversely, we show that any such Kelly–Mac Lane graph uniquely defines a combed tree. We show how to apply this to the construction of opetopes.