1. The aim of this note is to prove the

Theorem. Let

where the λnare real and

and let

Then

A similar result holds for infinite seriesconverging uniformly in [−T, T].

In a recent paper, we gave a topological description of Colombeau type algebras introducing algebras of sequences with exponential weights. Embeddings of Schwartz spaces into the Colombeau algebra $\G$ are well known, but for ultradistribution and periodic hyperfunction type spaces we give new constructions. We show that the multiplication of regular enough functions (smooth, ultradifferentiable or quasianalytic), embedded into corresponding algebras, is the ordinary multiplication.

In this paper we bring together two aspects of homological algebra, one modern and the other classical. MacLane(6) has suggested the desirability of avoiding superfluous references to the objects of the category in proving fundamental results in homology theory. Accordingly, we consider categories whose maps form a ringoid in the sense of Barratt(1) and impose on the ringoid axioms, expressed in the language of ring theory, but having a close affinity to the axioms of Buchsbaum(2) for an exact category. (See also Cartan and Eilenberg(3); Appendix.) The purpose of these axioms is to ensure that the standard concepts of homology theory are applicable to the ringoid. Consequently, we call such a ringoid a homological ringoid.

By means of a quaternion algebra over $\mathbb{F}$q(t), we construct an infinite series of torsion free, simply transitive, irreducible lattices in PGL2($\mathbb{F}$q((t))) × PGL2($\mathbb{F}$q((t))). The lattices depend on an odd prime power q = pr and a parameter τ ∈ $\mathbb{F}$q×, τ ≠ 1, and are the fundamental group of a square complex with just one vertex and universal covering Tq+1 × Tq+1, a product of trees with constant valency q + 1.

Our lattices give rise via non-archimedian uniformization to smooth projective surfaces of general type over $\mathbb{F}$q((t)) with ample canonical class, Chern numbers c12 = 2 c2, trivial Albanese variety and non-reduced Picard scheme. For q = 3, the Zariski–Euler characteristic attains its minimal value χ = 1: the surface is a non-classical fake quadric.

Let Nn be the number of real roots of a random algebraic equation The coefficients ξν are independent random variables identically distributed with expectation zero; the variance and third absolute moment are finite and non-zero. It is proved that

where εν tends to zero, but εν log n tends to infinity. The measure of the exceptional set tends to zero as n tends to infinity.

The very small water drops which are considered in this note are as a rule met with in the form of clouds with a high numerical density of drops. The interior of such a cloud will never be in temperature equilibrium with the walls of a containing vessel, and under these conditions measurements of the rate of fall of the top of a cloud are not measures of the Stokes free fall of a single drop. Optical methods of measurement assume, therefore, a particular importance. The angular diameter of the diffraction halo has long been used as a measure of drop size in clouds; the applicability of this method for small drops will be considered. The criterion of “Rayleigh scattering” has also been employed as showing that the diameter of the drops in question was very much smaller than the wave length of light. A third method which may be applied to drops of diameter of the order of one wave-length will be discussed.

In (5), Whitehead introduced a generalization of the Hopfinvariant in the form of a homomorphism H: πr(Sn) → πr(S2n−1), valid† if r < 3n − 3. In (2), the author extended Whitehead's definition to cover the cases r ≤ 4n − 4; r = 5, n = 2; r = 13, n = 4. He also defined, without restriction, a homomorphism

which was related to H, whenever H was defined, by the equation H* = FH, F being the Freudenthal suspension

In [6] and [9] the second author and Bill Richter showed that the natural ‘degree’ filtration on the homology of ΩSU(n) has a geometric realization, and that this filtration stably splits (as conjectured by M. Hopkins and M. Mahowald). The purpose of the present paper is to prove the real and quaternionic analogues of these results. To explain what this means, consider the following two ways of viewing the filtration and splitting for ΩSU(n). When n = ∞, ΩSU = BU. The filtration is BU(1)⊆BU(2)⊆… and the splitting BU≅ V1≤<∞is a theorem of Snaith[14]. The result for ΩSU(n) may then be viewed as a ‘restriction’ of the result for BU. On the other hand there is a well-known inclusion ℂPn−1. This extends to a map ΩΣℂPn−1→ΩSU(n), and the filtration (or splitting) may be viewed, at least algebraically, as a ‘quotient’ of the James filtration (or splitting) of ΩΣℂPn−1. It is now clear what is meant by the ‘real and quaternionic analogues’. In the quaternionic case, we replace BU by BSp=Ω(SU/SP), ΩSU(n) by Ω(SU(2n)/SP(n))and ℂPn−1 by ℍPn−1. The integral homology of Ω(SU(2n)/SP(n)) is the symmetric algebra on the homology of ℍPn−1, and may be filtered by the various symmetric powers. We show that this filtration can be realized geometrically, and that the spaces of the filtration are certain (singular) real algebraic varieties (exactly as in the complex case). The strata of the filtration are vector bundles over filtrations of Ω(SU(2n−2)/SP(n−1)), and the filtration stably splits. See Theorems (1·7) and (2·1) for the precise statement. In the real case we replace BU by Ω(SU/SO), Ω(SU(n)/SO(n)) and ℂPn−1 by ℝPn−1. Here integral homology must be replaced by mod 2 homology, and splitting is only obtained after localization at 2. (Snaith's splitting of BO in [14] can be refined [2, 8] so as to be exactly analogous to the splitting of BU:BO≅V1≤<∞MO(k).)

In 1922–23, Julia and Fatou proved that any 2 rational functions f and g of degree at least 2 such that f(g(z)) = g(f(z)), have the same Julia set. Baker then asked whether the result remains true for nonlinear entire functions. In this paper, we shall show that the answer to Baker's question is true for almost all nonlinear entire functions. The method we use is useful for solving functional equations. It actually allows us to find out all the entire functions g which permute with a given f which belongs to a very large class of entire functions.

An ideal of an integral group ring is divisible by a given integer if all of its elements share this common factor; the ideals most often encountered are rarely divisible in this sense. Only in the case of finite p-groups are powers of the augmentation ideal of the integral group ring ever divisible. For every e, there is some n for which the nth power of the augmentation ideal is divisible by pe. The smallest such integer n arises in many contexts; this paper describes its properties and interpretations.

For a class of kernels, we prove the Lp estimate for the exotic Riesz potential, with which a restriction theorem of the Fourier transform to surfaces of half the ambient dimension is proved. A simpler proof of Barcelo's result is given. We also find that it is possible to combine the Hausdorff–Young theorem with the Fefferman–Zygmund method to prove some optimal results on the restriction theorem.

We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some “Poisson-type” anti-concentration theorems that give bounds of the form 1/e + o(1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős–Littlewood–Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.

Some previous work on the theory of the electrical counter is extended by a detailed consideration of the ionic process involved. It is shown that with a positive point the total number of ions produced should be proportional to the initial ionisation. Evidence is adduced to show that with a negative point extinction of the discharge is probably brought about by a local increase of pressure, accompanied by an electrical wind along the sides of the point. The latter would bring about the discharge of a definite quantity of electricity, independent of the initial ionisation, provided the initial pressure was not too low. It is shown that these considerations account for a number of other phenomena connected with discharges from small electrodes, and that the quantities involved are of the right order of magnitude.

The experimental work in connection with this paper has been carried out in the Cavendish Laboratory. The author desires to thank Professor Sir E. Rutherford for his continual interest in its progress, and further to thank Professor Appleton, and Dr Chadwick for much helpful discussion of several of the points involved.

Let $G_{\bb C}$ be the algebraic group over ${\bb C}$ corresponding a simply connected complex Lie group $G$. The algebraic cobordism $\Omega^*(G_{\bb C)$ defined by Levine and Morel is computed explicitly.

In (1), Carter and Hawkes define the -normalizers of a finite soluble group for any saturated formation . These subgroups are conjugate, invariant under homomorphisms of the group, cover and avoid the chief factors of the group and may be characterized by means of the maximal chains of subgroups connecting them to the group. The first aim of the present paper is to generalize both the -normalizers and the relative system normalizers (Hall (5)) of a finite soluble group G. We choose an arbitrary normal subgroup X(p) of G for each prime p dividing the order of G, forming a normal system = {X(p)} of G. Each of these normal systems of G yields a conjugacy class of subgroups, called the -normalizers of G, which possess the above properties of -normalizers. However, they do not satisfy all the properties of the -normalizers unless is a so-called integrated normal system of G.

On certain assumptions as to the quantum numbers of the normal orbits of the successive electrons captured by an atom, the successive ionisation potentials can be estimated by making use of the relations between terms of the same quantum number for different atoms of the same electronic structure. Formulae for several of the successive ionisation potentials are given, the form of the formulae being suggested by theoretical reasoning and numerical values of the constants determined from known spectra; the successive ionisation potentials of O, Fe and Ag, calculated from these formulae, are tabulated. It is pointed out how far different these ionisation potentials are from the X-ray excitation potential of the same orbit of a neutral atom.

In some recent work Kodis* considers the scattering of a plane harmonic wave by a circular cylinder and uses variational methods to deduce an asymptotic formula for the scattering coefficient in the case of high-frequency waves. The scattering coefficient, σ, is denned as the total energy flux outward from the cylinder in the scattered wave divided by the energy flux in the beam of the incident wave which falls on the cylinder. In the limit, of geometrical optics, the scattered wave is made up of a wave reflected from the forward half of the cylinder with energy flux equal to that in the incident beam, and a wave behind the cylinder cancelling the incident wave there to form the shadow. Thus σ → 2 as ka → ∞, k being the wave number of the incident wave and a the radius of the cylinder. The next term in the asymptotic expansion for σ is proportional to . It has been determined already from the exact series expression for the field by White (6), Wu and Rubinow (7) and Kear (3). But Kodis gives an approximate method which also supplies this result and, since the coefficient is in good agreement with the values obtained from the exact solution, concludes that his approximate procedure could also be used for more general obstacles. In the present paper, we propose an alternative approximate method which is related to the one given by Kodis but is very much simpler. Also we go on to obtain the correction terms for general obstacles in detail. Various writers (see, for example, Keller (4) and van de Hulst(8)) have suggested previously that the correction is still proportional to , where a is some length defined by the body shape. However, the determination of a and of the numerical coefficient has been limited to two-dimensional obstacles.

1. It is well known that a game with perfect information has an equilibrium-point of pure strategies; this was first proved for two-person games by Zermelo (9), and later extended to n-person games by Kuhn(3). More recently, Dalkey(1) and Otter and Dunne (8) have published the stronger result (Theorem 6 below): If in the complete inflation of a game Γ every player has complete information about every other player, then Γ has an equilibrium-point of pure strategies.

The investigation of the isotropic scattering of radiation in an infinite spherically symmetric atmosphere, which was carried out in a previous paper, is extended to a finite atmosphere bounded by a totally absorbing shell.

It is shown that the solution may be expressed as a series which will certainly converge when the radius of the shell is large enough, and whose terms are found by iteration.

Numerical investigation shows that even for quite modest values of the radius, the convergence is so fast that for most practical purposes all but the first term can be neglected. Except at the very ends of the range, a very simple expression then gives an excellent approximation.