Let S be a semigroup with a compact topology in which multiplication is continuous on the left (i.e. xi→x implies xiy→xy for each y in S). Then S has a minimal left ideal L which is compact; each idempotent e in L is a right identity for L (xe = xfor each x ∈ L)and L = Se; Ge = eL is a group and L is the union of all such groups; and if f is a second idempotent in L, the canonical map x ↦ fx of Ge to Gf is an algebraic isomorphism (see Ruppert(2) for these facts). Baker and Milnes(1), §4(A), have observed that, in the case in which S is the Stone–Cech compactification of a discrete abelian group, the canonical map from Ge to Gf may not be a homeomorphism. (This contrasts with the situation in compact semigroups with separately continuous multiplication.) We present a simple proof of a more definitive result.

The following theorem is proved. For any positive integer n there exists t depending only on n such that if the word w is a multilinear commutator, then the class of all groups G in which w(G) is locally finite and the product of any t values of w has order dividing n is a variety.

The purpose of this note is to draw attention to a certain correspondence between the melting-points of normal paraffins and of fatty acids and to indicate a simple interpretation of this phenomenon. If the number of carbon atoms in a normal paraffin is plotted against the corresponding melting temperature, all the points in the diagram lie very close to a smooth curve drawn among them. If a similar diagram is made for the fatty acids, the points corresponding to an even number of carbon atoms lie on a curve of the same character as that found for paraffins. The points for an odd number of carbon atoms lie on a separate but similar curve, exemplifying the well-known alternation property.

We formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.

In the theory of functions of a complex variable a positive quantity g(r), denned in terms of a circle |z| = r, can often be estimated by means of another similar quantity f(R) denned in terms of a circle |z| = R with R > r. Inequalities of the form

arise in this connexion. Putting R = r + h we have for every h > 0

.

In [10] it was shown that the element Sq2n has nilpotence height 2n+2 in the mod2 Steenrod algebra. Here the method and result are generalised to show that for an odd prime p the element Ppn has nilpotence height p(n+1) in the modp Steenrod algebra. Some of the other results of [10] are also generalised to odd primes p.

A technique is developed for improving the speed and range of convergence of iteration procedures. The method is shown to include, as special cases, Newton's method and Aitken's δ2-acceleration process.

Let the eight points in space common to three arbitrary quadric surfaces be called eight associated points. These points possess the property that any one of them is uniquely determined if the other seven are given. It is interesting to give the analytical expressions for eight associated points in terms of symbols entirely free from coordinate systems. Thus we take the first eight digits 1, 2,…8 to denote the points; a group of two digits to denote the line joining these two; a group of three to denote a plane, and so on. Also a conjunction of two groups denotes the point, line or plane common to the two groups, thus (34, 678) is the point common to the line 34 and the plane 678.

This paper asks: given a vector bundle ξ and a line bundle λ over the same base space, are λ[otimes ]ξ and ξ equivalent? We concentrate on real bundles ξ. Although the question is sensible in its own right, we explain in Section 2 our immediate motivation for studying it. In Section 3 we make some general comments about the question, the most significant being that under certain restrictions the answer depends on the stable class of ξ rather than on ξ itself (Proposition 3·4).

The rest of the paper tackles an interesting special case. To state the main result, let P(ℝn+1) denote n-dimensional real projective space, H the Hopf line bundle over it, and an+1 the order of the reduced Grothendieck group [wavy overbar]KO0(P(ℝn+1)).

This paper is a continuation of (6). We will use the definitions and notation of (6) freely. We continue our study of the restrictions imposed on the mod p cohomology of finite H-spaces by the requirement that is a U(M) algebra. As in (6) we will restrict our attention to the case where p is an odd prime. As we observed in (6), the restriction proved there, that the pth power map on is trivial, may not really require the hypothesis of . However, the restrictions proved in this paper are not valid without the assumption that . Indeed, they actually characterize the known mod odd finite H-spaces which have U(M) algebras.

Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K by

L can be viewed as a module over the ring , and a fortiori over the group ring KG.

Let {α} denote the fractional part of the real number α. Write χ(x, y) = 1 for {x} < y and χ(x, y) = 0 otherwise. A real sequence (xn) is uniformly distributed (mod 1) if

It is a consequence of (1·1) that DN = o(N),

Where

is the discrepancy of the sequence (xn).

Let Σ be a closed surface of genus ≥ 1, G a compact Lie group, not necessarily connected with Lie algebra g, ξ,: P → Σ a principal G-bundle, and suppose Σ equipped with a Riemannian metric and g with an invariant scalar product so that the Yang—Mills equations on ξ are defined. Further, let

be the universal central extension of the fundamental group π of Σ and ΓR the group obtained from Γ when its centre Z is extended to the additive group R of the reals. We show that there are bijective correspondences between various spaces of classes of Yang—Mills connections on ξ and spaces of representations of Γ and ΓR (as appropriate) in G. In particular, we show that the holonomy establishes a homeomorphism between the moduli space N(ξ) of central Yang–Mills connections on ξ and the space Repξ(Γ, G) of representations of Γ in G determined by ξ. Our results rely on a detailed study of the holonomy of a central Yang–Mills connection and extend corresponding ones of Atiyah and Bott for the case where G is connected.

Consider the eigenvalue problem

where q(x) is a (bounded) function in the class L(a, b). We may suppose without loss of generality that 0 ≤ α < π and 0 ≤ β < π. Then as is well known there are infinitely many eigenvalues λr (r = 0, 1, 2, …) and λn ∽ n2π2 (b − a)−2 as n → ∞.

In this paper we list the elliptic curves defined over Q(√− 3) with good reduction away from the prime dividing 3. As in [8] and [9] a discriminant estimate is used to show that such a curve must have a subgroup of order 3 defined over Q(√−3).

1. The object of this paper is to evaluate two integrals involving Fox's H-function and employ them to establish two Fourier series for the H-function. Some Fourier series for Meijer's G-function and MacRobert's E-function are obtained as particular cases. Some results recently given by MacRobert(6, 7), Jain (4) and Kesarwani(5) are shown as particular cases.

Let R be a set of points in n-dimensional Euclidean space, and let Δ′(R) denote the lower bound of the determinants of non-homogeneous lattices which have no point in R. For Δ′(R) to be non-zero it is necessary, as Macbeath has shown (2), that R should have infinite volume.

Let G(d) be a given simple d-regular graph on n labelled vertices, where dn is even. Such a graph will be called an initial graph. Denote by Gp(d) a random subgraph of G(d) obtained by removing edges, each with the same probability q — 1 —p, independently of all other edges (i.e. each edge remains in Gp(d) with probability p). In a recent paper [10] the asymptotic distributions of the number of vertices of a given degree in a random graph Gp(d) were given. The aim of this sequel is to present a wide variety of results devoted to probability distributions of the maximum and minimum degree of Gp(d) with respect to different values of the edge probability p and degree of regularity d. It should be noted here that very detailed results on a similar subject in the case when the initial graph is a complete graph (i.e. when d = n – 1) have already been obtained by Bollobás in the series of papers [2]–[4] (some additional information to the paper [4] was given in [9]). Also, in proving our results we will make use of some ideas given by Bollobás in these papers.

Let e be an integer greater than 1 and let pbe a prime such that p ≡ 1 (mod e). Criteria for 2 to be a residue of degree e modulo p have been obtained in various forms for e = 2, 3, 4 and 5. Thus, Euler and Lagrange proved that 2 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 8). Gauss showed that 2 is a cubic residue mod p if and only if p is representable as p = l2 + 27m2 with integers l and m, and that 2 is a quartic residue mod p if and only if p is representable as p = 12 + 64m2. Lehmer (5) and Alderson (1) have found similar but more complicated conditions for 2 to be a quintic residue mod p. There are analogous results about 3. For example, it follows from quadratic reciprocity that 3 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 12), and Jacobi(4) showed that 3 is a cubic residue mod p if and only if 4p is representable as l2 + 35m2.

A Fourier integral will be described as of finite type if the range of integration of the integrals by means of which the coefficients in the Fourier integral are defined is a finite interval instead of the usual (−∞ ∞). Thus is of finite type (p, q), such that where ƒ(x) is the generating function of the series.