Let A ⊃ ℤ be an integral domain which is finitely generated over ℤ and let a,b,c be non-zero elements of A. Extending earlier work of Siegel, Mahler and Parry, in 1960 Lang proved that the equation (*) aϵ +bη = c in ϵ, η ∈ A* has only finitely many solutions. Using Baker's theory of logarithmic forms, Győry proved, in 1979, that the solutions of (*) can be determined effectively if A is contained in an algebraic number field. In this paper we prove, in a quantitative form, an effective finiteness result for equations (*) over an arbitrary integral domain A of characteristic 0 which is finitely generated over ℤ. Our main tools are already existing effective finiteness results for (*) over number fields and function fields, an effective specialization argument developed by Győry in the 1980's, effective results of Hermann (1926) and Seidenberg (1974) on linear equations over polynomial rings over fields, and similar such results by Aschenbrenner, from 2004, on linear equations over polynomial rings over ℤ. We prove also an effective result for the exponential equation aγ1v1···γsvs+bγ1w1 ··· γsws=c in integers v1,…,ws, where a,b,c and γ1,…,γs are non-zero elements of A.

We give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closures of (cabled) string links. We show that finite type invariants classify string links up to Ck-moves for k ≤ 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar classification of string links up to Ck-moves and concordance for k ≤ 6.

Dual integral equations involving Bessel functions occur in the solution of some boundary-value problems in potential theory with conditions prescribed on a circular disc and a considerable amount of attention has been given to the solution of such equations (cf. (1)). The method of solving these dual integral equations is very similar to that employed in the solution of certain dual series equations involving Legendre functions. Equations of this type occur in problems in potential theory with conditions prescribed on a spherical cap and their solution has been obtained by Collins (2). No definite mathematical connexion has, however, been established between these dual series and dual integral equations and the object of this note is to establish such a connexion.

I am indebted to Mr L. Kassel of the Department of Commerce, U.S.A., for pointing out an error of computation in a recent paper in these Proceedings.

The asymptotic expansion for the partition function should read

where

For X a subset of a group G, the smallest normal subgroup of G which contains X is called the normal closure of X and is denoted by ngp (X; G) or simply by ngp (X) if there is no possibility of ambiguity. By a surface group we mean the fundamental group of a compact surface. We are interested in determining when a normal subgroup of a surface group contains a simple loop – the homotopy class of an embedding of S1 in the surface, or more generally, a power of a simple loop. This is significant to the study of 3-manifolds since a Heegaard splitting of a 3-manifold is reducible (cf. [2]) if and only if the kernel of the corresponding splitting homomorphism contains a simple loop. We give an answer in the case that the normal subgroup is the normal closure ngp (α) of a single element α: if ngp (α) contains a (power of a) simple loop β then α is homotopic to a (power of a) simple loop and β±1 is homotopic either to (a power of) α or to the commutator [α, γ] of a with some simple loop γ meeting a transversely in a single point. This implies that if a is not homotopic to a power of a simple loop, then the quotient map π1(S) → π1(S)/ngp (α) does not factor through a group with more than one end. In the process we show that π1(S)/ngp (α) is locally indicable if and only if α is not a proper power and that α always lifts to a simple loop in the covering space Sα of S corresponding to ngp (α). We also obtain some estimates on the minimal number of double points in certain homotopy classes of loops.

In this paper we elucidate the structure of locally soluble absolutely irreducible skew linear groups, and more generally of locally soluble normal subgroups of arbitrary absolutely irreducible skew linear groups. The conclusions are similar to those for soluble such subgroups, but the proofs are not. We also give a couple of applications to the theory of group algebras. Much of our argument works with considerably weaker hypotheses and therefore our methods may well have wider application than just to absolutely irreducible groups.

Given a static Schwarzschild spacetime of ADM mass M, it is well known that no ingoing causal geodesic starting in the outer domain r > 2M will cross the event horizon r = 2M in finite Schwarzschild time. We show that in gravitational collapse of Vlasov matter this behaviour can be very different. We construct initial data for which a black hole forms and all matter crosses the event horizon as Schwarzschild time goes to infinity, and show that this is a necessary condition for geodesic completeness of the event horizon. In addition to a careful analysis of the asymptotic behaviour of the matter characteristics our proof requires a new argument for global existence of solutions to the spherically symmetric Einstein–Vlasov system in an outer domain, since our initial data have non-compact support in the radial momentum variable and previous methods break down.

A unital JB-algebra is a Jordan algebra A with identity together with a complete norm satisfying, for all a, b ∈ A,

(i) (a2b) a = a2(ba),

(ii) ∥a2∥ = ∥a∥2,

(iii) ∥ab∥ ≤ ∥a∥ ∥b∥,

(iv) ∥a2 + b2∥ ≥ ∥a2∥, ∥b2∥.

(It should be noted that axiom (iii) is a consequence of (ii) and (iv).) Such spaces have been studied by several authors (3, 6, 11), and as a consequence their structure is now quite well understood. Many of the results of these papers, while relying on the existence of an identity for their proofs, can be formulated for algebras which lack this property. C*-algebra theory and operator theory abound in examples of spaces which fail to be unital JB-algebras only in this one respect, and this motivates the study of the general case undertaken in this note.

We use the Jones–Wenzl idempotents to construct a basis of the Temperley–Lieb algebra TLn. This allows a short calculation for a Gram determinant of Lickorish's bilinear form on the Temperley–Lieb algebra.

In this paper we discuss smooth and locally smooth cyclic actions on cohomology complex projective spaces. We shall show that in low dimensions the presence of a codimension-two fixed-point component forces many algebraic invariants to behave as they do in the linear examples.

1. Introduction and recapitulation. In a previous paper (4) we have defined a ringoid (with identities) as a set of elements α, β,… in which addition and multiplication are defined for certain pairs of elements in such a way that, whenever operations are defined, the usual ring axioms are satisfied. Precisely, is an abstract category ((3), p. 108) with an addition operation such that the following axioms hold:

(R 1) For any two identity elements ι, ι′ of, let ℋ(ι, ι′) be the set of elements ofwhose left identity is ι and right identity is ι′; then ℋ(ι, ι′) is empty or an Abelian group under addition.

1. Introduction. This paper is concerned with proving identities in commutative Moufang loops. Many such identities were derived in chapter VIII of (1) in the course of demonstrating the local nilpotence of commutative Moufang loops. The results there are regarded as constituting the ‘first grammar of associators’: the reader is assumed to have a good knowledge of them. The current paper develops additional material required for the determination in (5) of the precise nilpotence class of the free commutative Moufang loop on any given finite number of generators. It is called a ‘grammar’ because it lists formal ways in which the language of associators works, and is merely meant to serve a reader of ‘literature’ in the language such as (5). However, it may be of interest for other purposes, such as answering Manin's question ((3), Vopros 10·3; (4), problem 10·2) on the 3-rank of the free commutative Moufang loop of exponent 3. There is also the problem raised below as to whether the Triple Argument Hypothesis is a consequence of the commutative Moufang loop laws. Finally, the Möbius function in Section 9 may tempt someone to look at lattice-theoretical aspects of associators.

Let $E$ be a rearrangement invariant (r.i.) function space on ‘0, 1’. We consider the space $\Lambda({\cal R},E)$ of measurable functions $f$ such that $fg \in E$ for every a.e. converging series $g =\sum a_nr_n \in E$, where $(r_n)$ are the Rademacher functions. Curbera ‘4’ showed that, for a broad class of spaces $E$, the space $\Lambda({\cal R},E)$ is not order-isomorphic to a r.i. space. We study cases when $\Lambda({\cal R},E)$ is order-isomorphic to a r.i. space. We give conditions on $E$ so that $\Lambda({\cal R},E)$ is order-isomorphic to $L_{\infty}$. This includes certain classes of Lorentz and Marcinkiewicz spaces. We study further when $\Lambda({\cal R},E)$ is orderisomorphic to a r.i. space different from $L_{\infty}$. This occurs for the Orlicz spaces $E = L_{\Phi_q}$ with $\Phi_q(t)$ asymptotically equivalent to $\exp |t|^q-1$ and $0 < q < 2$.

Let f be a hyperbolic transcendental meromorphic function such that the finite singularities of f−1 are in a bounded set. We show that there exists 0<s(f)[les ]2 such that

formula here

for each point a in the Julia set of f, where

formula here

We then show that s(f)[les ]dimJ(f), the Hausdorff dimension of the Julia set, and give examples of such functions for which dimJ(f)>s(f). This contrasts with the situation for a hyperbolic rational function f where it is known that dimJ(f) = s(f).

Let G be a finite group, p a prime divisor of |G| and suppose that G is not a p-group. In this note, we show that the number of elements x ∈ G such that xp = 1 is at most (p|G|)/(p + 1). This answers a question posed by D. MacHale. When G is a Frobenius group of order p(p + 1), p a Mersenne prime, the above bound is attained.

For a given function f, regular in the unit disk D(0, 1), logarithmic means of order p may be defined in terms of integrals by the formulae

for 0 < p < , and by

when 0 < r < 1. Associated p-orders are subsequently defined by

for 0 < p.

This paper continues the investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we now present a rigorous method for the special case of a half-immersed circular cylinder. The aim is to justify, in one particular case, those assumptions made in a previous paper giving a non-rigorous method for an arbitrary smooth cylinder which intersects the free surface normally, concerning the asymptotic values on the cylinder of the two related auxiliary potentials which were introduced to enable an evaluation of the coefficients describing wave-making and virtual mass. This is done by constructing an integral equation of the second kind whose kernel is small and deducing the leading term in the iteration solution. The method is an extension of the method for infinite depth and it is found that there is a simple relationship between the kernels. In conclusion, computations for these asymptotic results are presented and compared with the trend of known non-asymptotic results.

1. The cross-flow type of heat interchanger is shown diagrammatically in Fig. 1. One of the fluids involved is arranged to pass through a nest of small tubes, while the other streams past the tubes at right angles to them. To fix ideas we suppose that the tubes contain cooling water which enters at a temperature T1. The temperature of the water leaving a tube depends upon its position, but the water from all the tubes is mixed and is led to a single exit pipe in which the temperature is T2. In the same way the fluid to be cooled enters at a temperature t1 and leaves at a temperature t2. If the cooling water is not used again by being led through another nest of tubes also exposed to the fluid to be cooled, the arrangement is termed a single-pass heat interchanger.

1. Let f(x) be a real function of period 2π, integrable L over (0, 2π), and let

By sn(x) and σn (x) we denote respectively the partial sums and the first arithmetic means of the series (1·1). Similarly, by and we denote the partial sums and the first arithmetic means of the series

conjugate to (1·1). By we mean the function conjugate to f(x), that is

where the integral is taken in the principal-value sense.

In 1933 Borsuk(2) made the following conjecture: Every bounded set of points in Euclidean n-space En can be represented as the union of n + 1 sets of smaller diameter. He proved it for n = 2. Hadwiger (5) proved Borsuk's conjecture assuming the additional condition that the surface of the set is sufficiently smooth. On the other hand, up to now the conjecture has been proved for n ≤ 3 only (3, 4).