In this paper, we look at the Diophantine equation(0.1)Under the ABC conjecture, we show that it has only finitely many nontrivial solutions. Unconditionally, we show that the set of n for which the above equation admits an integer solution a1,. . .,at is of asymptotic density zero.

Suppose that f(x) is non-negative for x ≥ 0 and let

In a recent paper (4) inequalities were proved relating fn(x) with the nth derivative f(n)(x) of f(x). However, the two main results, Theorems 3 and 4, were proved only for n = 1 and 2. In the first part of this paper we shall prove the corresponding results for all positive integral n under a slightly less restrictive hypothesis (Theorem 3). We shall then give an application showing that from the hypothesis f(x) ≤ μ(x) for all x it is sometimes possible to deduce f(n)(x) = O{μ(n)(x)} for a set of values x depending only on μ(x).

We study the existence of complements to a partial tilting module over an arbitrary ring. As a consequence, we show that a finitely generated partial tilting module over an artin algebra has a (possibly infinitely generated) complement.

1. A well-known theorem on probability may be stated as follows. Let λdt be the chance that a certain event will happen in time dt, so that λT is the average number of occurrences of the event in an interval T. If we take a large number of repetitions of the interval T the actual numbers of occurrences will fluctuate about the mean λT, being say λT + x′, λT + x″,…. Then the theorem states that

The ejection of an α-particle by a radioactive substance is an event of the type contemplated and the theorem was applied to this physical problem by von Schweidler in 1905. Much similar work, both theoretical and experimental, has been done at intervals since then*.

It is unknown whether there can exist a family of functions from Ω1 into Ω of size less than that dominates all functions from Ω1 into Ω. We show that there is no such family if the continuum is real-valued measurable, and that the existence of such a family has consequences for cardinal arithmetic, and is related to large cardinal axioms.

Every closed connected orientable three-manifold can be constructed by surgery along an appropriately chosen link in the three-sphere, S3 ((10), (15)). Such a link-surgery description is never unique, but the equivalence between different descriptions has been explicitly identified ((8),(3)). However, the situation with regard to manifolds that can be obtained by surgery along a knot in S3 is less clear ((6), p. 47): some manifolds are known to have at least two knot-surgery descriptions ((12), (14), p. 270, (1), (11)), whilst certain manifolds (e.g. S3 and S2 × S1) are widely suspected to have just one (see problems 1·15, 1·16 of (9)).

Barbour [l] invented an ingenious method of establishing the asymptotic distribution of the number X of specified subgraphs of a random graph. The novelty of his method relies on using the first two moments of X only, despite the traditional method of moments that involves all moments of X (compare [8, 10, 11, 14]). He also adjusted that new method for counting isolated trees of a given size in a random graph. (For further applications of Barbour's method see [4] and [10].) The main goal of this paper is to show how this method can be extended to a general setting that enables us to derive asymptotic distributions of subsets of vertices of a random graph with various properties.

In [6], we considered the equation

where z ∈ ℂ and the pi are real-valued functions; abstract word-problem concepts and techniques were applied to the local problem of the bifurcation of periodic solutions out of the solution Z ≡ 0. This paper is a sequel to [6]; we present an extension of certain concepts given in that paper, and give a global version of some of our word-problem results.

McCallum and Smith (1) have developed a logical computer which solves problems which are essentially of the following kind.

Motivated by discussion on practical stability in LaSalle and Lefschetz (3), Weiss and Infante (5), have discussed various notions of stability over finite time interval of a given differential system. This theory of stability differs from the usual stability theory mainly by the preassigned limits to which any given solution of the differential system must adhere. Sufficient conditions for these notions of stability are given in (5) in terms of certain Lyapunov-like functions satisfying some appropriate differential inequalities. The purpose of this article is to introduce some complementary notions of finite time stability and weaken the conditions on the differential inequalities involving Lyapunov-like functions by the use of comparison principles.

Experiments with a helical method of focussing β-rays are described. Although the method does not appear to be so generally useful for obtaining ordinary β-ray spectra, it possesses certain advantages, of which may be mentioned:

(1) Large dispersion;

(2) Greater intensity;

(3) Good screening.

The method produces some evidence on the question of the continuous background, and experiments with it in this connection are described.

In 1914 Mitchell (13) underook the determination of the finite primitive collineation groups in more than three dimensions which are generated by homologies. Among these groups are five which are remarkable as not being members of an infinitely extended series, and these five isolated groups fall naturally into two sets. One set comprises three groups of orders 27.34:5, 29.34.5.7 and 213.35.52.7, belonging respectively to space of five, six and seven dimensions, the centres of the generating homologies of the smaller groups being among those of the larger ones; these groups had been considered earlier by Burnside (3) as groups of linear substitutions with rational coefficients, and both the groups and the configurations formed by the centres of the generating homologies are closely related to polytopes in Euclidean space of six, seven and eight dimensions first described by Gosset (9) and later studied in detail by Coxeter (6, 7). The two remaining groups, of orders 26.34.5 and 28.36.5.7, belong respectively to four and five dimensions, and the centres of the forty-five generating homologies of the former appear among the centres of the 126 generating homologies of the latter. The smaller group is the well-known simple group of order 25920, and is probably the most extensively studied of all special groups of finite order. Its representation as a collineation group in four dimensions was first obtained by Burkhardt (2), who determined the invariants of the group; the simplest of these is a quartic primal with nodes at the 45 centres of homologies, which was later studied by Coble (5) and myself (14). The geometry of the configuration formed by these 45 nodes has been investigated in detail by Baker (1), whose results I have used elsewhere (15) to classify the operations of the group and obtain their distribution into conjugate sets.

1. The plane quartic curves which pass through twelve fixed points g, of which no three lie on a straight line, no six on a conic and no ten on a cubic, form a net of quartics represented by the equation

Suppose that a function f(z, n) depends on a large parameter n. A proposed expression g(z, n) is an asymptotic approximation for f(z, n) if it can be shown that the error (i.e. the difference between f(z, n) and g(z, n)) is small of an appropriate order when n → ∞. Effective error bounds are particularly useful in numerical work with asymptotic expansions. Most of the existing derivations of error bounds involve complicated calculations.

We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the ‘twistor’ bundle of conformal inner products having holomorphic image. The second solution allows to use standard results in algebraic geometry to show that the Weyl connections on the two-sphere whose geodesics are the great circles are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane.

In this article, we study the amenability of Banach algebras in general, and that of Lipschitz algebras in particular. After introducing an alternative definition of amenability, we extend a result of [5], thereby proving a new characterization of amenability for Banach algebras. This characterization relates the amenability of a Banach algebra A to the space of bounded homomorphisms from A into another Banach algebra B (Theorem 4). This result allows us to solve the problem of amenability for virtually all Lipschitz algebras (of complex or Banach algebra valued functions), a class of algebras which has been studied in [2], [4] and [5].

Results of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

Examples are given of legendrian links in the manifold of cooriented contact elements of the plane, or equivalently, in the 1-jet space of the circle which are not equivalent via an isotopy of contact diffeomorphisms. These examples have generalizations to linked legendrian spheres in contact manifolds diffeomorphic to ℝn×Sn−1. These links are distinguished by applying the theory of generating functions to contact manifolds.

In this paper, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge 24 is uniquely determined by its weight one Lie algebra if the Lie algebra has the type A3,43A1,2, A4,52, D4,12A2,6, A6,7, A7,4A1,13, D5,8A1,2 or D6,5A1,12 by using the reverse orbifold construction. Our result also provides alternative constructions of these vertex operator algebras (except for the case A6,7) from the Leech lattice vertex operator algebra.

We give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. We then discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that p-solvable groups can only admit blocks of Loewy length 4 if p=2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all p ≡ 1 (mod 3). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.