A longstanding conjecture claims that the Diophantine equation

formula here

has finitely many solutions and, maybe, only those given by

formula here

Among the known results, let us mention that Ljunggren [9] solved (1) completely when q = 2 and that very recently Bugeaud et al. [3] showed that (1) has no solution when x is a square. For more information and in particular for finiteness type results under some extra hypotheses, we refer the reader to Nagell [11], Shorey & Tijdeman [16, 17] and to the recent survey of Shorey [15]. One of the main tools used in most of the proofs is Baker's theory of linear form in archimedean logarithms of algebraic numbers and especially a dramatic sharpening obtained when the algebraic numbers involved are closed to 1. This was first noticed by Shorey [14], and has been applied in numerous works relating to (1) or to the Diophantine equation

formula here

see for instance [1, 5, 10]. The purpose of the present work is to prove a similar sharpening for linear forms in p-adic logarithms and to show how it can be applied in the context of (1). Further, following previous investigations by Sander [12] and Saradha & Shorey [13], we derive from our results an irrationality statement for Mahler's numbers.

1. The following sentence occurs in a recent paper ((1), p. 54): we do not understand precisely what it means, but we shall show that the idea behind it can be put on a reasonably quantitative basis: ‘In the vocabulary of solvers of simultaneous equations, the operator (1, −2, 1) tends to lead to ill-conditioning when the number of equations is large.’

Given any positive integer M, we show that there are infinitely many real quadratic fields that do not admit universal quadratic forms with even cross coefficients in M variables.

The type of Markoff process which is considered in this paper corresponds to a system capable of n states, the time being regarded as a continuously varying parameter. At any instant t the probability distribution is represented by the vector

Let $\mathcal{P}$ be a partition of a finite set X. We say that a transformation f : X → X preserves (or stabilises) the partition $\mathcal{P}$ if for all P ∈ $\mathcal{P}$ there exists Q ∈ $\mathcal{P}$ such that Pf ⊆ Q. Let T(X, $\mathcal{P}$) denote the semigroup of all full transformations of X that preserve the partition $\mathcal{P}$.

In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to solve Pei Huisheng's conjecture.

The aim of this paper is to solve the more complex problem of finding the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.

The paper ends with a number of problems for experts in group and semigroup theories.

In this paper I give an algorithm to find all ‘small’ S-integral points on an elliptic curve. I would like to thank N. Stephens for suggesting I consider such equations and the Wingate Foundation for supporting me whilst carrying out the research. As is usual c1, c2, …, will denote positive real constants which are effectively computable.

Many facts in the theory of general Riesz spaces are easily verified by thinking in terms of spaces of functions. A proof via this insight is said to use representation theory. In recent years a growing number of authors has successfully been trying to bypass representation theorems, judging them to be extraneous. (See, for instance, [9,10].) In spite of the positive aspects of these efforts the following can be said. Firstly, avoiding representation theory does not always make the facts transparent. Reading the more cumbersome constructions and procedures inside the Riesz space itself one feels the need for a pictorial representation with functions, and one suspects the author himself of secret heretical thoughts. Secondly, the direct method leads to repeating constructions of the same nature over and over again.

1. Introduction. The object of this investigation is to obtain by means of contour integrals exact solutions of certain nth-order differential equations, together with their n independent power-series solutions, their asymptotic solutions and the relationship between these two types of solution. The Stokes phenomenon associated with the changing constants in these asymptotic solutions will then be investigated by various methods. Paper I is concerned with the properties of the various contour-integral solutions of the equations under consideration. The manner in which the arbitrary constants in the general asymptotic solution must change as the argument of the independent variable z varies is dealt with in paper II. This phenomenon is considered in a way that has proved profitable for the approximate solution of more general linear differential equations of the nth order that are approximately identical with those considered here hi certain regions of the complex z–plane. In later publications, these approximations will be described together with their application to explicit physical problems.

Modify the usual percolation process on the infinite binary tree by forbidding infinite clusters to grow further. The ultimate configuration will consist of both infinite and finite clusters. We give a rigorous construction of a version of this process and show that one can do explicit calculations of various quantities, for instance the law of the time (if any) that the cluster containing a fixed edge becomes infinite. Surprisingly, the distribution of the shape of a cluster which becomes infinite at time t > ½ does not depend on t; it is always distributed as the incipient infinite percolation cluster on the tree. Similarly, a typical finite cluster at each time t > ½ has the distribution of a critical percolation cluster. This elaborates an observation of Stockmayer [12].

Let E denote the set of non-identity idempotent matrices in the full matrix ring Mn(R) over a principal ideal domain R. A necessary and sufficient condition is found for the subsemigroup generated by E to be the set of all matrices in Mn(R) of rank less than n. The condition is satisfied when R is a discrete valuation ring and when R is the ring of integers. Thus every n × n matrix of rank less than n is a product of idempotent integer matrices.

Zeeman has drawn attention to the sequences in which catastrophes, or modes of instability, can be linked, and it is a common observation that sequences of catastrophes of low order are always found in the environment of catastrophes of higher order. In this paper, a simple buckling model is presented that generates in an elegant manner a complete sequence of the umbilic catastrophes as represented by the semi-symmetric branching points. The scan of a single fundamental parameter of this model is shown to trace a route through all regimes of the umbilic bracelet, giving in turn the hyperbolic, symbolic, elliptic, parabolic and hyperbolic umbilic catastrophes.

Rollo Davidson conjectured in 1968 that every stationary second order line process in the plane which has a.s. no parallel lines is necessarily a Cox (or doubly stochastic Poisson) process. This conjecture is disproved here. An affirmative answer is further given to the question whether there exists a lattice type point process in the plane which is stationary under arbitrary area preserving affine transformations.

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

The problem of the collision between an electron and an atom was first considered on the Quantum theory by Born, who has worked out in detail the case of atomic hydrogen, and has obtained formulae giving the variation of scattering with angle both for elastic and inelastic collisions. Born's solution is only approximate. The purpose of this note is to discuss the physical nature of the approximations used by him, and also to extend his results to Helium, for which experimental evidence is now available. It is found that the theoretical curve agrees with the experimental as well as the approximations used would lead one to expect. We shall confine ourselves to elastic collisions.

We consider classes of second order boundary value problems with a nonlinearity f(t, x) in the equations and subject to a multi-point boundary condition. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. The symmetry of solutions is also studied. Conditions are determined by the relationship between the behavior of the quotient f(t, x)/x for x near 0 and ∞ and the largest positive eigenvalue of a related linear integral operator. Our analysis mainly relies on the topological degree and fixed point index theories.

Let be a closed 3-braid in S3 with axis Lγ. The exterior of γ U Lγ in S3 is a fibre bundle over S1 with fibre a disc with 3 holes. The monodromy is given by a matrix in PSL(2,) and the braid γ is called hyperbolic if its matrix is hyperbolic.

In this note we introduce a general class of finite ramified coverings π X˜ ↓ X. Examples of ramified covers in our sense include: finite covering spaces, branched covering spaces and the orbit map Y ↓ Y/G where G is a finite group and Y an arbitrary G-space. For any d-fold ramified covering π: X˜ ↓ X we construct a transfer homomorphism

with the expected property that

is multiplication by d. As a consequence we obtain a simple proof of the Conner conjecture; viz. the orbit space of an arbitrary finite group action on a ℚ-acyclic space is again ℚ acyclic.

In [7], the second author proved that there is an integer k such that every element of a finite non-abelian simple group S is a product of k commutators in S. The motivation for proving this result came from a model-theoretic question about simple groups. The proof depended on the classification of the finite simple groups, a theorem of Malle, Saxl and Weigel [5] which shows that in many finite simple classical groups S there is a real conjugacy class R such that S=R3∪{1}, and an ultraproduct argument. Here we shall use a similar combination of ideas to prove the following result.

The use of a coordinate system convected with the moving medium for describing its mechanics, first proposed by Hencky (5), has since been extended by several authors, and has several advantages over the more conventional use of a coordinate system fixed in space; Brillouin(1) has shown that the relation between the strain-energy function for an ideally elastic solid and the stress tensor takes a very simple form when the latter is referred to a convected coordinate system; Oldroyd(8) has given a very general discussion of the formulation of rheological equations of state and has shown that the right invariance properties are most readily recognized when the equations are referred to a convected coordinate system; Green and Zerna (4) have similarly expressed the equations of motion and boundary conditions; and Gleyzal (2), and Green and Shield (3) have applied the formalism to certain problems in elasticity theory.

1. Let H = (aij) be a positive-definite Hermitian matrix of order n. For any k distinct integers i1, i2, …, ik between 1 and n, we shall use the symbol (i1, i2, …, ik) to denote the k-rowed principal submatrix of H corresponding to the rows and columns with indices i1, i2, …, ik. It is well known thatM

and more generally,