The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the G-representation naturally associated to the p∞-Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.

Let{N(x), x ≥ 0} be a Poisson process with intensity parameter λ > 0, and introduce

When looking for the changepoint in the Land's End data, Kendall and Kendall [7] proved for all 0 < ε1 < 1 − ε2 < 1 that

where {V(s), − ∞ < s < ∞} is an Ornstein–Uhlenbeck process with covariance function exp (−|t − s|). D. G. Kendall has posed the problem of replacing ε1 and ε2 by zero or by sequences εi(n) → 0 (n → ∞) (i = 1, 2), in (1·1). In this paper we study the latter problem and also its L2 version. The proofs will be based on the following weighted approximation of Zn.

The complex hyperbolic version of Shimizu's lemma gives an upper bound on the radii of isometric spheres of maps in a discrete subgroup of PU(n, 1) containing a vertical Heisenberg translation. The purpose of this paper is to show that in a neighbourhood of this bound radii of isometric spheres only take values in a particular discrete set. When the group contains certain ellipto-parabolic maps this upper bound can be improved and the set of values of the radii is more restricted. Examples are given that show that these results cannot be improved.

A number of formulae of an integral geometric character are presented for the most general planar, homogeneous line-segment process, one in which a given segment length and orientation may depend upon (a) the point process of segment midpoints and (b) the lengths and orientations of other segments. The sense in which these formulae have a probabilistic/statistical interpretation is made precise. For the general process, two interpretations are given; one requires the theory of Palm distributions whilst the other depends upon ergodic results. When additional structure for the process is assumed, the integral geometric formulae lead to interesting, non-intuitive sampling formulae.

In this paper we are dealing with the oscillatory and asymptotic behaviour of the nth order, (n > 1), retarded differential equation

where g is differentiable on the half-line (t0,∞) with (I) g(t) ≤ t for every t ≥ t0 (II) g'(t) ≥ 0 for every t ≥ t0 (III) = ∞.

The notion of orthogonality is axiomatically defined on a poset. Various notions of orthocomplementation are distinguished and conditions are given in order to induce an orthocomplementation from an orthogonality and vice versa. Subsequently ⊥-modular ⊥-poset are defined and the set of morphisms between two posets with orthogonality is briefly discussed. Given the notions of additive monoid and of positive semi-ring, an orthogonality relation is introduced on the set of idempotent elements of a positive semi-ring. Finally, the obtained results are applied to the set of idempotent and absorbent endomorphisms of an additive monoid.

In a previous paper it was shown that a necessary and sufficient condition for two regular polynomial matrices T and U to have relatively prime determinants is that a certain matrix R be non-singular. This generalization of the resultant of two scalar polynomials is extended to give an expression for the sum of the degrees of the greatest common divisors of all pairs of invariant factors of T and U in terms of the rank of R. Further, if rank R exceeds a given value then specified pairs of factors are relatively prime.

Let M be a complete hyperbolic n-manifold of finite volume. By a systole of M we mean a shortest closed geodesic in M. By the systole length of M we mean the length of a systole. We denote this by sl (M). In the case when M is closed, the systole length is simply twice the injectivity radius of M. In the presence of cusps, injectivity radius becomes arbitrarily small and it is for this reason we use the language of ‘systole length’.

In the context of hyperbolic surfaces of finite volume, much work has been done on systoles; we refer the reader to [2, 10–12] for some results. In dimension 3, little seems known about systoles. The main result in this paper is the following (see below for definitions):

In this work we generalize Markov's inequality for any derivative of a polynomial on a real Hilbert space and provide estimates for the second and third derivatives of a polynomial on a real Banach space. Our result on a real Hilbert space answers a question raised by L. A. Harris in his commentary on problem 74 in the Scottish Book [20]. We also provide generalizations of previously obtained inequalities of the Bernstein and Markov-type for polynomials with curved majorants on a real Hilbert space.

The phase space of a finite dimensional classical Hamiltonian system is a C∞ differentiable manifold M which carries a C∞ differential 2-form ω which is closed, dω = 0, and non-singular in the sense that there is a bijective map α→ Xα from covariant vector fields to contravariant vector fields satisfying the identity

Such a pair (M, μ) is called a symplectic manifold.

In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we give a geometric proof of a Theorem due to Alvarez and Rodriguez that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half-planes attached along simple open geodesic components. We next consider gluing infinitely many pairs of pants along their cuffs to obtain an infinite hyperbolic surface. We prove that there always exists a choice of twists in the gluings such that the surface is complete regardless of the size of the cuffs. This generalises the examples of Matsuzaki.

In the second part we consider complete hyperbolic flute surfaces with rapidly increasing cuff lengths and prove that the corresponding quasiconformal Teichmüller space is incomplete in the length spectrum metric. Moreover, we describe the twist coordinates and convergence in terms of the twist coordinates on the closure of the quasiconformal Teichmüller space.

Stopping times are a powerful tool in the theory of stochastic processes, so it is natural to ask whether they have a counterpart in the theory of non-commutative processes. This paper is a part answer to that question. We show that the ‘formalism’ of stopping times carries over to a non-commutative context and prove an Optional Stopping Theorem.

In this paper, we study the structure of certain infinite-dimensional Jordan algebras admitting an inner product. These algebras, called J*-algebras in the sequel, have already been considered in (4) in connexion with the norm uniqueness problem for non-associative algebras. We deal here with the structure and classification of these algebras. Existence of self-adjoint idempotents plays a central role in the classification problem.

It is well known that the β-ray type of disintegration is usually accompanied by the emission of γ-rays, and it is a matter of great importance to decide which of these two phenomena occurs first, that is, whether the γ-rays are emitted before the nucleus disintegrates or afterwards. It is not practicable to attempt a direct solution of this problem, but since the emission of the disintegration electron results in a change of the nuclear charge of + 1 it would be sufficient to determine the atomic number of the radioactive body at the moment of emission of the γ-rays. For instance, an atom of the β-ray body radium B has an atomic number 82, but after disintegration when it becomes radium C it has an atomic number 83, so that if the γ-rays were known to be emitted from a body of atomic number 82 it would mean that they were emitted before the disintegration, whereas if the atomic number 83 were found it would show that they were emitted afterwards.

The magnetisation of a single crystal of a ferromagnetic substance is accompanied by a distortion of the crystal which depends on the intensity and direction of magnetisation. Webster has shown that the magnetostriction in unsaturated states may be accounted for in an entirely satisfactory manner by making the assumption (for which the evidence is entirely convincing) that at ordinary temperatures an apparently unmagnetised crystal consists of small regions magnetised to saturation in various directions, so as to give no resultant magnetisation to the whole crystal, and that these regions possess natural directions of easy magnetisation.

It is shown that the algebra of the Duffin-Kemmer matrices can be handled quite neatly by introducing an extra dimension and by using the Γ-formalism of an earlier paper (3). By this means it is possible to derive identities similar to the De Broglie-Pauli identities for the Dirac matrices. To complete the analogy a Γ-formalism is also set up for the Dirac matrices.

Let Fn be a free group of rank n and let Out Fn be its outer automorphism group. The main result of this paper is that Out F3 has a faithful representation as a group of automorphisms of the polynomial ring in seven variables over the integers. This extends a similar result for n = 2 (see Helling [3], Horowitz [5] and Rosenberger [12]), and provides a partial answer to a conjecture attributed in [5] to W. Magnus. As an application of the special nature of the representing polynomials, we obtain our second result, that Out F3 is virtually residually torsion-free nilpotent.

For the group of real numbers R, an exponential monomial is defined as a function of the form xr(−ixz), for some non-negative integer r and some complex number z. Similarly, an exponential polynomial is a function P(x) exp (−ixz), for a polynomial P. In a now famous paper ((15)), Schwartz proved that every closed translation invariant subspace (variety) of the space of continuous functions on R is determined by the exponential monomials it contains. His techniques do not generalize to groups other than R as they use the theory of functions of one complex variable. A shorter proof of this result, using the Carleman transform of a function, was given by Kahane in his thesis ((9)). Ehrenpreis ((5)) proved results similar to those of Schwartz for certain varieties in the space of analytic functions of n complex variables, and Malgrange ((13)) proved the related result that any solution in ℰ(Rn) (for the notation see (16)) of the homogeneous convolution equation μ*f = 0, for some μ∈ℰ′, belongs to the closure of the exponential polynomial solutions of the equation.

1. Introduction. Hajnal (5) showed that under wide conditions a sequence of products H(1, q) = Mq…M1q = 1,2,…, of square non-negative matrices Mq approaches a sequence of positive matrices of rank 1. We call a product H(1,q) inhomogeneous if its factors M1,…, Mq are not necessarily all equal to one another. When the matrices Mq are members of an ‘ergodic set’, and x and y are positive vectors, the projective distance d(H(1,q)x, H(1,q)y) decays at least exponentially fast as q increases. An important condition on an ergodic set is that any product of g members from the set be a matrix in which all elements are (strictly) positive, where g is some fixed positive integer.