The theory of heights for rational points on arithmetic elliptic curves is becoming well known. An important fact in the basic theory is the relationship between the naïve and the canonical height of a rational point; in fact, they differ by a uniformly bounded amount. The paper provides a generalisation of this fact from the point of view of heights of polynomials, rather than heights of points. This concept is extended to polynomials in several variables.

Given the Diophantine equation a2+b2+c2=3abc, a solution triple of natural numbers (a, b, c) can be arranged in ascending order so that a[les ]b[les ]c. Then, given the largest element c, one can ask whether this uniquely determines the triple. This is referred to as the Markoff conjecture. The paper proves that, if c is prime, then there is indeed only one triple that solves the equation with c as the largest element. The proof uses only standard algebraic number theory, but it was prompted by geometric considerations.

It is shown that the general plane section of a double structure on an integral curve C⊆P3 has a connected numerical character (that is, its Hilbert function is of decreasing type). The paper gives applications, particularly to the stability of the normal bundle of space curves.

All groups and rings in this paper are finite, and p denotes a prime number.

R. Shepherd [28] and C. R. Leedham-Green and S. McKay [16] showed that any p-group of maximal class contains a subgroup of class at most 2 the index of which is bounded above by a function of p. These papers gave rise to a program for the classification of p-groups that used the notion of coclass proposed by C. R. Leedham-Green and M. F. Newman [21] in 1980. They made several conjectures. The strongest conjecture, Conjecture A, asserted that every p-group of coclass r contains a subgroup of class at most 2 the index of which is bounded above by a function depending only on p and r. These conjectures were proved in a long series of papers by C. R. Leedham-Green, S. McKay, S. Donkin, W. Plesken, A. Shalev and E. Zelmanov (cf. [3, 16–20, 27]). In a recent paper, A. Shalev [26] gave a proof of Conjecture A for all primes p with explicit bounds.

Pansu has shown that the growth function of every virtually nilpotent group Γ with respect to any finite generating set has asymptotics γ(n)∼αnd, where d is the degree of growth of Γ. The paper refines his result in the special case of 2-step nilpotent groups to obtain γ(n)=αnd +O(nd−1).

We develop a theory of modules of type FP∞ over group algebras of hierarchically decomposable groups. This class of groups is denoted H[Fscr ] and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an h[Fscr ]-group and Mis a ℤG-module of type FP∞ then M has finite projective dimension over ℤH for all torsion-free subgroups H of G. We also show that if G is an h[Fscr ]-group of type FP∞ and M is a ℤG-module which is ℤF-projective for all finite subgroups F of G, then M has finite projective dimension over ℤG. Both of these results have as a special case the striking fact that if G is an h[Fscr ]-group of type FP∞ then the torsion-free subgroups of G have finite cohomological dimension. A further result in this spirit states that every residually finite h[Fscr ]-group of type FP∞ has finite virtual cohomological dimension.

Let G be a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p[ges ]0. In this paper, we classify all pairs (X, Y) of reductive subgroups of G which have a dense (X, Y)-double coset in G. In fact, we show that there is a dense (X, Y)-double coset in G precisely when G=XY is a factorisation. The possible factorisations that can occur have recently been determined in [11, Theorem A].

We now state our main result.

The paper finds best constants in a class of multiplicative inequalities with one-dimensional spatial variables

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where f is a periodic function with zero mean value, and the norms in the right-hand side of the expression are the L2-norms.

Let

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be a formal power series. In 1913, G. Pólya [7] proved that if, for all sufficiently large n, the sections

formula here

have real negative zeros only, then the series (0.1) converges in the whole complex plane C, and its sum f(z) is an entire function of order 0. Since then, formal power series with restrictions on zeros of their sections have been deeply investigated by several mathematicians. We cannot present an exhaustive bibliography here, and restrict ourselves to the references [1, 2, 3], where the reader can find detailed information.

In this paper, we propose a different kind of generalisation of Pólya's theorem. It is based on the concept of multiple positivity introduced by M. Fekete in 1912, and it has been treated in detail by S. Karlin [4].

Functionals of higher derivative type are linear combinations of functionals of the form f→f(n)(ζ), where n[ges ]2 and 0<[mid ]ζ[mid ]<1. The paper shows that, if L is a functional of higher derivative type and f is a function in the class S of univalent functions that maximises Re{L} over S, then L(f)≠0. In addition, if the function f is a rational function, then it must be a rotation of the Koebe function k(z)=z(1−z)−2. These results are applied to establish several cases of the two-functional conjecture for functionals of higher derivative type.

Suppose that f(z) is a meromorphic function of bounded characteristic in the unit disk Δ[ratio ][mid ]z[mid ]<1. Then we shall say that f(z)∈N. It follows (for example from [3, Lemma 6.7, p. 174 and the following]) that

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where h1(z), h2(z) are holomorphic in Δ and have positive real part there, while Π1(z), Π2(z) are Blaschke products, that is,

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where p is a positive integer or zero, 0<[mid ]aj[mid ]<1, c is a constant and

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We note in particular that, if c≠0, so that f(z)[nequiv ]0,

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so that f(z)=0 only at the points aj. Suppose now that zj is a sequence of distinct points in Δ such that

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If f(zj)=0 for each j and f∈N, then f(z)≡0.

N. Danikas [1] has shown that the same conclusion obtains if f(zj)→0 sufficiently rapidly as j→∞. Let εj, λj be sequences of positive numbers such that

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Danikas then defines

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and proves Theorem A.

Let f[ratio ][Copf ]n, 0→[Copf ]p, 0 be a [Kscr ]-finite map germ, and let i=(i1, …, ik) be a Boardman symbol such that [sum ]i has codimension n in the corresponding jet space Jk(n, p). When its iterated successors have codimension larger than n, the paper gives a list of situations in which the number of [sum ]i points that appear in a generic deformation of f can be computed algebraically by means of Jacobian ideals of f. This list can be summarised in the following way: f must have rank n−i1 and, in addition, in the case p=6, f must be a singularity of type [sum ]i1,i2.

A Hilbert module over a C*-algebra B is a right B-module X, equipped with an inner product 〈·, ·〉 which is linear over B in the second factor, such that X is a Banach space with the norm ∥x∥[ratio ]=∥〈x, x〉∥1/2. (We refer to [8] for the basic theory of Hilbert modules; the basic example for us will be X=B with the inner product 〈x, y〉=x*y.) We denote by B(X) the algebra of all bounded linear operators on X, and we denote by L(X) the C*-algebra of all adjointable operators. (In the basic example X=B, L(X) is just the multiplier algebra of B.) Let A be a C*-subalgebra of L(X), so that X is an A-B-bimodule. We always assume that A is nondegenerate in the sense that [AX]=X, where [AX] denotes the closed linear span of AX.

Denote by AX the algebra of all mappings on X of the form

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where m is an integer and ai∈A, bi∈B for all i. Mappings of form (1.1) will be called elementary, and this paper is concerned with the question of which mappings on X can be approximated by elementary mappings in the point norm topology.

The paper proves two univalence criteria involving harmonic mappings in multiply connected domains. These generalise a classical result of Radó, Kneser and Choquet and a recent result of Bshouty and Hengartner.

It is shown that the composition operator w[map ]w∘u generated by an inner function u is a self-map of the Muckenhoupt class A2, but that it need not preserve the Muckenhoupt classes Ap for p∈(1, 2)∪(2, ∞).

Let ω=(ωn)n[ges ]1 be a log concave sequence such that lim infn→+∞ ωn/nc>0 for some c>0 and ((log ωn)/nα)n[ges ]1 is nonincreasing for some α<1/2. We show that, if T is a contraction on the Hilbert space with spectrum a Carleson set, and if ∥T−n∥=O(ωn) as n tends to +∞ with [sum ]n[ges ]11/(n log ωn)=+∞, then T is unitary. On the other hand, if [sum ]n[ges ]11/(n log ωn)<+∞, then there exists a (non-unitary) contraction T on the Hilbert space such that the spectrum of T is a Carleson set, ∥T−n∥=O(ωn) as n tends to +∞, and lim supn→+∞ ∥T−n∥=+∞.

The paper gives a condition that is sufficient for E. Bishop's property (β), and it applies this result to the Cesàro operator on the Hardy spaces Hp, 1<p<∞. In the process, it identifies the approximate point spectrum of the Cesàro operator.

Ideas from string theory and quantum field theory have been the motivation for new invariants of knots and 3-dimensional manifolds which have been constructed from complex algebraic structures such as Hopf algebras [17, 22], monoidal categories with additional structure [24], and modular functors [14, 23]. These constructions are closely related. Here we take a unifying categorical approach based on a natural 2-dimensional generalisation of a topological field theory in the sense of Atiyah [1], and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces.

The paper gives simple necessary and sufficient conditions for a closed 4-manifold to be homotopy equivalent to the mapping torus of a self homotopy equivalence of a PD3-complex. This is a homotopy analogue of the Stallings and Farrell fibration theorems available in other dimensions. The paper also considers 4-manifolds which admit a geometry of Euclidean factor type and complex surfaces which fibre over S1.

The paper shows that the times spent in [0, +∞) by certain processes Y which are defined by perturbations of Brownian motion involving reflection at maxima and minima are beta distributed. This result relies heavily on Ray–Knight theorems for such perturbed Brownian motions.