Let P(α, β)n (x) be the Jacobi polynomials which are defined by

formula here

or given as hypergeometric series by

formula here

They satisfy the orthogonality relations

formula here

Let p be an odd prime number. Let k be a [pfr ]-number field and [ofr ] the ring of all integers of k with a prime element π. Let K/k be a cyclic extension with Galois group G, and [Afr ] the associated order of the ring [Ofr ] of all integers in K:

[Afr ]={f∈kG[mid ]f[Ofr ]⊆[Ofr ]}.

F. Bertrandias and M-J. Ferton [1] obtained necessary and sufficient conditions that [Ofr ] is [Afr ]-free in the case K/k is of degree p. The purpose of this paper is to study such conditions in case K/k is a cyclic totally ramified Kummer extension of degree n=pm.

The purpose of this note is to present examples of nilpotent group extensions which split at every p-localization but fail to split. These examples disprove the conjecture by Peter Hilton [4, 5] that any nilpotent group extension with finitely generated quotient which splits at every prime must necessarily split. In [3] Hilton proves his conjecture for extensions with abelian kernel. Moreover, some further special cases of the conjecture are established by Casacuberta and Hilton in [1].

We describe two families of counterexamples to Hilton's conjecture. In both families the groups are torsion-free and finitely generated. The first family consists of groups of class 2 and rank 6. In the second family, the groups have rank 5, which, in view of Hilton's result in [3], is the minimal possible rank for a torsion-free counterexample to the conjecture. However, this reduction in rank is obtained at the expense of increasing the class from 2 to 3. Theorem 2·3 establishes that this increase in class is unavoidable: there are no torsion-free counterexamples of rank 5 and class 2.

The proof of Theorem 2·3 is based on a fact, Lemma 2·4, about quadratic forms. The author would like to thank J. W. S. Cassels for providing him with this crucial fact together with an outline of its proof.

Baumslag [Bm1] and Remeslennikov [R1] independently showed that the abstract group with finite presentation

〈a, x, y[mid ]ax =aay, [x, y] =1=[a, ay]〉.

is metabelian; the derived subgroup is free abelian of countably infinite rank. Both generalized this example to show that every finitely generated metabelian group embeds in a finitely presented metabelian group. Thomson [T] generalized this result still further to show that every finitely generated group of upper triangular matrices over an abstract ring A lies inside a finitely presented group of upper triangular matrices over some extension ring of A. For a valuable survey of these results, see [St].

Baumslag proved analogous theorems for Lie algebras in [Bm2]. In this paper we shall prove some analogous results for pro-p groups.

The Bianchi groups are the groups PSL2([Oscr ]d), where [Oscr ]d denotes the ring of integers of the field ℚ (√−d) for each square-free positive integer d. These groups have long been of interest, not only because of their intrinsic interest as abstract groups, but also because they arise naturally in number theory and geometry. For a discussion of their algebraic properties we refer the reader to Fine [4]. Among the groups PSL2(R), with R the ring of integers of an algebraic number field, they are distinguished by the nature of their normal subgroup structure. It was shown by Serre [20] that if R is not isomorphic to ℤ or [Oscr ]d, then for every normal subgroup K of SL2(R) there is an ideal I of R such that the image in SL2(R)/K of the kernel of the natural map from SL2(R) to SL2(R/I) is central and isomorphic to a subgroup of the group of roots of 1 in R. On the other hand, the group PSL2(ℤ) and the Bianchi groups have many subgroups of finite index which are not of the above type: this follows easily from the fact that PSL2(ℤ) is a free product of a group of order 2 and a group of order 3, and the fact, proved by Grunewald and Schwermer [6], that each Bianchi group has a normal subgroup of finite index which can be mapped epimorphically to a non-abelian free group.

We introduce in this paper the notion of a normal system for a compact, Hausdorff, right topological group, G. This generalizes the notion of a strong normal system as introduced in [7], in which it was shown that closed subgroups of compact right topological groups with dense topological centres possess strong normal systems and have Haar measure. Their method involved the construction of a strong normal system, the Furstenberg–Namioka system. In this paper we study a variation of the Furstenberg–Namioka system which we call the N-system. The N-system for a group G is a normal system and we show that if G has dense topological centre, it possesses an N-system, though the two systems do not in general coincide. In Section 2 we give examples of groups with arbitrarily long N-systems and show that a dense topological centre is not a necessary requirement for the existence of normal systems. In Section 3 we introduce the notion of minimality of normal systems and we show that if the N-system for a group, G, is minimal then the N-system and the Furstenberg–Namioka system, if the latter exists, coincide (and therefore G has Haar measure!). Finally in our main theorem, we show that if G has a minimal N-system then the closed normal subgroups of G must lie between successive terms of the N-system.

In [11], we considered the question of existence of elliptic fibre space structures on a smooth Calabi–Yau threefold X. Necessary conditions are that there exists a nef divisor D on X with D3=0; D2[nequiv ]0 and D·c2[ges ]0. In the case when D·c2[ges ]0, it was observed that a result from [9] implies that there is an elliptic fibre space structure determined by D, and so [11] concerned itself with the case when D·c2=0. In this case, we were not able to deduce in general the existence of an elliptic fibre space structure, but only in the presence of a further condition. In particular, it follows from the Theorem in [11] that, if r denotes the number of rational surfaces E on X with D[mid ]E≡0, and if the Euler characteristic e(X)≠2r, then some positive multiple of D determines an elliptic fibre space structure on X (the theorem in fact proves a slightly stronger result, which gives the elliptic fibre space structure if there is any non-rational surface E on X with D[mid ]E≡0). The purpose of this note is to clarify the meaning of the above rather mysterious condition on the non-vanishing of e(X)/2−r.

Every nonsingular projective real algebraic curve C has a unique, up to isomorphism over ℝ, nonsingular projective complexification V. If V\C is disconnected, then C is said to be dividing (we identify C with the set of real points of V). Classical results supply numerous examples of dividing real curves. This class of curves was first studied by Felix Klein [11]. A characterization of dividing real curves is given in [6].

In higher dimensions the situation is more complicated. First of all, every nonsingular projective real algebraic variety X of dimension d always has several nonisomorphic nonsingular projective complexifications, provided that d[ges ]2. Furthermore, if d[ges ]2 and W is a nonsingular projective complexification of X, then W\X is connected (we identify X with the set of real points of W). What does it then mean for X to be dividing? An answer can be given in terms of homology theory. Let K be a principal ideal domain. Assume that X, regarded as a topological manifold, is orientable over K. We say that X is dividing over K if there exists a fundamental homology class of X over K whose image in Hd(W, K) under the homomorphism induced by the inclusion map X[rarrhk ]W is zero. In the present paper we prove that this definition does not depend on the choice of W, and give a characterization of real varieties dividing over ℚ. For more information on real varieties dividing over ℤ/2 the reader may consult [18] (please note that terminology in [18] is different than here). We also discuss the relationship between real varieties dividing over ℚ (or ℤ) and dividing over ℤ/2. It follows from well-known facts that a real curve is dividing over K if and only if it is dividing.

The space of circles in S3 completed to include points is identified with a complex quadric hypersurface [Hscr ] of CP4. A conformal foliation by arcs of circles of a domain in R3 now corresponds to a complex holomorphic curve in [Hscr ]. The space [Hscr ] minus a hyperplane section fibres in a natural way over the space of lines in R3, each fibre consisting of the well known family of circles of Villarceau transformed by a homothety. As a consequence we show that there is a correspondence between harmonic morphisms on the 3-dimensional space forms and demonstrate the existence of a distinguished surface associated to a conformal foliation by arcs of circles.

We generalize Cartan's isoparametric functions to complex valued functions and classify all those horizontally conformal complex isoparametric functions defined on domains of R3 with arcs of circles or lines as fibres.

We define a classifying space for contractible homotopy pushout diagrams and then study its homotopy type.

Let u be harmonic on the unit ball B of the Euclidean space ℝn, where n[ges ]2. There exist homogeneous harmonic polynomials uj of degree j such that [sum ]∞j=0uj converges absolutely and locally uniformly to u on B. We refer to the series [sum ]∞j=0uj, which is unique, as the polynomial expansion of u. By h∞ we denote the Hardy class of all bounded harmonic functions on B, and if u∈h∞, we write ∥u∥∞=supB[mid ]u[mid ].

Let L0=L0(Ω, [sum ], μ) denote the vector space of (equivalence classes of) measurable functions on a measure space (Ω, [sum ], μ), taking values in a finite-dimensional Hilbert space H. We give L0 the topology τ0 of local convergence in measure[ratio ]τ0 is a complete vector space topology, with base of neighbourhoods

formula here

where

formula here

In this paper we introduced atomic Hardy spaces on product domains Sn−1×Sm−1 and given the L2-boundedness of the Marcinkiewicz integral operator with Hardy space function kernel on Rn×Rm.

This paper relates cyclic cohomology, as obtained from traces with suitable domains on free product algebras, to the construction, by using the quantum stochastic calculus of Hudson and Parthasarathy, of quantum (stochastic) flows. As an application of the methods of the present article, we show how the characters of finitely summable Fredholm modules, as constructed by Connes, can be recovered from quantum flows in both the specialisation to the pure gauge and to the Brownian case. We also relate the regularized trace of Connes, used in the construction of these characters, to Lindblad generators.

We prove an existence theorem for a steady planar flow of an ideal fluid past an obstacle, containing a bounded symmetric vortex pair and approaching a uniform flow at infinity. The vorticity is a rearrangement of a prescribed function.

The stream function ψ for the flow satisfies the equation

−Δψ=ϕ∘ψ

in a region bounded by the line of symmetry, where ϕ is an increasing function that is unknown a priori.

The result is obtained from a variational principle in which a functional related to the kinetic energy is maximised over all flows whose vorticity fields are rearrangements of a prescribed function.