We consider IA-endomorphisms (i.e. Identical in Abelianization) of a free metabelian group of finite rank, and give a matrix characterization of their fixed points which is similar to (yet different from) the well-known characterization of eigenvectors of a linear operator in a vector space. We then use our matrix characterization to elaborate several properties of the fixed point groups of metabelian endomorphisms. In particular, we show that the rank of the fixed point group of an IA-endomorphism of the free metabelian group of rank n[ges ]2 can be either equal to 0, 1, or greater than (n−1) (in particular, it can be infinite). We also point out a connection between these properties of metabelian IA-endomorphisms and some properties of the Gassner representation of pure braid groups.

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.

In [10] it was shown that the element Sq2n has nilpotence height 2n+2 in the mod2 Steenrod algebra. Here the method and result are generalised to show that for an odd prime p the element Ppn has nilpotence height p(n+1) in the modp Steenrod algebra. Some of the other results of [10] are also generalised to odd primes p.

Let [Ascr ]* be the mod-2 Steenrod algebra of cohomology operations and χ its canonical antiautomorphism. For all positive integers f and k, we show that the excess of the element χ[Sq (2k−1f)· Sq (2k−2f)… Sq (2f)·Sq (f)] is (2k−1)μ(f), where μ(f) denotes the minimal number of summands in any representation of f as a sum of numbers of the form 2i−1. We also interpret this result in purely combinatorial terms. In so doing, we express the Milnor basis representation of the products Sq (a1)…Sq (an) and χ[Sq (a1)…Sq (an)] in terms of the cardinalities of certain sets of matrices.

For s[ges ]1, let ℙs=[ ]2= [x1, …, xs] be the mod-2 cohomology of the s-fold product of ℝP∞ with itself, with its usual structure as an [Ascr ]*-module. A polynomial P∈ℙs is hit if it is in the image of the action [Ascr ]*×ℙs→ℙs, where [Ascr ]* is the augmentation ideal of [Ascr ]*. We prove that if the integers e, f, and k satisfy e<(2k−1)μ(f), then for any polynomials E and F of degrees e and f respectively, the product E·F2k is hit. This generalizes a result of Wood conjectured by Peterson, and proves a conjecture of Singer and Silverman.

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.

Let G be a locally compact group and let L1(G) be its group algebra. This paper demonstrates that the order structure of the second dual algebra L1(G)** identifies the topological group G.

Let Γ be a connected G-symmetric graph of valency r, whose vertex set V admits a non-trivial G-partition [Bscr ], with blocks B∈[Bscr ] of size v and with k[les ]v independent edges joining each pair of adjacent blocks. In a previous paper we introduced a framework for analysing such graphs Γ in terms of (a) the natural quotient graph Γ[Bscr ] of valency b=vr/k, and (b) the 1-design [Dscr ](B) induced on each block. Here we examine the case where k=v and Γ[Bscr ]=Kb+1 is a complete graph. The 1-design [Dscr ](B) is then degenerate, so gives no information: we therefore make the additional assumption that the stabilizer G(B) of the block B acts 2-transitively on B. We prove that there is then a unique exceptional graph for which [mid ]B[mid ]=v>b+1.

We give a condition on a full coaction (A, G, δ) of a (possibly) non-amenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A×δ[mid ]G/N is Morita equivalent to A×δG×δˆrN. This condition obtains if N is amenable or δ is normal. It is preserved under Morita equivalence, inflation of coactions, the stabilization trick of Echterhoff and Raeburn, and on passing to twisted coactions.

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

Analysis of projections of a convex body is a familiar topic in tomography. However, instead of considering standard projection bodies, this work investigates a convex body introduced by Schneider [8] which is a Minkowski average of projections. The question addressed here is similar to that posed by Goodey and Weil [4] with respect to Minkowski averages of sections, as opposed to projections, that is, can the shape of a convex body be determined from random sections? Their main result shows that a body K is determined by the average of its two-dimensional sections, but not by the average of its one-dimensional sections. The goal of this study is to uncover the extent to which a convex body is determined by the average of its projections.

In [10] Loeb and Osswald develop a Daniell–Stone integration theory in topological vector lattices. As an application they obtain an extension of the Bochner integral for functions with values in a topological vector lattice. The Daniell–Stone approach requires an order structure for definition of a null function. Here we build on the ideas in [10], but abandon the lattice structure. The internal representatives of the extended integrable functions in [10] turn out to be S-integrable liftings. This is our starting point: we define the class of extended integrable functions as the class of functions that have S-integrable liftings; a definition that is independent of any order structure. As a consequence, several of the proofs in the construction of the Banach space of extended integrable functions are almost identical to the analogous proofs in [10] and we omit them. After defining the integral we show that each extended integrable function induces a countably additive vector measure of bounded variation. This permits us to use the tools of the theory of vector measures. If the nonstandard hull of the Banach space has the Radon–Nikodym property, then the vector measure induced by an extended integrable function has a Bochner integrable Radon–Nikodym derivative; this yields a characterization of the Banach space of extended integrable functions. On the other hand for some spaces we can construct internal functions and standardize them to get very ‘spiky’ extended integrable functions whose induced measure is not differentiable on any set of positive measure. With this method we show that if a Banach space is not superreflexive, then it does not have the super-Radon–Nikodym property.

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 ].

In an earlier paper [MR] the authors introduced the inverse measure μ[dagger](dt) of a given measure μ(dt) on [0, 1] and presented the ‘inversion formula’ f[dagger](α)=αf(1/α) which was argued to link the respective multifractal spectra of μ and μ[dagger]. A second paper [RM2] established the formula under the assumption that μ and μ[dagger] are continuous measures.

Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation μ[map ]μ[dagger] creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the ‘fine multifractal spectra’ and not for the ‘coarse’ ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate Hölder exponents 0 and ∞.

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

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where

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We show that a centrally symmetric convex body is determined, up to translation, by any Radon transform of one of its projection functions. It is also proved that all Radon transforms are injective when restricted to projection functions of polytopes. Both results were known previously in the case of Radon transforms Ri,j with i<j. Here we use harmonic analysis on Grassmann manifolds to show that these results extend to all Radon transforms.

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.

The problem is raised of finding the best possible constant in the maximal Khintchine inequality for Rademacher sequence ε=(εk)k[ges ]1:

formula here

being valid for all a1, …, an∈R with n[ges ]1, where 0<p<∞ is given and fixed. We conjecture that the best possible constant is

formula here

where B=(Bt[ges ]0) is standard Brownian motion. For simplicity, we consider only the case p=1 and prove that this conjecture is as close to the truth as desired in the following asymptotic sense:

formula here

being valid for all [mid ]a1[mid ][les ]1, …, [mid ]an[mid ][les ]1 and all n[ges ]1, where Sk =[sum ]ki=1aiεi and ∥a→n∥2 =([sum ]nk=1 [mid ]ak[mid ]2) ½[ges ]2. It should be noted here that

formula here

The method of proof relies upon Skorohod's imbedding. Motivated by consequences of this result we deduce in a purely computational way that:

formula here

whenever a1=1, a2=λ, a3=λ2, …, an=λn−1 and λ belongs to ]0, 1/2] with n[ges ]1. The constant 2/√3 is shown to be the best possible in this inequality.

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.

We consider the scattering of time-harmonic electromagnetic plane waves by a piecewise homogeneous obstacle. In the interior of the obstacle there exists a core which may be a dielectric, a perfect conductor or an imperfect conductor. First, we obtain integral representations for the scattered fields incorporating all the boundary and transmission conditions. These representations are then used to derive the corresponding electric far field patterns. We prove reciprocity and scattering theorems and study the spectrum of the electric far field operator for these problems. Finally, we prove the optical theorem, that is a connection of the electric far field pattern to the scattering cross-section.