In this paper we establish a theory of (quasi-)Fredholm elements in the setting of Jordan pairs. As for associative algebras, Barnes idempotents are introduced and a new numerical invariant, namely the cocapacity, is defined.

The study of the differential geometry of surfaces in 3-space has a long and celebrated history. Over the last 20 years a new approach using techniques from singularity theory has yielded some interesting results (see, for example [3, 5, 19] for surveys).

Of course surfaces arise in a number of ways: they are often defined explicitly as the image of a mapping f: R2→R3. Since the subject is differential geometry one normally asks that these defining mappings are smooth, that is infinitely differentiable, however it is not true, in any sense, that most such parametrisations will yield manifolds. For such mappings have self-intersections, and more significantly they may have crosscaps (also known as Whitney umbrellas). Moreover if we perturb the maps these singularities will persist; that is they are stable. (See [14, 20] for details.)

Consequently when studying the differential geometry of surfaces in 3-space there are good reasons for studying surfaces with crosscaps.

In this paper we carry out a classification of mappings from 3-space to lines, up to changes of co-ordinates in the source preserving a crosscap. We can apply our results to the geometry of generic crosscap points. In [10] we computed geometric normal forms for the crosscap and used them to study the dual of the crosscap. We shall see that the approach here yields more information than that obtained in [10], although the latter has the advantage of being more explicit (in particular various aspects of the geometry can be compared using the normal forms).

We refer the reader to [16, 18] for background material concerning singularity theory.

The arc index α(L) of a link L is shown by a direct combinatorial argument to be related to sprv(FL(v, z)), the Laurent degree of its Kauffman polynomial, by the inequality

α(L)[ges ]sprv (FL(v, z))+2.

Equality is conjectured for alternating links.

In this paper, we prove the following:

Theorem. Let C⊂ℙ3be a complex reduced irreducible space curve which is non- degenerate (i.e., C is not contained in any plane). Let p: C→ℙ2be a general projection. Then the topological fundamental group π1(ℙ2/p(C)) is abelian.

K-loops have their origin in the theory of sharply 2-transitive groups. In this paper a proof is given that K-loops and Bruck loops are the same. For the proof it is necessary to show that in a (left) Bruck loop the left inner mappings L(b)L(a) L(ab)−1 are automorphisms. This paper generalizes results of Glauberman [3], Kist [8] and Kreuzer [9].

Our main purpose is to explore the relationship between normal subgroups of Hecke groups and regular maps on compact orientable surfaces. We use regular maps to find all the normal subgroups of Hecke groups of index [les ]60. In Sections 1–4 we review the basic results concerning Hecke groups and normal subgroups of Fuchsian groups and in Section 5 we outline the basic facts we need about regular maps. The regular maps of genus 0 are the Platonic solids and in Section 6 we use these to completely determine the genus zero normal subgroups of Hecke groups. The regular maps of genus 1 were determined this century by Brahana [1]. We use them in Sections 7 and 8 to determine the genus 1 normal subgroups of Hecke groups. We give alternative proofs and extend theorems of M. Newman and also Kern-Isberner and Rosenberger. Unlike the genus 0 and 1 cases there are only finitely many regular maps of genus g[ges ]2. In Section 9 we use more recent results concerning their classification to find all normal subgroups of Hecke groups of index [les ]60. This was done for the modular group in [13].

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.

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 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.

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.

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.

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 ∞.

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.

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 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.

(Volume 116 (1994), 147–149)

This note is to correct [2], where the proofs of the theorems stated require extra conditions on the manifolds. This is because it is assumed tacitly in [2] that a locally square integrable ∂-closed (p, q)-form on a manifold Ω with a piecewise strictly pseudoconvex boundary is ∂-exact. This is equivalent to Hq(Ω, Dp)=0, q[ges ]1, where Dp is the sheaf of germs of holomorphic p-forms on Ω. This is not true in general, but true if Ω is Stein; but to assume Ω is Stein is too restrictive a condition.