In continuation of earlier work [7], we study in this paper the structure of cocommutative Hopf algebras of finite representation type. Since every such algebra is the group algebra H([Gscr ]) of a finite algebraic group [Gscr ], this problem has two interrelated aspects. On the one hand, one has to understand the structure of the group scheme [Gscr ], while on the other the Morita equivalence class of H([Gscr ]) has to be determined. Although we shall here be mainly concerned with the latter, structural information on the underlying group schemes enters crucially at several points.

We show that if G is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group G*, such that G is a subgroup of G* but G is not quasiconvex in G*. We also prove that any non-elementary subgroup of a torsion-free word hyperbolic group G contains a free group of rank 2 which is malnormal and quasiconvex in G.

Let Aθ be the universal C*-algebra generated by a pair U, V of unitaries satisfying VU = e2πiθUV where θ = p/q with p, q∈ℤ, 0 < p < q and (p, q) = 1. It is shown that every endomorphism of Aθ is unital, injective and centre preserving. The corresponding surjections of T2 are characterized and it is shown that the determinants of the associated endomorphisms of K1(Aθ)≅ℤ2 are congruent to 1 modulo q.

In his celebrated paper [7], Martinet showed the equivalence between the infinitesimal versality and the versality for [Ascr ]- and [Kscr ]-morphisms. By using this theorem, he obtained the following Theorem 1·1 which played one of the key roles for the classification of C∞ stable map-germs ([1, 7]).

We show that in the finite dimensional space ℝd provided with a metric induced by a norm, the collection of Borel sets is the smallest collection containing the open balls and closed under complements and countable disjoint unions.

A recent result of Bestvina and Brady [1, theorem 8·7], shows that one of two outstanding questions has a negative answer; either there exists a group of cohomological dimension 2 and geometric dimension 3 (a counterexample to the Eilenberg–Ganea Conjecture [4]), or there exists a nonaspherical subcomplex of an aspherical 2-complex (a counterexample to the Whitehead Conjecture [11]). More precisely, Bestvina and Brady construct a family of groups which are potential counterexamples to the Eilenberg–Ganea Conjecture, each of which has cohomological dimension 2. These are also examples of groups of type FP2 which are not finitely presented (see [1]). Dicks and Leary [3] give an explicit way of obtaining presentations (on finite generating sets) for these groups. For some of these examples, it is shown in [1] that any 2-dimensional classifying space would give rise to a counterexample to the Whitehead conjecture.

We will refer to the examples cited above as Bestvina–Brady groups. These come equipped with natural, nonpositively curved cubical 3-dimensional classifying complexes, which we will call Bestvina–Brady complexes. In this short note, we show that these Bestvina–Brady complexes are (up to homotopy equivalence) formed by applying the Quillen plus construction to certain finite 2-complexes. From this, together with known facts about 2-complexes with aspherical plus constructions, we recover the result of Bestvina and Brady [1] that the Bestvina–Brady groups act freely on acyclic 2-complexes, and hence have cohomological dimension at most 2. It also follows that these groups have free relation modules of finite rank and so are of type FF. Finally, we use our construction to give an alternative proof of the cited theorem of Bestvina and Brady; at least one of the Eilenberg–Ganea and Whitehead conjectures is false.

We do not use the full force of the Morse-theoretical techniques developed in [1], but will assume two results form that paper; the asphericity of the Bestvina–Brady complexes and the non-finite presentability of the Bestvina–Brady groups.

The sphere SX = {x∈X: 〈x, x〉 = 1} of a right Hilbert C*-module X over a unital C*-algebra B is studied using differential geometric techniques. An action of the unitary group of the algebra [Lscr ]B(X) of adjointable B-module operators of X makes SX a homogeneous space of this group. A reductive structure is introduced, as well as a Finsler metric. Metric properties of the geodesic curves are established. In the case B a von Neumann algebra and X self-dual, the fundamental group of SX is computed.

The set of asymptotic values of a continuous function on the open unit disc in ℝ2 forms an analytic set, in the sense of being a continuous image of a Polish space (complete, separable metric space). This was proved in [9] by J. E. McMillan, who had earlier given versions of this result for holomorphic and meromorphic functions. We extend his method to the case of a function on the open unit ball of ℝn which is continuous merely in the fine topology, the coarsest topology making all subharmonic functions continuous. In particular, we use a version of McMillan's ingenious metric on a certain space of equivalence classes of asymptotic paths. McMillan also proved in [9] that the set of point asymptotic values of a continuous function in the unit disc forms an analytic set. We use a modification of the McMillan metric to extend this result to fine continuous functions in the unit ball and deduce that the set of boundary points of the unit ball at which the function has an asymptotic value forms an analytic set.

The automorphism group of a p-group acts naturally on the mod-p cohomology of the group. This action need not be faithful, but we show that it contains every simple module as a composition factor.

We consider arbitrary polynomial perturbations

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of the harmonic oscillator. In (1), f and g are polynomials of x, y with coefficients depending analytically on the small parameter ε. Let us denote n = max (deg f, deg g), H = ½(x2 + y2). Using the energy level H = h as a parameter, we can express the first return mapping of (1) in terms of h and ε. For the corresponding displacement function d(h, ε) = [Pscr ](h, ε)−h we obtain the following representation as a power series in ε:

formula here

which is convergent for small ε. The Melnikov functions Mk(h) are defined for h[ges ]0. Each isolated zero h0∈ (0, ∞) of the first non-vanishing coefficient in (2) corresponds to a limit cycle of (1) emerging from the circle x2 + y2 = 2h0 when ε increases from zero. Our main result in this paper is the following.

We define a sequence of purely algebraic invariants – namely, classes in the Quillen cohomology of the Π-algebra π∗X – for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Π-algebra can be realized as the homotopy Π-algebra of a space.

Numerical simulations and group invariance considerations point to the existence of similarity solutions of the form

formula here

of the generalized Korteweg–de Vries equation

formula here

Here, x∗, t∗ and c are real parameters, x and t are real variables with t≠t∗, p is a positive integer and interest is focussed on the case where p[ges ]4 for which solutions of the initial-value problem for (**) are not known to be always globally defined. It is shown that smooth solutions of (**) of the form appearing in (*) do indeed exist. Some detailed properties of the function ψ appearing in (*) are also obtained.

We study Δ(x; ϕ), the error term in the asymptotic formula for [sum ]n[les ]xcn, where the cns are generated by the Rankin–Selberg series. Our main tools are Voronoï-type formulae. First we reduce the evaluation of Δ(x; ϕ) to that of Δ1(x; ϕ), the error term of the weighted sum [sum ]n[les ]x(x−n)cn. Then we prove an upper bound and a sharp mean square formula for Δ1(x; ϕ), by applying the Voronoï formula of Meurman's type. We also prove that an improvement of the error term in the mean square formula would imply an improvement of the upper bound of Δ(x; ϕ). Some other related topics are also discussed.

Given a uniformly bounded representation of a locally compact group, we consider the closed circled convex hull K of the orbit of a vector. We call K a simple motion system (SMS) and endow its linear hull with the Minkowski functional of K. The representation theory on these ‘SMS-spaces’ is discussed, in particular for C0-representations, for irreducible representations of connected groups and for integrable representations. As an application we give a criterion for the decomposibility of representations.

We construct univalent functions in the unit disc, whose coefficient sequences (an) have arbitrarily long intervals of zeros and at the same time arbitrarily long intervals where [mid ]an[mid ] > nεn holds, (εn) being an arbitrary prescribed sequence of positive numbers tending to zero. Furthermore we show that the initial interval of coefficients of such a function can be prescribed to be any interior point of the coefficient region.

By studying dressing orbits a complete account is given of the (full) super-horizontal holomorphic maps of finite type from the 2-plane into a full flag manifold. It is shown that precisely one dressing orbit contains all 2-spheres of finite type. From this follows a simple description of all harmonic 2-spheres in CPn of finite type.

This paper is devoted to a global existence theorem of meromorphic solutions of the form Z(z)=Zo(z)+R(z) of a nonlinear Riemann–Hilbert problem (RHP) for multiply connected domains Gq(q[ges ]1), where Zo(z) is the singular part of the solution, R(z) is the regular part which is a holomorphic solution of some appropriate nonlinear RHP for Gq(q[ges ]1). Under appropriate conditions on the characteristics of both the singular part Zo(z) (number of poles) and regular part (winding number) we prove the existence of meromorphic solutions Z(z) of the form Z(z)=Zo(z)+R(z). The proof is based on a special construction of the singular part Zo(z) and an adequate formulation of Newton's method for the regular part R(z).

We study the weak star accumulation points of sequences of probability measures of the form [XKn(x) ρn(x)dσ(x)]/ [∫Kn ρn(t)dσ(t)], where ρ(x)>0 is continuous on Rκ, σ denotes Lebesgue measure in Rκ and the sequence of compact sets Kn⊂Rκ converges in the sense of Hausdorff towards a compact set K. The motivation of our study was given by a result relating interval averages with the winding number. Similar probability measures are considered in partial differential equation problems and we extend our study to this case.