In this note we obtain a sharp estimate for a radial derivative of bounded harmonic functions in the ball.

Mutants of boundary links may not be boundary links, not even homology boundary links. Hence mutants of homology boundary links may not be homology boundary links.

Let R be a prime ring of characteristic not equal to two and let T be an automorphism of R. If U is a Lie ideal of R such that T is nontrivial on U and xxT — xTx is in the center of R for every x in U, then U is contained in the center of R.

For any X and any q > 0, one has natural inclusions where the groups S1 and S3 act on S4q-1 in the standard way and are the G-invariant homotopy subsets, G = S1 or G = S3. It is proved here that for any space X of the homotopy type of a CW-complex and for π4q-1 (X) in the c3 cl stable range, the inclusion is m fact an equality when localized away from the prime 2.

It is known that the n-th denominators Qn (α, β, z) of a real J-fraction

where

form an orthogonal polynomial sequence (OPS) with respect to a distribution function ψ(t) on ℝ. We use separate convergence results for continued fractions to prove the asymptotic formula

the convergence being uniform on compact subsets of

It is proved that a compact subset of a finite product of Brelot harmonic spaces is multipolar if it is a locally multipolar set.

Dans ce travail, on étudie les propriétés de la conorme, c(a) d'un élément a d'une C*-algèbre A. Nous montrons que c(a) est aussi douée de la condition de B*- algèbre, c'est-à-dire qu'elle vérifie l'égalité: c(a)2 = c(a*a) = c(aa*) = c(a*)2. Nous remarquons que la conorme est invariante par restriction à toute sous-C*-algèbre de A, propriété connue pour le spectre.

Soit {an} une suite d'éléments réguliers telle que an → a régulier. Nous donnons des conditions nécessaires et suffisantes pour que où a† désigne l'inverse de Moore-Penrose de a.

We consider uncountable products of nontrivial compact, convex subsets of normed linear spaces. We show that these products do not have the complete invariance property i.e. they include a nonempty, closed subset which is not a fixed point set (i.e. the set of all fixed points) for any continuous mapping from the product into itself. In particular we give an answer to W.Weiss' question whether uncountable powers of the unit interval have the complete invariance property.

Using second epi derivatives, we introduce a generalized second fundamental form for Lipschitzian hypersurfaces. In the case of a convex hypersurface, our approach leads back to the classical second fundamental form, which is usually obtained from the second fundamental forms of the outer parallel surfaces by means of a limit procedure.

We study the relative trace formula of a reductive group over an algebraic number field. Following Langlands we stabilize the geometric side of the relative trace formula contributed by the elliptic regular double cosets.

In general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

We consider a piecewise monotonie and piecewise continuous map T on the interval. If T has a derivative of bounded variation, we show for an ergodic invariant measure μ with positive Ljapunov exponent λμ that the Hausdorff dimension of μ equals hμ / λμ.

It is shown that if a solvable group is not locally finite, then the group algebra over a field of characterisitc 0 has no nonzero algebraic ideals.

The u-invariant u(K) of a field K is the smallest number such that every k-dimensional regular quadratic form over K is universal. Let O(V,f) be the orthogonal group of a finite-dimensional regular metric vector space over a field K of characteristic distinct from 2. Let π ∊ 0(V), B(π) := V(π - 1), dim[B(π) ∩ kernel(π - 1)] > u(K). Given λ1,..., λm ∊ K* where m := dim B(π) - u(K) + 1. Then π = σ1······σk where k := dim B(π) and σi is a symmetry with negative space Kai and f(ai, ai) = λi for i ∊ { 1 , . . . , m}. We prove similar theorems also for symplectic groups where transvections are taken as generators.

A complete lattice L is constructively completely distributive,(CCD)(L), if the sup map defined on down closed subobjects has a left adjoint. We characterize preservation of this property by left exact functors between toposes using a "logical comparison transformation". The characterization is applied to (direct images of) geometric morphisms to show that local homeomorphisms (in particular, product functors) preserve (CCD) objects, while preserving (CCD) objects implies openness.

In their book Singularities and Groups in Bifurcation Theory M. Golubitsky, I. Stewart and D. Schaeffer have introduced an equivariant version of Martinet's notion of V (for variety)-equivalence with parameter. In this paper we give a unified proof that, in this context, infinitesimal stability is equivalent to stability at the local level of germs and that stability in the unfolding category is equivalent to versality.

Let be polynomials orthogonal with respect to different distributions. Conditions are given which imply that the coefficients in the expansion of Pn in terms of Q0, Q1,..., Qn are non-negative.

The annihilating polynomials for trace forms, as discovered recently by Conner, are shown to also annihilate many other classes of positive quadratic forms over a field F provided that F satisfies suitable conditions.

In this paper we obtain the bv0-norms for several well-known classes of matrix operators.

We give a characterization of reflexive modules over QF-3 rings generalizing the concept of linearly compact modules. Further, we study necessary and sufficient conditions for left QF-3 rings to be right QF-3.