On the structure of level sets of uniform and Lipschitz quotient mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{R} $$

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient mappings from . We show that if , is a uniform quotient mapping then for every has a bounded number of components, each component of separates and the upper bound of the number of components depends only on and the moduli of co-uniform and uniform continuity of .Next we prove that all level sets of any co-Lipschitz uniformly continuous mapping from to are locally connected, and we show that for every pair of a constant and a function with , there exists a natural number , so that for every co-Lipschitz uniformly continuous map with a co-Lipschitz constant and a modulus of uniform continuity , there exists a natural number and a finite set with card so that for all has exactly components, has exactly components and each component of is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of for are also described - they have a finite tree structure.     

Some remarks on isologic extensions of groups

In this paper we study the concept of -isologisms among the -marginal extensions of groups, with respect to a given variety of groups . We also give some equivalent conditions under which two extensions are -isologic. Received: 9 January 2002     

Set of involutions arising from an egglike inversive plane

To every egglike inversive plane there is associated a family of involutions of the point set of such that circles of are the fixed point sets of the involutions in . Korchmaros and Olanda characterized a family of involutions on a set of size n² + 1to be for an egglike inversive plane of order n by four conditions. In this paper, we give an alternative proof where the Galois space PG(3,n) in which is embedded is built up directly by using concepts and results on finite linear spaces.     

Investigation of subalgebra lattices by means of Hasse constants

Hasse constants and their basic properties are introduced to facilitate the connection between the lattice of subalgebras of an algebra and the natural action of the automorphism group Aut( ) on . These constants are then used to describe the lattice of subloops of the smallest nonassociative simple Moufang loop.     

Cyclicity of zeros of families of analytic functions

Let be a family of holomorphic functions in the unit disk , which are also holomorphic in a parameter . We express cyclicity (=generalized multiplicity) of a zero of at via some algebraic characteristics of the ideal generated by the Taylor coefficients of . As an example we estimate the cyclicity of the family of generalized exponential polynomials.     

A cohomological property of <Emphasis Type="Italic">p</Emphasis>-groups

Let p be a prime, a finite p-group, any finite group with order divisible by p, and any action of on . We show that the cardinality of the set of all derivations with respect to this action is a multiple of p. This generalises theorems of Frobenius and Hall. Received: 16 June 2003     

A classification of martingale Hardy spaces associated with rearrangement-invariant function spaces

Let X be a rearrangement-invariant Banach function space over a complete probability space , and denote by the Hardy space consisting of all martingales such that . We prove that implies for any filtration if and only if Doobs inequality holds in X, where denotes the martingale defined by , n = 0, 1, 2, ..., and a.s.Received: 1 August 2000     

The Laplacian on $$ C(\overline \Omega) $$ with generalized Wentzell boundary conditions

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in defined by generates an analytic semigroup of angle on for every > 0 and (for the definition of cf. (1.3)).Received: 13 July 2002     

Equidistribution of Kronecker sequences along closed horocycles

It is well known that (i) for every irrational number the Kronecker sequence m (m = 1,...,M) is equidistributed modulo one in the limit , and (ii) closed horocycles of length become equidistributed in the unit tangent bundle of a hyperbolic surface of finite area, as . In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant the Kronecker sequence embedded in along a long closed horocycle becomes equidistributed in for almost all , provided that . This equidistribution result holds in fact under explicit diophantine conditions on (e.g. for = 2) provided that , with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for , our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence n² modulo one.     

Transformations of Grassmannians preserving the class of base subsets

Let be a finite-dimensional projective space and be the Grassmannian consisting of all k-dimensional subspaces of . In the paper we show that transformations of sending base subsets to base subsets are induced by collineations of to itself or to the dual projective space . This statement generalizes the main result of the authors paper [19].     

A class of bounded operators on Sobolev spaces

We describe a class of nonlinear operators which are bounded on the Sobolev spaces , for and 1 < p < . As a corollary, we prove that the Hardy-Littlewood maximal operator is bounded on , for and 1 < p < ; this extends the result of J. Kinnunen [7], valid for s = 1. Received: 5 December 2000     

The modular automorphisms of the Drinfeld modular curve $$ X_{1}(\frak n) $$

In this work we determine the group of modular automorphisms of the Drinfeld modular curve for any nonconstant polynomial .Received: 19 September 2002     

Cancellative residuated lattices

Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( -groups). Although cancellative monoids are defined by quasi-equations, the class of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to -groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of .We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of -groups, hence the latter form a variety, denoted by . Furthermore we prove that the map that sends a subvariety of -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of to the lattice of subvarieties of . Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzies characterization of categorically equivalent varieties.     

The dimension of the kernel in finite and infinite intersections of starshaped sets

Summary. We establish the following Helly-type result for infinite families of starshaped sets in Define the function f on {1, 2} by f(1) = 4, f(2) = 3. Let be a fixed positive number, and let be a uniformly bounded family of compact sets in the plane. For k = 1, 2, if every f(k) (not necessarily distinct) members of intersect in a starshaped set whose kernel contains a k-dimensional neighborhood of radius , then is a starshaped set whose kernel is at least k-dimensional. The number f(k) is best in each case. In addition, we present a few results concerning the dimension of the kernel in an intersection of starshaped sets in Some of these involve finite families of sets, while others involve infinite families and make use of the Hausdorff metric.     

A note on the isotopy problem for $$ \mathbb{R}^2 $$ -planes

For a class of stable planes we define a notion of isotopy equivalence with respect to that class and prove that any two planes of a certain class of -planes comprising all affine -planes are isotopy equivalent. Furthermore we obtain that all affine -planes are isotopy equivalent in the class of affine -planes. Finally we give an example which shows that this approach cannot be easily generalized to 2-dimensional projective planes, and we outline a different way for a possible generalization.Received: 27 April 2001     

Spectral triangles of Schrödinger operators with complex potentials

Consider the Schrödinger operator with a complex-valued potential v of period Let and be the eigenvalues of L that are close to respectively, with periodic (for n even), antiperiodic (for n odd), and Dirichelet boundary conditions on [0,1], and let be the diameter of the spectral triangle with vertices We prove the following statement: If then v(x) is a Gevrey function, and moreover