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     

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.     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

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     

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     

A characterization of $$ \cal N $$ -constrained groups

We prove that a finite group G is -constrained if and only if it contains a nilpotent subgroup I satisfying for all .Received: 22 July 2002     

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.     

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     

Join congruence relations and closure relations

In this paper we show that, given a complete lattice , the following three lattices are the same: (1) the lattice of closure relations on , (2) the lattice of meet-closed subsets of , and (3) the lattice of complete join congruence relations on .     

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.     

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 note on the 3-rank of quadratic fields

The difference between the 3-rank of the ideal class group of an imaginary quadratic field and that of the associated real quadratic field is equal to 0 or 1. In this note, we give an infinite family of examples in each case.Received: 9 September 2002     

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     

Coloured Permutations Containing and Avoiding Certain Patterns

Let be the set of all coloured permutations on the symbols 1, 2, . . . , n with colours 1, 2, . . . , r, which is the analogous of the symmetric group when r = 1, and the hyperoctahedral group when r = 2. Let be a subset of d colours; we define to be the set of all coloured permutations . We prove that the number of -avoiding coloured permutations in . We then prove that for any , the number of coloured permutations in which avoid all patterns in except for and contain exactly once equals . Finally, for any , this number equals . These results generalize recent results due to Mansour, Mansour and West, and Simion.AMS Subject Classification: 05A05, 05A15.     

Linear spaces with many projective planes

We assume that in a linear space there is a non-empty set M of points with the property that every plane containing a point of M is a projective plane. In section 3 an example is given that in general is not a projective space. But if M can be completed by two points to a generating set of P, then is a projective space.