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.     

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.     

Estimates of Marcinkiewicz Integrals with Bounded Homogeneous Kernels of Degree Zero

Under the cancellation property and a certain Dini-type condition on kernels, we prove that Marcinkiewicz integrals with kernels which are homogeneous functions of degree zero, are bounded from to , from to , and from to for .     

Stability of the Cauchy congruence on restricted domain

Let S be a real interval with , and be a function satisfying We show that if h is Lebesgue or Baire measurable, then there exists such that That result is motivated by a question of E. Manstaviius. Received: 11 February 2003     

Finite splitting fields of normal subgroups

Let G be a finite group, a normal subgroup, p a prime, a finite splitting field of characteristic p for G and We prove that is a splitting field for N, using the action of the Galois group of the field extension on the irreducible representations of N. As is a splitting field for the symmetric group S_n we get as a corollary that is a splitting field for the alternating group A_n. Received: 31 July 2003     

A Change-of-Variable Formula with Local Time on Curves

Let be a continuous semimartingale and let be a continuous function of bounded variation. Setting and suppose that a continuous function is given such that F is C^1,2 on and F is on . Then the following change-of-variable formula holds: where is the local time of X at the curve b given by and refers to the integration with respect to . A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.     

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     

On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

We consider the three-dimensional Schrödinger operators and where , A is a magnetic potential generating a constant magnetic field of strength , and where decays fast enough at infinity. Then, A. Pushnitskis representation of the spectral shift function (SSF) for the pair of operators is well defined for energies We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. Communicated by Bernard HelfferSubmitted 23/09/03, accepted 15/01/04     

Extensions of distance preserving mappings in euclidean and hyperbolic geometry

Suppose that X is a real inner product space of (finite or infinite) dimension at least 2. A distance preserving mapping , where is a (finite or infinite) subset of a finite-dimensional subspace of X, can be extended to an isometry of X. This holds true for euclidean as well as for hyperbolic geometry. To both geometries there exist examples of non-extentable distance preserving , where S is not contained in a finite-dimensional subspace of X.     

Remarks on ample vector bundles of curve genus two

Let be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a -bundle over and that for any fiber F of the bundle projection . The pairs with = 2 are classified, where is the curve genus of . This allows us to improve some previous results. Received: 13 June 2006     

Uniformly summing sets of operators on spaces of vector–valued continuous functions

Let X, Y be Banach spaces. We say that a set is uniformly p–summing if the series is uniformly convergent for whenever (x_n) belongs to . We consider uniformly summing sets of operators defined on a -space and prove, in case X does not contain a copy of c₀, that is uniformly summing iff is, where T (φ x) = (T^#φ) x for all and x∈X. We also characterize the sets with the property that is uniformly summing viewed in . Received: 1 July 2005     

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 class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

The Ideal Structure of Toeplitz Algebras

We investigate the ideal structure of the Toeplitz algebra of a totally ordered abelian group . We show that the primitive ideals of are parametrised by the disjoint union of the duals of the order ideals of , and identify the hull-kernel topology on when the chain of orderideals in is isomorphic to a subset of      

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      

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.     

Proof of the Ergodic Hypothesis for Typical Hard Ball Systems

We consider the system of hard balls with masses and radius r in the flat torus of size . We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case . The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic. Communicated by Eduard ZehnderSubmitted 17/10/02, accepted 01/12/03