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.     

Weingarten rotation surfaces in 3-dimensional de Sitter space

In the 3-dimensional de Sitter Space , a surface is said to be a spherical (resp. hyperbolic or parabolic) rotation surface, if it is a orbit of a regular curve under the action of the orthogonal transformations of the 4-dimensional Minkowski space which leave a timelike (resp. spacelike or degenerate) plane pointwise fixed. In this paper, we give all spacelike and timelike Weingarten rotation surfaces in .     

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.     

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      

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].     

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.     

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     

Sharp Conditions for the CLT of Linear Processes in a Hilbert Space

In this paper we study the behavior of sums of a linear process associated to a strictly stationary sequence with values in a real separable Hilbert space and are linear operators from H to H. One of the results is that satisfies the CLT provided are i.i.d. centered having finite second moments and . We shall provide an example which shows that the condition on the operators is essentially sharp. Extensions of this result are given for sequences of weak dependent random variables under minimal conditions.     

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     

Two-weight Weak-type Norm Inequalities for the Commutators of Fractional Integrals

In this paper, we obtain a sufficient condition for two-weight weak type norm inequality for the commutators of fractional integral , where and .     

Rigidity of minimal hypersurfaces of spheres with two principal curvatures

Let M be a compact oriented minimal hypersurface of the unit n-dimensional sphere Sⁿ. It is known that if the norm squared of the second fundamental form, , satisfies that for all , then M is isometric to a Clifford minimal hypersurface ([2], [5]). In this paper we will generalize this result for minimal hypersurfaces with two principal curvatures and dimension greater than 2. For these hypersurfaces we will show that if the average of the function is n - 1, then M must be a Clifford hypersurface. Received: 24 December 2002     

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.     

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

Note on a Theorem of Munkres

We prove that given a Riemannian manifold with boundary, having a finite number of compact boundary components, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result extends to manifolds and to embedded PL manifolds of dimensions 2, 3 and 4. We employ these results to prove that manifolds of the types above admit quasimeromorphic mappings onto As an application we prove the existence of G-automorphic quasimeromorphic mappings, where G is a Kleinian group acting on Dedicated to the memory of Robert BrooksThis paper represents part of the authors Ph.D. thesis written under the supervision of Prof. Uri Srebro.     

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