Nonlinear Liouville theorems for Grushin and Tricomi operators

The aim of this paper is to study necessary conditions for existence of weak solutions of the inequality

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Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

Combings of groups and the grammar of reparameterization

A new construction of combings is used to distinguish between several previously indistinguishable classes of groups associated to the theory of automatic groups and non-positive curvature in group theory. We construct synchronously bounded combings for a class of groups that are neither bicombable nor automatic. The linguistic complexity of these combings is analysed: in many cases the language of words in the combing is an indexed language.     

Tame kernels and further 4-rank densities

There has been recent progress on computing the 4-rank of the tame kernel 458-si65.gif"> for F a quadratic number field. For certain quadratic number fields, this progress has led to “density results” concerning the 4-rank of tame kernels. These results were first mentioned in Conner and Hurrelbrink (J. Number Theory 88 (2001) 263) and proven in Osburn (Acta Arith. 102 (2002) 45). In this paper, we consider some additional quadratic number fields and obtain further density results of 4-ranks of tame kernels. Additionally, we give tables which might indicate densities in some generality.     

Volumes and diophantine inequalities associated with decomposable forms

For homogeneous decomposable forms in n variables with real coefficients, we consider the associated volume of all real solutions to the inequality . We relate this to the number of integral solutions to the Diophantine inequality in the case where F has rational coefficients. We find quantities which bound the volume and which yield good upper bounds on the number of solutions to the Diophantine inequality in the rational case.     

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .     

Unitals in Topological Affine Translation Planes Need Not Be Strictly Convex

We study the relationship of two incidence geometric convexity notions, namely, ovoids in real affine spaces and compact unitals of codimension 1 in topological affine translation planes. In [3] we showed that every ovoid in a translation plane is a unital, and we asked if the converse is true. Here we introduce the notion of a shell, which is distinctly weaker than that of an ovoid and still implies the unital property if the translation plane is properly chosen (and the shell is not too degenerate). We give an explicit example of a shell that is not an ovoid. The question remains whether or not conversely, every compact unital of codimension 1 in a translation plane is a shell.This paper was written while the third author was supported by a grant from DFG and TÜBITAK.Received March 12, 2002 Published online November 18, 2002     

Additive Functions with Respect to Numeration Systems on Regular Languages

Asymptotic formulae for the summatory function of additive arithmetic functions related to numeration systems given by regular languages are derived.This author is supported by the START project Y96-MAT of the Austrian Science Fund.This author warmly thanks P. Grabner for his kind invitation to Graz and his hospitality.Received February 18, 2002; in revised form September 20, 2002 Published online May 15, 2003     

Asymptotic behaviour of linear evolutionary integral equations

The asymptotic behaviour of bounded solutions of evolutionary integral equations in a Banach spaceX

Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh _n that imply that the sequence (1+h _n (x))e ^inx is a Riesz basis forL ²[–,]. These conditions involve theL ²-norm of the second derivativesh _n . The new results replace the differential operatoryy by more general differential operators inL ²-spaces, in particular, by the Hermite differential operator inL ²(R), ande ^inx by arbitrary orthonormal bases.