A B S T R A C T S

L_p EXTREMAL POLYNOMIALS. RESULTS AND PERSPECTIVES
Yamina Laskri minalaskri@yahoo.fr
Rachid Benzine rabenzine@yahoo.fr

2000 Mathematics Subject Classification: 30C40, 30D50, 30E10, 30E15, 42C05.
Key words: Asymptotic behaviour, orthogonal polynomials, L_p extremal polynomials, curve, arc, circle, segment.

Let α = β+γ be a positive finite measure defined on the Borel sets of C, with compact support, where β is a measure concentrated on a closed Jordan curve or on an arc (a circle or a segment) and γ is a discrete measure concentrated on an infinite number of points.
In this survey paper, we present a synthesis on the asymptotic behaviour of orthogonal polynomials or L_p extremal polynomials associated to the measure α. We analyze some open problems and discuss new ideas related to their solving.

MODULI STACKS OF POLARIZED K3 SURFACES IN MIXED CHARACTERISTIC
Jordan Rizov jordan.rizov@nl.abnamro.com

2000 Mathematics Subject Classification: 14J28, 14D22.
Key words: K3 surfaces, Moduli spaces.

In this note we define moduli stacks of (primitively) polarized K3 spaces. We show that they are representable by Deligne-Mumford stacks over Spec(Z). Further, we look at K3 spaces with a level structure. Our main result is that the moduli functors of K3 spaces with a primitive polarization of degree 2d and a level structure are representable by smooth algebraic spaces over open parts of Spec(Z). To do this we use ideas of Grothendieck, Deligne, Mumford, Artin and others.
These results are the starting point for the theory of complex multiplication for K3 surfaces and the definition of Kuga-Satake abelian varieties in positive characteristic given in our Ph.D. [J. Rizov. Moduli of K3 Surfaces and Abelian Variaties. Ph. D. thesis, University of Utrecht, 2005]. thesis.

HAUSDORFF MEASURES OF NONCOMPACTNESS AND INTERPOLATION SPACES
Eduardo Brandani da Silva ebsilva@uem.br
Dicesar L. Fernanadez dicesar@ime.unicamp.br

2000 Mathematics Subject Classification: 46B50, 46B70, 46G12.
Key words: Banach spaces, measures of noncompactness, interpolation.

A new measure of noncompactness on Banach spaces is defined from the Hausdorff measure of noncompactness, giving a quantitative version of a classical result by R. S. Phillips. From the main result, classical results are obtained now as corollaries and we have an application to interpolation theory of Banach spaces.

SOME GENERALIZATION OF DESARGUES AND VERONESE CONFIGURATIONS
Malgorzata Prazmowska malgpraz@math.uwb.edu.pl
Krzysztof Prazmowski krzypraz@math.uwb.edu.pl

2000 Mathematics Subject Classification: 51E14, 51E30.
Key words: Desargues configuration, Veblen configuration, partial Steiner triple system, graph, combinatorial Grassmannian, combinatorial Veronesian.

We propose a method of constructing partial Steiner triple system, which generalizes the representation of the Desargues configuration as a suitable completion of three Veblen configurations. Some classification of the resulting configurations is given and the automorphism groups of configurations of several types are determined.

ON THE RESIDUUM OF CONCAVE UNIVALENT FUNCTIONS
K.-J. Wirths kjwirths@tu-bs.de

2000 Mathematics Subject Classification: 30C25, 30C45.
Key words: Concave univalent functions, domain of variability, residuum.

Let D denote the open unit disc and f:D→[`C] be meromorphic and injective in D. We further assume that f has a simple pole at the point p О (0,1) and is normalized by f(0) = 0 and f′(0) = 1. In particular, we are concerned with f that map D onto a domain whose complement with respect to [`C] is convex. Because of the shape of f(D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p).
We determine for fixed p ∈ (0,1) the set of variability of the residuum of f, f ∈ Co(p).

MULTIPLIERS ON SPACES OF FUNCTIONS ON A LOCALLY COMPACT ABELIAN GROUP WITH VALUES IN A HILBERT SPACE
Violeta Petkova petkova@math.u-bordeaux1.fr

2000 Mathematics Subject Classification: Primary 43A22, 43A25.
Key words: multipliers, translations, spaces of vector-valued functions.

We prove a representation theorem for bounded operators commuting with translations on L²_ω(G,H), where G is a locally compact abelian group, H is a Hilbert space and ω is a weight on G. Moreover, in the particular case when G = R, we characterize completely the spectrum of the shift operator S_1,ω on L_ω²(R,H).

KADEC NORMS ON SPACES OF CONTINUOUS FUNCTIONS
Maxim R. Burke burke@upei.ca
Wiesaw Kubis wkubis@pu.kielce.pl
Stevo Todorcevic stevo@logique.jussieu.fr

2000 Mathematics Subject Classification: Primary: 46B03, 46B26. Secondary: 46E15, 54C35.
Key words: t_p-Kadec norm, Banach space of continuous functions, compact space.

We study the existence of pointwise Kadec renormings for Banach spaces of the form C(K). We show in particular that such a renorming exists when K is any product of compact linearly ordered spaces, extending the result for a single factor due to Haydon, Jayne, Namioka and Rogers. We show that if C(K₁) has a pointwise Kadec renorming and K₂ belongs to the class of spaces obtained by closing the class of compact metrizable spaces under inverse limits of transfinite continuous sequences of retractions, then C(K₁×K₂) has a pointwise Kadec renorming. We also prove a version of the three-space property for such renormings.

FINITE GROUPS AS THE UNION OF PROPER SUBGROUPS
Jiping Zhang jzhang@pku.edu.cn

2000 Mathematics Subject Classification: 20D60,20E15.
Key words: Finite Group, simple group, covering number.

As is known, if a finite solvable group G is an n-sum group then n − 1 is a prime power. It is an interesting problem in group theory to study for which numbers n with n-1 > 1 and not a prime power there exists a finite n-sum group. In this paper we mainly study finite nonsolvable n-sum groups and show that 15 is the first such number. More precisely, we prove that there exist no finite 11-sum or 13-sum groups and there is indeed a finite 15-sum group. Results by J. H. E. Cohn and M. J. Tomkinson are thus extended and further generalizations are possible.

CAYLEY-HAMILTON THEOREM FOR MATRICES OVER AN ARBITRARY RING
Jeno Szigeti jeno.szigeti@uni-miskolc.hu

2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.
Key words: The preadjoint and the right characteristic polynomial of an n×n matrix, the commutator subgroup [R,R] of a ring R.

For an n×n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients:

(λ0I+C0)+A(λ1I+C1)+… +An-1(λn-1I+Cn-1)+An (n!I+Cn) = 0,

where λ₀+λ₁x+…+λ_n-1 x^n-1+n!xⁿ is the right characteristic polynomial of A in R[x], I ∈ M_n(R) is the identity matrix and the entries of the n×n matrices C_i, 0 ≤ i ≤ n are in [R,R]. If R is commutative, then

C0 = C1 = … = Cn-1 = Cn = 0

and our identity gives the n! times scalar multiple of the classical Cayley-Hamilton identity for A.