Szigeti, J.
CayleyHamilton theorem for matrices over an arbitrary ring
(pp. 269276)
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 DeligneMumford
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 KugaSatake
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@tubs.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 Î (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.ubordeaux1.fr
2000 Mathematics Subject Classification:
Primary 43A22, 43A25.
Key words:
multipliers, translations, spaces of vectorvalued functions.
We prove a representation theorem for bounded operators commuting
with translations on L^{2}_{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_{ω}^{2}(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(K_{1}) has a pointwise
Kadec renorming and K_{2} 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(K_{1}×K_{2}) has a pointwise Kadec renorming. We also prove a version of the threespace 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 nsum group
then n − 1 is a prime power. It is an interesting problem in group
theory to study for which numbers n with n1 > 1 and not a prime
power there exists a finite nsum group. In this paper we mainly
study finite nonsolvable nsum groups and show that 15 is the
first such number. More precisely, we prove that there exist no
finite 11sum or 13sum groups and there is indeed a finite 15sum
group. Results by J. H. E. Cohn and M. J. Tomkinson are thus extended and further
generalizations are possible.
CAYLEYHAMILTON THEOREM FOR MATRICES OVER AN ARBITRARY RING
Jeno Szigeti
jeno.szigeti@unimiskolc.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 CayleyHamilton identity with right matrix coefficients:
(λ_{0}I+C_{0})+A(λ_{1}I+C_{1})+…
+A^{n1}(λ_{n1}I+C_{n1})+A^{n}
(n!I+C_{n}) = 0, 

where λ_{0}+λ_{1}x+…+λ_{n1}
x^{n1}+n!x^{n}
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
C_{0} = C_{1} = … = C_{n1} = C_{n} = 0 

and our identity gives the n! times scalar multiple of the
classical CayleyHamilton identity for A.
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