Bedratyuk, L.
A note about the Nowicki conjecture
on Weitzenböck derivations
(pp. 311316)
A B S T R A C T S
INDICE DE POINT FIXE POUR LES MORPHISMES DE CHAÎNES
Robert Cauty
cauty@math.jussieu.fr
2000 Mathematics Subject Classification:
54H25, 55M20.
Key words:
Fixed point index.
The aim of this paper is to define a fixed point index for compact maps
in the class of algebraic ANRs. This class, which we introduced in [2], contains all open
subsets of convex subsets of metrizable topological vector spaces. In this class, it is
convenient to study the fixed points of compact maps with the help of the chain morphisms
that they induce on the singular chains. For this reason, we first define a fixed point
index for a certain class of chain morphisms, and then define the fixed point index of
compact maps as the fixed point index of the induced chain morphism. This fixed point
index has all the usual properties of an index, including the mod ptheorem. The
results of this paper are thus, in the metrizable case, a vast generalization of the
Schauder conjecture.
ON THE VERTEX FOLKMAN NUMBERS F_{v}(2,...,2;q)
Nedyalko Nenov
nenov@fmi.unisofia.bg
2000 Mathematics Subject Classification:
05C55.
Key words:
Folkman numbers, vertex coloring, edge coloring.
For a graph G the symbol
(a_{1},...,a_{r}) means
that in every rcoloring of the vertices of G
for some i {1,...,r} there exists a monochromatic
a_{i}clique of color i.
The vertex Folkman numbers
F_{v}(a_{1},...,a_{r};q) = 
min
 {V(G):G (a_{1},...,a_{r}) and K_{q} G} 

are considered. In this paper we shall compute the Folkman numbers
when k
12 and r is sufficiently
large. We prove also new bounds for some vertex and edge Folkman numbers.
GEOMETRY OF WARPED PRODUCT SEMIINVARIANT SUBMANIFOLDS OF A LOCALLY RIEMANNIAN PRODUCT MANIFOLD
Mehmet Atçeken
matceken@gop.edu.tr
2000 Mathematics Subject Classification:
53C42, 53C15.
Key words:
Riemannian warped product, Riemannian product and warped product semiinvariant.
In this article, we have studied warped product semiinvariant
submanifolds in a locally Riemannian product manifold and introduced
the notions of a warped product semiinvariant submanifold. We have
also proved several fundamental properties of a warped product
semiinvariant in a locally Riemannian product manifold.
CLASS NUMBER TWO FOR REAL QUADRATIC FIELDS OF RICHAUDDEGERT TYPE
R. A. Mollin
ramollin@math.ucalgary.ca
2000 Mathematics Subject Classification:
Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.
Key words:
Quadratic fields, primeproducing polynomials, class numbers, continued fractions, cycles of ideals, RichaudDegert types.
This paper contains proofs of conjectures made in [16] on class number 2 and what this author has dubbed the EulerRabinowitsch polynomial for real quadratic fields. As well, we complete the list of RichaudDegert types given in [16] and show how the behaviour of the EulerRabinowitsch polynomials and certain continued fraction expansions come into play in the complete determination of the class number 2 problem for such types. For some values the determination is unconditional, and for others, the wide RichaudDegert types, the determination is conditional on the generalized Riemann hypothesis (GRH).
COMPOUND COMPOUND POISSON RISK MODEL
Leda D. Minkova
leda@fmi.unisofia.bg
2000 Mathematics Subject Classification:
60K10, 62P05.
Key words:
Compound Poisson process, PólyaAeppli
risk model, ruin probability, CramérLundberg approximation.
The compound Poisson risk models are widely used in practice. In
this paper the counting process in the insurance risk model is a
compound Poisson process. The model is called Compound Compound Poisson Risk Model. Some basic properties and ruin probability are given. We analyze the model under the proportional reinsurance. The optimal retention level and the corresponding adjustment coefficient are obtained. The particular case of the PólyaAeppli risk model is discussed.
A NOTE ABOUT THE NOWICKI CONJECTURE
ON WEITZENBÖCK DERIVATIONS
Leonid Bedratyuk
leonid.uk@gmail.com
2000 Mathematics Subject Classification:
13N15, 13A50, 16W25.
Key words:
Classical invariant theory, covariants of binary form, derivations.
We reduce the Nowicki conjecture on Weitzenböck derivations of polynomial algebras to a well known problem of classical invariant theory.
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