Serdica Mathematical Journal
Volume 28, Number 2, 2002
C O N T E N T S
A B S T R A C T S
CERAMI (C) CONDITION AND MOUNTAIN PASS THEOREM FOR
MULTIVALUED MAPPINGS
A. Kristály
akristal@math.ubbcluj.ro,
Cs. Varga
csvarga@cs.ubbcluj.ro
2000 Mathematics Subject Classification:
58E05, 58E30.
Key words:
Critical point theory, mountain pass, multivalued
functions, deformations, Palais-Smale condition, Cerami condition.
We prove a general minimax result for multivalued mapping.
As application, we give existence results of critical point
of this mapping which satisfies the Cerami (C) condition.
POROSITY AND VARIATIONAL PRINCIPLES
Asit Kumar Sarkar
asit_kumar_sarkar@yahoo.com and
sarkar@123india.com
2000 Mathematics Subject Classification:
33A65.
Key words:
Generating relation, quasi-bilateral generating function.
A group-theoretic method of obtaining more general
class of generating functions from a given class of partial quasi-bilateral
generating functions involving Hermite, Laguerre and
Gegenbaur polynomials are discussed.
DISCRIMINANT SETS OF FAMILIES OF HYPERBOLIC
POLYNOMIALS OF DEGREE 4 AND 5
Vladimir Petrov Kostov
kostov@math.unice.fr
2000 Mathematics Subject Classification:
12D10, 14P05, 26C10.
Key words:
Hyperbolic polynomial, hyperbolicity domain,
overdetermined stratum.
A real polynomial of one real variable is hyperbolic (resp.
strictly hyperbolic) if it has
only real roots (resp. if its roots are real and distinct). We prove that
there are 116 possible non-degenerate configurations between the roots
of a degree 5 strictly
hyperbolic polynomial and of its derivatives (i.e. configurations without
equalities between roots). The standard Rolle theorem allows 286 such
configurations. To obtain the result we study the hyperbolicity
domain of the family P(x;a,b,c) = x^{5}-x^{3}+ax^{2}+
bx+c (i.e. the set of values of a,b,c Î R
for which the polynomial is hyperbolic) and its stratification defined by the
discriminant sets Res(P^{(i)},P^{(j)}) = 0,
0 £ i < j
£ 4.
GROUPS WITH THE MINIMAL CONDITION
ON NON-``NILPOTENT-BY-FINITE'' SUBGROUPS
O. D. Artemovych
artemovych@franko.lviv.ua
2000 Mathematics Subject Classification:
20E16, 20F18, 20F22.
Key words:
Nilpotent-by-finite group, minimal
non-``nilpotent-by-finite'' group, minimal condition.
We characterize the groups which do not have non-trivial perfect sections
and such that any strictly descending chain of
non-``nilpotent-by-finite'' subgroups is finite.
A PRODUCT TWISTOR SPACE
David E. Blair
blair@math.msu.edu
2000 Mathematics Subject Classification:
53C15, 53C26, 53C28.
Key words:
Almost product structures, almost
quaternionic structures of the second kind, product twistor space.
In previous work a hyperbolic twistor space over a paraquaternionic
Kähler manifold was defined, the fibre being the hyperboloid model
of the hyperbolic plane with constant curvature -1. Two almost
complex structures were defined on this twistor space
and their properties studied. In the present paper we consider a
twistor space over a
paraquaternionic Kähler manifold with fibre given by the
hyperboloid of 1-sheet, the
anti-de-Sitter plane with constant curvature -1. This twistor
space admits two natural
almost product structures, more precisely almost para-Hermitian
structures, which form the
objects of our study.
IDEAL CRITERIA FOR BOTH X^{2}-DY^{2} = m_{1}
AND
AND x^{2}-Dy^{2} = m_{2}
TO HAVE PRIMITIVE SOLUTIONS FOR ANY INTEGERS
m_{1}, m_{2} PRIME TO D > 0
R. A. Mollin
ramollin@math.ucalgary.ca
2000 Mathematics Subject Classification:
11D09, 11A55, 11R11.
Key words:
continued fractions, Diophantine equations,
fundamental units, simultaneous solutions, ideals, norm form equations.
This article provides necessary and sufficient conditions for both
of the Diophantine equations
X^{2}-DY^{2} = m_{1} and x^{2}-Dy^{2} =
m_{2} to have primitive solutions when
m_{1},m_{2} Î Z, and D
Î N is not
a perfect square. This is given in terms of the ideal theory of
the underlying real quadratic order Z[ÖD].
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