Serdica Mathematical Journal
Volume 25, Number 4, 1999
C O N T E N T S
A B S T R A C T S
A DIFFERENTIAL GAME DESCRIBED BY A HYPERBOLIC SYSTEM
Diko M. Souroujon
1991 Mathematics Subject Classification: 49N55.
Key words:
differential game, eSlater
saddle point, eSlater maximin and minimax,
hyperbolic dynamic system, hyperbolic boundaryvalue problem,
approximat model (scheme).
An antagonistic differential game of hyperbolic
type with a separable linear vector payoff function is considered.
The main result is the description of all eSlater saddle
points consisting of program strategies, program eSlater
maximins and minimaxes for each eÎ
R^{N}_{ > } for this game.
To this purpose,
the considered differential game is reduced to find the optimal
program strategies of two multicriterial problems of hyperbolic
type. The application of approximation enables us to relate these
problems to a problem of optimal program control, described by a
system of ordinary differential equations, with a scalar payoff
function. It is found that the result of this problem is not
changed, if the players use positional or program strategies. For
the considered differential game, it is interesting that the
eSlater saddle points are not equivalent
and there exist two eSlater saddle points
for which the values of all components of
the vector payoff function at one of them are greater than the
respective components of the other esaddle point.
REPRESENTATIONS AND POSITIVE DEFINITE FUNCTIONS ON HYPERGROUPS
A. NasrIsfahani
isfahani@math.ui.ac.ir
1991 Mathematics Subject Classification:
Primary 43A65, 43A35. Secondary 43A62, 43A10.
Key words:
Hypergroup,
representation, positive definite function.
Some relationships between representations of a hypergroup X,
its algebras, and positive definite functions on X are studied.
Also, various types of convergence of positive
definite functions on X are discussed.
TOPOLOGICAL DICHOTOMY AND UNCONDITIONAL CONVERGENCE
Pascal Lefèvre
lefevre@euler.univartois.fr
1991 Mathematics Subject Classification:
42A20, 42A55, 42C10, 43A46, 43A77.
Key words:
Banach space, unconditional
convergence, Sidon sets, quasisure convergence.
In this paper, we give a criterion for unconditional convergence with
respect to some summability methods, dealing with the topological size
of the set of choices of sign providing convergence. We obtain similar
results for boundedness. In particular, quasisure unconditional convergence
implies unconditional convergence.
A CAUCHY INTEGRAL
RELATED TO A ROBOTSAFETY DEVICE SYSTEM
E. J. Vanderperre
evanderperre@hotmail.com
S. S. Makhanov
makhanov@siit.tu.ac.th
1991 Mathematics Subject Classification:
60K10.
Key words:
robot, safety device, invariant measure, availability,
riskcriterion.
We introduce a robotsafety device system attended by two
different repairmen. The twin system is characterized by the natural feature
of cold standby and by an admissible ``risky'' state. In order to analyse
the random behaviour of the entire system (robot, safety device, repair
facility) we employ a stochastic process endowed with probability measures
satisfying general Hokstadtype differential equations. The solution
procedure is based on the theory of sectionally holomorphic functions,
characterized by a Cauchytype integral defined as a Cauchy principal
value in double sense. An application of the SokhotskiPlemelj formulae
determines the longrun availability of the robotsafety device. Finally,
we consider the particular but important case of deterministic repair.
NULL CONDITION FOR SEMILINEAR WAVE EQUATION
WITH VARIABLE COEFFICIENTS
Fabio Catalano
catalano@univaq.it
1991 Mathematics Subject Classification:
35L15.
Key words:
wave equation, conformal group, energy estimates,
hyperbolic rotations, nonlinear Cauchy problem, null condition,
Sobolev spaces, Von Wahl estimates.
In this work we analyse the nonlinear Cauchy problem
(¶_{tt}  D) u(t,x) =
á( lg
+O(  1
(1+t+x )
^{a}
 ))Ñ
_{t,x} u(t,x), Ñ_{t,x} u(t,x)
ñ, 

whit initial data
u(0,x) = e u_{0}(x),
u_{t}(0,x) = e u_{1}(x) .
We assume a ³ 1, x
Î R^{n}
(n ³ 3) and g
the matrix related to the Minkowski space. It can be considerated
a pertubation of the case when the quadratic term has constant
coefficient lg (see Klainerman [6])
We prove a global existence and uniqueness theorem
for very regular initial data. The proof avoids a direct
application of Klainermann method (Null condition, energy conformal method),
because the result is obtained by a combination beetwen the
energy estimate (norm L^{2}) and the decay estimate
(norm L^{¥}).
ALTERNATIVE CHARACTERIZATION OF THE CLASS kUCV
AND RELATED CLASSES OF UNIVALENT FUNCTIONS
Stanislawa Kanas
skanas@prz.rzeszow.pl
1991 Mathematics Subject Classification:
30C45.
Key words:
univalent functions, convex functions, starlike
functions of order b, aconvex
functions, uniformly convex
functions, uniformly starlike functions,
kuniformly convex functions, kstarlike functions.
In this paper an alternative characterization of the class of functions called
kuniformly convex is found. Various relations concerning connections
with other classes of univalent functions are given. Moreover a new class
of univalent functions, analogous to the 'Mocanu class' of functions,
is introduced. Some results concerning this class are derived.
Back