Serdica Mathematical Journal
Volume 23, Number 34, 1997
C O N T E N T S

The Fiftieth Anniversary of the Institute of Mathematics and Informatics

Augustynowicz, A.
Existence and uniqueness of solutions for partial
differentialfunctional equations of the first order with deviating argument
of the derivative of unknown function
(pp. 203210)

Danchev, P. V.
Isomorphism of commutative modular group algebras
(pp. 211224)

Kolev, D.
Involutivity and symple waves in R^{2}
(pp. 225232)

Choban, M. M.
Functionally countable spaces and Baire functions
(pp. 233242)

Elabbasy, E. M.
Oscillation theorems for second order sublinear ordinary
differential equations
(pp. 243254)

De Blasi, F. S.
On typical compact convex sets in Hilbert spaces
(pp. 255268)

Matthes, K., K. Nawrotzki, R. SiegmundSchultze
On the structure of spatial branching processes
(pp. 269312)

Kirchev, K. P., G. S. Borisova
Commuting nonselfadjoint operators and
their characteristic operatorfunctions
(pp. 313334)

Plichko, A. M.
Decomposition of Banach space into a direct sum
of separable and reflexive subspaces and Borel maps
(pp. 335350)

Fabian, M., P. Hájek, V. Zizler.
Uniform Eberlein compacta and uniformly Gâteaux smooth norms
(pp. 351362)

Holá, L'.
Coincidence of Vietoris and Wijsman topologies: a new proof
(pp. 363366)
A B S T R A C T S
EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR PARTIAL
DIFFERENTIALFUNCTIONAL EQUATIONS OF THE FIRST ORDER WITH
DEVIATING ARGUMENT OF THE DERIVATIVE OF UNKNOWN FUNCTION
A. Augustynowicz
1991 Mathematics Subject Classification:
35F25.
Key words:
differentialfunctional equations, partial of the
first order, existence and uniqueness, deviating argument of derivative
We consider the existence and uniqueness problem for partial
differentialfunctional equations of the first order with the initial
condition for which the righthand side depends on the derivative of
unknown function with deviating argument.
ISOMORPHISM OF COMMUTATIVE MODULAR GROUP ALGEBRAS
P. V. Danchev
1991 Mathematics Subject Classification:
Primary: 20C07; Secondary 20K10, 20K21.
Key words:
isomorphism, commutative group algebras, units, direct sum of cyclics,
splitting groups.
Let K be a field of characteristic p > 0 and let G
be a direct sum of cyclic groups, such that its torsion part is a
pgroup.
If there exists
a Kisomorphism KH @ KG for some group H,
then it is shown that H @ G.
Let G be a direct sum of cyclic groups, a divisible group or a simply
presented torsion abelian group. Then KH @ KG
as Kalgebras for all fields K
and some group H if and only if H @ G.
INVOLUTIVITY AND SIMPLE WAVES IN R^2
Dimitar Kolev
1991 Mathematics Subject Classification:
35L40, 35L45, 35L60, 35L65.
Key words:
simple wave, simple state, involutivity, Riemann inveriant.
A strictly hyperbolic quasilinear 2 x 2 system in two
independent variables with C^{2} coefficients is considered.
The existence of a simple wave solution in the sense that the solution
is a 2dimensional vectorvalued function of the so called Riemann
invariant is discussed. It is shown, through a purely geometrical approach,
that there always exists simple wave solution
for the general system when the coefficients are arbitrary
C^{2} functions depending on both, dependent and independent
variables.
FUNCTIONALLY COUNTABLE SPACES AND BAIRE FUNCTIONS
M. M. Choban
1991 Mathematics Subject Classification:
54H05, 54C20, 28A20.
Key words:
functionally countable space, Baire space,
Baire set, distinguished set, Baire function, measurable mapping.
The concept of the distinguished sets is applied to the
investigation of the functionally countable spaces. It is
proved that every Baire function on a functionally countable
space has a countable image. This is a positive answer to
a question of R. Levy and W. D. Rice.
OSCILLATION THEOREMS FOR SECOND ORDER SUBLINEAR ORDINARY
DIFFERENTIAL EQUATIONS
E. M. Elabbasy
1991 Mathematics Subject Classification:
34C15.
Key words:
second order differential equation, oscillation.
Oscillation criteria are given for the second order sublinear nonautonomous
differential equation.
(r(t)y(x)x¢(t))¢+q(t)g(x(t)) = f(t). 

These criteria extends and improves earlier oscillation criteria of Kamenev,
Kura, Philos and Wong. Oscillation criteria are also given for second
order sublinear damped nonautonomous differential equations.
ON TYPICAL COMPACT CONVEX SETS IN HILBERT SPACES
F. S. De Blasi
1991 Mathematics Subject Classification:
Primary 41A65, 54E52; secondary 46B20.
Key words:
compact convex set, metric
antiprojection, multivalued locus, Baire category.
Let E be an infinite dimensional separable space and
for e Î E and X a nonempty compact convex subset of E,
let q_{X}(e) be the metric antiprojection of e on X. Let n ³ 2
be an arbitrary integer. It is shown that for a typical (in the sence of the
Baire category) compact convex set X Ì E the metric antiprojection
q_{X}(e) has cardinality at least n for every e in a
dense subset of E.
ON THE STRUCTURE OD SPATIAL BRANCHING PROCESSES
Klaus Matthes, Kurt Nawrotzki, Rainer SiegmundSchultze
1991 Mathematics Subject Classification:
Primary: 60J80; Secondary: 60J10, 60G60.
Key words:
Branching particle systems, twosided infinite Markov sequences of a
random populations, genealogy, Poisson distribution
The paper is a contribution to the theory of branching processes with
discrete time and a general phase space in the sense of [2]. We
characterize the class of regular, i.e. in a sense sufficiently
random, branching processes (F_{k})_{k Î Z}
by almost sure properties of their
realizations with
out making any assumptions about stationarity or existence
of moments. This enables us to classify the clans of (F_{k}) into
the regular part and the completely nonregular part. It turns
out that the completely nonregular branching processes are built up from
singleline processes, whereas the regular ones are mixtures of
lefttail trivial processes with a Poisson family structure.
COMMUTING NONSELFADJOINT OPERATORS AND
THEIR CHARACTERISTIC OPERATORFUNCTIONS
K. P. Kirchev, G. S. Borisova
1991 Mathematics Subject Classification:
Primary 47A48, Secondary 60G12.
Key words:
operator colligation, complete characteristic operatorfunction, joint
characteristic function, trunk
In this paper we present some generalizations of
results of M. S. Liv sic [4, 6], concerning regular colligations
(A_{1},A_{2}, H, F, E, s_{1}, s_{2},g, (~g))
(s_{1} > 0) of a pair of commuting nonselfadjoint
operators A_{1}, A_{2} with finite dimensional imaginary parts, their complete
characteristic functions and a class W(s_{1}, s_{2}) of operatorfunctions
W(x_{1},x_{2},z) : E® E
in the case of an inner function W(1,0,z) of the class
W(s_{1}).
We consider regular colligations
(A_{1},..., A_{n},H,F,E,s_{1},...,s_{n},{g_{k1}}_{2}^{n},{(~g)_{k1}}^{n}_{2}) (s_{1} > 0) of ntuples (n > 2) of
commuting nonselfadjoint operators A_{1},A_{2},...,A_{n}
with finite dimensional
imaginary parts, their complete characteristic functions and a description
of a class W_{G}(s_{1},...,s_{n})
of operatorfunctions W(x_{1}, ...,x_{n},z):E® E in the
case when W(1,0,... ,0,z) is not inner function of the class
W(s_{1})
(s_{1} > 0, n ³ 2). We
essentially use the conditions for the operators {s_{k}}_{1}^{n},
{g_{k1}}^{n}_{2}, {(~g)_{k1}}^{n}_{2}
that V. A. Zolotarev has considered in [9].
DECOMPOSITION OF BANACH SPACE INTO A DIRECT SUM
OF SEPARABLE AND REFLEXIVE SUBSPACES AND BOREL MAPS
Anatolij M. Plichko
1991 Mathematics Subject Classification:
46B26.
Key words:
Banach space, Borel map.
The main results of the paper are:
Theorem 1. Let a Banach space E be decomposed into a direct sum of
separable and reflexive subspaces. Then for every Hausdorff locally
convex topological vector space Z and for every linear continuous
bijective operator T:E® Z, the inverse T^{1} is a Borel map.
Theorem 2. Let us assume the continuum hypothesis. If a Banach space
E cannot be decomposed into a direct sum of separable and reflexive
subspaces, then there exists a normed space Z and a linear continuous
bijective operator T:E® Z such that T^{1} is not a Borel map.
UNIFORM EBERLEIN COMPACTA
AND UNIFORMLY GÂTEAUX SMOOTH NORMS
Marián Fabian, Petr Hájek, Václav Zizler
1991 Mathematics Subject Classification:
46B03, 46B20.
Key words:
uniform Eberlein compacta, uniform Gâteaux smooth
norms, weak compact generating.
It is shown that the dual unit ball B_{X*} of a Banach space
X^{*} in its weak star topology is a uniform Eberlein compact
if and only if X admits a uniformly Gâteaux smooth norm and
X is a subspace of a weakly compactly generated space. The
bidual unit ball B_{X**} of a Banach space X^{**}
in its weak star topology is a uniform Eberlein compact if and only
if X admits a weakly uniformly rotund norm. In this case X
admits a locally uniformly rotund and Fréchet differentiable norm.
An Eberlein compact K is a uniform Eberlein compact if and only
if C(K) admits a uniformly Gâteaux differentiable norm.
COINCIDENCE OF VIETORIS AND
WIJSMAN TOPOLOGIES: A NEW PROOF
L'. Holá
1991 Mathematics Subject Classification:
54B20.
Key words:
Vietoris topology, Wijsman topology, metric space, compact space.
Let (X,d) be a metric space and CL(X) the family of all
nonempty closed subsets of X. We provide a new proof of
the fact that the coincidence of the Vietoris and Wijsman
topologies induced by the metric d forces X to be a compact
space. In the literature only a more involved and indirect proof
using the proximal topology is known. Here we do not need this
intermediate step. Moreover we prove that (X,d) is boundedly
compact if and only if the bounded Vietoris and Wijsman topologies
on CL(X) coincide.
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