Serdica Mathematical Journal

Volume 23, Number 3-4, 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 differential-functional equations of the first order with deviating argument of the derivative of unknown function (pp. 203-210)
• Danchev, P. V. Isomorphism of commutative modular group algebras (pp. 211-224)
• Kolev, D. Involutivity and symple waves in R² (pp. 225-232)
• Choban, M. M. Functionally countable spaces and Baire functions (pp. 233-242)
• Elabbasy, E. M. Oscillation theorems for second order sublinear ordinary differential equations (pp. 243-254)
• De Blasi, F. S. On typical compact convex sets in Hilbert spaces (pp. 255-268)
• Matthes, K., K. Nawrotzki, R. Siegmund-Schultze On the structure of spatial branching processes (pp. 269-312)
• Kirchev, K. P., G. S. Borisova Commuting nonselfadjoint operators and their characteristic operator-functions (pp. 313-334)
• Plichko, A. M. Decomposition of Banach space into a direct sum of separable and reflexive subspaces and Borel maps (pp. 335-350)
• Fabian, M., P. Hájek, V. Zizler. Uniform Eberlein compacta and uniformly Gâteaux smooth norms (pp. 351-362)
• Holá, L'. Coincidence of Vietoris and Wijsman topologies: a new proof (pp. 363-366)

A B S T R A C T S

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR PARTIAL DIFFERENTIAL-FUNCTIONAL EQUATIONS OF THE FIRST ORDER WITH DEVIATING ARGUMENT OF THE DERIVATIVE OF UNKNOWN FUNCTION
A. Augustynowicz

1991 Mathematics Subject Classification: 35F25.
Key words: differential-functional equations, partial of the first order, existence and uniqueness, deviating argument of derivative

We consider the existence and uniqueness problem for partial differential-functional equations of the first order with the initial condition for which the right-hand 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 p-group. If there exists
a K-isomorphism 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 K-algebras 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 quasi-linear 2 x 2 system in two independent variables with C² coefficients is considered. The existence of a simple wave solution in the sense that the solution is a 2-dimensional vector-valued 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 C² 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 non-autonomous 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 non-autonomous 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 Siegmund-Schultze

1991 Mathematics Subject Classification: Primary: 60J80; Secondary: 60J10, 60G60.
Key words: Branching particle systems, two-sided 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 non-regular part. It turns out that the completely non-regular branching processes are built up from single-line processes, whereas the regular ones are mixtures of left-tail trivial processes with a Poisson family structure.

COMMUTING NONSELFADJOINT OPERATORS AND THEIR CHARACTERISTIC OPERATOR-FUNCTIONS
K. P. Kirchev, G. S. Borisova

1991 Mathematics Subject Classification: Primary 47A48, Secondary 60G12.
Key words: operator colligation, complete characteristic operator-function, joint characteristic function, trunk

In this paper we present some generalizations of results of M. S. Liv sic [4, 6], concerning regular colligations (A₁,A₂, H, F, E, s₁, s₂,g, (~g)) (s₁ > 0) of a pair of commuting nonselfadjoint operators A₁, A₂ with finite dimensional imaginary parts, their complete characteristic functions and a class W(s₁, s₂) of operator-functions W(x₁,x₂,z) : E® E in the case of an inner function W(1,0,z) of the class W(s₁). We consider regular colligations (A₁,..., A_n,H,F,E,s₁,...,s_n,{g_k1}₂ⁿ,{(~g)_k1}ⁿ₂) (s₁ > 0) of n-tuples (n > 2) of commuting nonselfadjoint operators A₁,A₂,...,A_n with finite dimensional imaginary parts, their complete characteristic functions and a description of a class W_G(s₁,...,s_n) of operator-functions W(x₁, ...,x_n,z):E® E in the case when W(1,0,... ,0,z) is not inner function of the class W(s₁) (s₁ > 0, n ³ 2). We essentially use the conditions for the operators {s_k}₁ⁿ, {g_k1}ⁿ₂, {(~g)_k1}ⁿ₂ 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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