African Journals Online
Quaestiones Mathematicae

VOLUME 24 ISSUE 2 (2001)

Quaestiones Mathematicae 24(2) 2001, 129-132
© 2001 NISC Pty Ltd, www.nisc.co.za  

Absolute Retracts as Reduced Products

M Jenner, P Jipsen, P Ouwehand and H Rose

Department of Mathemetics and Applied Mathematics, University of Cape Town, 7701, Rondebosch, South Africa.

e-mails: {mary, jipsen, peter0, rose}@maths.uct.ac.za

Abstract: We show that in finitely generated congruence distributive varieties every absolute retract is a product of reduced powers of maximal subdirectly irreducibles.

Mathematics Subject Classification (1991): 08B15, 08B25

Keywords: absolute retracts, reduced products, maximal irreducibles, lattices of varieties, products,amalgamated products,and other kinds of limits and colimits, distributive, product

Quaestiones Mathematicae 24(2) 2001, 133-146
© 2001 NISC Pty Ltd, www.nisc.co.za 

Optimal Systems of Symmetry Subalgebras for Big Models of Gasdynamics

SV Khabirov

Institute for Symmetry Analysis and Mathematical Modelling, University of North-West, Mmabatho 2735, South Africa.

e-mail: [email protected] 

Abstract: The gasdynamics equations admit 11 dimensional Lie algebra of transformations for the arbitrary state equations. For special state equations the algebra may be expanded. There are 13 special state equations for which the gasdynamics equations are named big models. Some of the algebras are similar.
The algebra structure may be different under further specialization of the state equations. It was noted in the calculation process of the expansion of optimal systems that only 2 such specializations need to be added.
We make a review of extensions for which the optimal systems of subalgebras are calculated. Each of the extensions are presented as semidirect sums of Abelian 6 dimensional ideal and subalgebra. The normalized optimal systems of these subalgebras are calculated. This will be helpful in the calculation of the full optimal systems.

Mathematics Subject Classification (1991): 35Q35

Keywords: symmetry, other equations arising in fluid dynamics, gasdynamics, state equation, Lie algebra of transformations, optimal systems of subalgebras, Lie, algebra, transformations, extensions

Quaestiones Mathematicae 24(2) 2001, 147-155
© 2001 NISC Pty Ltd, www.nisc.co.za 

Is the Construct of L-Topological Spaces a Co-Tower Extension of Some Simpler Construct?

Horst Herrlich ¹ and Dexue Zhang ²

¹ Fachbereich 3, Mathematik, University of Bremen, 28334, Germany.

e-mail:  [email protected] 

² Department of Mathematics, Sichuan University, Chengdu 610064, China.

e-mail: [email protected]

Abstract: In an earlier paper, the second author associated with any fibre-small topological construct $\bold A$ and any completely distributive lattice L, an extension of $\bold A$, called the (L)-co-tower extension of $\bold A$. He demonstrated that many familiar constructs in fuzzy topology can be expressed as co-tower extensions of more basic constructs and asked whether the construct L-$\bold Top$ of L-topological spaces (the stratified Chang-Goguen spaces) can be expressed as a co-tower extension. In this note we answer this question with "no" and "yes". In particular we present the following theorems:
(1) If L is a 3-element linearly ordered lattice (or, more generally, any linearly ordered finite lattice with at least 3 elements), then L-$\bold Top$ cannot be expressed as an L-co-tower extension.
(2) If L is a 4-element Boolean algebra (or, more generally, any atomic complete Boolean algebra), then L-$\bold Top$ is (up to concrete isomorphism) the L-co-tower extension of the construct $\bold Top$ of topological spaces.

Mathematics Subject Classification (1991): 54B30, 54A40, 18B30

Keywords: topological construct, L-Topological space, co-tower extension, completely distributive lattice, categorical methods, fuzzy topology, categories of topological spaces and continuous mappings, distributive, lattice, extensions, Boolean algebra, algebra, complete, topological space

Quaestiones Mathematicae 24(2) 2001, 157-163
© 2001 NISC Pty Ltd, www.nisc.co.za 

Spread Uniformities and Uniform Spreads

Hlengani J Siweya

Department of Mathematics, University of the Norht, Private Bag X1106, Sovenga, 0727, South Africa.

e-mail: [email protected] 

Abstract: We discuss a point free analog of the topological fact that every uniformly continuous function on a dense subspace of a uniform space into a complete uniform space has a unique continuous extension. It is shown that a uniform frame homomorphism $\h : L \rightarrow M$ from a complete uniform frame L into a uniform frame M (not necessarily complete) onto which a dense surjection $\rho : N \rightarrow M$ maps has a unique uniform extension $\g : L \rightarrow N$ such that $\rho \circ g = h$. We introduce spread uniformities and uniform spreads into the theory of uniform frames, and, then, show that the extension so constructed is a uniform spread.

Mathematics Subject Classification (1991): 54D15, 54D20, 54E15

Keywords: uniform frame, uniform local connectedness, spread uniformity, uniform spread, spread omomorphism, higher separation axioms, noncompact covering properties, uniform structures and generalization, dense, uniformspace, complete, frame, frames

Quaestiones Mathematicae 24(2) 2001, 165-173
© 2001 NISC Pty Ltd, www.nisc.co.za 

Bases for Cones and Reflexivity

Ioannis A Polyrakis

Department of Mathematics, National Technical University of Athens, Zographou 157 80, Athens, Greece.

e-mail: [email protected] 

Abstract: It is proved that a Banach space E is non-reflexive if and only if E has a closed cone with an unbounded, closed, dentable base. If E is a Banach lattice, the same characterization holds with the extra assumption that the cone is contained in $\E_{+}$. This article is also a survey of the geometry (dentability) of bases for cones.

Mathematics Subject Classification (1991): 46A25, 46A40, 46B10, 46B22, 46B42

Keywords: Radon-Nikodym property, dentability/unbounded convex sets, reflexivity and semi-reflexivity, ordered topological linear spaces,vector lattices, duality and reflexivity, Krein-Milman, Banach lattices, Banach, banach space, Banach lattice, lattice

Quaestiones Mathematicae 24(2) 2001, 175-181
© 2001 NISC Pty Ltd, www.nisc.co.za 

On Projective Motion in Finsler Spaces II

SP Singh and JK Gatoto

Department of Mathematics, Egerton University, P.O. Box 536, Njoro, Kenja.

e-mail: [email protected] 

Abstract: K. Takano [4] in a series of papers has studied and developed affine motion in non-Riemann $\K^{*}$-spaces. R.S. Sinha [6] has studied affine motions in recurrent Finsler spaces. The existence of projective motion in a asymmetric Finsler space was discussed by F. Meher [7].
Let $\xi^{i}$ be a contravariant vector generating an infinitesimal mapping $\eta : F_{n} \rightarrow \bar{F}_{n}$ of type $\bar{x}^{i} = x^{i} + \xi^{i}(x)dt$. Such a mapping, when preserving the parallelism of a pair of any two vectors in $\F_{n}$, defines an affine motion. The above infinitesimal transformation defines a projective if it transforms the system of geodesics into that of geodesics. The present authors [8] have discussed projective curvature collineation by considering vanishing of the Lie-derivature of the curvature tensor in Finsler space. The purpose of this paper is to study projective motion in Finsler spaces for the curvature tensor $\R_{jkh}^{\ast \i}$. The notations used in the sequel are due to E. Cartan [1] and H. Rund [3].

Mathematics Subject Classification (1991): 53B40

Keywords: projective motion, Finsler space, Finsler spaces and generalizations, Lie-derivative, contra field, Concurrent field, curvature collineation, tensor, study

Quaestiones Mathematicae 24(2) 2001, 183-191
© 2001 NISC Pty Ltd, www.nisc.co.za 

Well-Bounded Operators on Non-Reflexive Banach Spaces II

Qingping Cheng ¹ and Ian Doust ²

¹ Institute of Mathematics, Hebrew University of Jerusalem, Givat-Ram, Jerusalem 91904, Isreal.

e-mail: [email protected] 

² Institute of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia.

e-mail: [email protected] 

Abstract: In this paper we examine the relationship between the various subclasses of well-bounded operators. Although some of these subclasses have been shown to coincide on certain families of Banach spaces, we show here that apart from these previously known cases, the subclasses are typically distinct.

Mathematics Subject Classification (1991): Primary 47B40

Keywords: Banach, well-bounded operators, decompositions of the identity, non-converging nest property, spectral operators, decomposable operators, well bounded operators, operators, Banach spaces, banach space

Quaestiones Mathematicae 24(2) 2001, 193-200
© 2001 NISC Pty Ltd, www.nisc.co.za 

Traces of Epimorphism Classes

Lutz Schröder

Department of Mathematics and Computer Science, University of Bremen, P.O. Box 330440, 28334 Bremen, Germany.

e-mail: [email protected] 

Abstract: Classes of morphisms that occur as preimages of the class of all epimorphisms, of all extremal epimorphisms, respectively of all retractions under some faithful functor are characterized by suitable closure properties. The corresponding question for regular or strict epimorphisms is presented as an open problem.
A further application of the methods developed in this context yields a similar characterization of classes of morphisms that are final with respect to some faithful functor.

Mathematics Subject Classification (1991): 18A10, 18A20, 18A22, 18A32

Keywords: category, faithful functor, (extremal) epimorphism, retraction, semicategory, graphs, diagram schemes, precategories, neocategories, epimorphisms, monomorphisms, special classes of morphisms, null morphisms, factorization, categories, special properties of functors, factorization of morphisms, substructures, quotient structures, congruences, amalgams, open

Quaestiones Mathematicae 24(2) 2001, 201-211
© 2001 NISC Pty Ltd, www.nisc.co.za 

Stably Continuous s- Frames

Phethiwe Matutu

Department of Mathematics, University of Stellenbosch, Private Bag X1, Matieland, &602, South Africa.

e-mail: [email protected] 

Abstract: It is shown that the categories of stably continuous $\sigma$-frames and compact regular $\sigma$-biframes are equivalent. This is the analogue of Banaschewski and Brümmer [1], linking the stably continuous frames and compact regular biframes.

Mathematics Subject Classification (1991): 54D30, 54D20, 06B35, 06D05

Keywords: stably continuous s-frames, compact regular s-biframes, compactness, noncompact covering properties, continuous lattices,generalizations,applications, structure and representation theory, categories, category, stably continuous frame, frames, frame

Quaestiones Mathematicae 24(2) 2001, 213-216
© 2001 NISC Pty Ltd, www.nisc.co.za 

A Note on Boundedness of an Integro-Differential Equation

Juan E Nápoles Valdés

Universidad de la Cuenca del Plata, Lavalle 50, (3400) Corrientes, Argentina and

Universidad Tecnologica Nacional, French 414, (3500) Resistencia, Chaco, Argentina.

e-mail:  [email protected]  & [email protected] 

Abstract: In this paper we study the prolongability and certain properties of the integro-differential equation
(1) $\x^{''} + \a(t)f(t,x,\x^{'})\x^{'} + g(t,\x^{'}) + h(x) = \int\limits_{0}^{t} C(t,s)\x^{'}(s)ds$

Mathematics Subject Classification (1991): Primary Classification 45J05; Secondary 45MXX

Keywords: Boundedness, Lyapunov functionals, integro-ordinary differential equations, study

Quaestiones Mathematicae 24(2) 2001, 217-223
© 2001 NISC Pty Ltd, www.nisc.co.za 

The Nielsen Coincidence Number of Maps into Tori.

Jerzy Jezierski

Department of Mathematics, University of Agriculture, Nowoursynowska 166, 02 766 Warszawa, Poland.

e-mail: [email protected] 

Abstract: We give a formula for the Nielsen coincidence number of a pair of maps from a surface into the 2-torus. We also show that a similar formula gives the lower bound of the number of connected components of the coincidence set of a pair of maps between two tori (possibly of different dimensions).

Mathematics Subject Classification (1991): 55M20

Keywords: coincidence index, coincidence Nielsen number, fixed points and coincidences tori, Nielsen

Quaestiones Mathematicae 24(2) 2001, 225-235
© 2001 NISC Pty Ltd, www.nisc.co.za 

Bounded Subsets and Weak Realcompactness Conditions

S Kundu ¹ AB Raha ² and MA Swardson ³

¹ Department of Mathematics, Indian Institue of Technology, Delhi , New Delhi 110016, India.

e-mail: [email protected] 

² Stat–Math Division, Indian Statistical Institute 203, B.T. Road, Calcutta 700035, India.

e-mail: [email protected] 

³ Department of Mathematics, Ohio University, Athens, Ohio 45701, USA.

e-mail: [email protected] 

Abstract: A subset A of X is bounded if every continuous real-valued function on X is bounded on A. A completely regular Hausdorff space X is said to have the bz-property if every bounded subset of X is contained in a bounded zero subset of X. In this paper, we study the bz-property and its relation to other well known topological properties. We also introduce some new topological properties, all weaker than realcompactness, that are related to the bz-property. The origin of the bz-property lies in a measure-theoretic problem.

Mathematics Subject Classification (1991): 5450, 54F99, 54G20

Keywords: bounded set, zero set, (weak) bz-space, (weak) m-space, space of pseudo countable type, unbounded point, untractable point, special sets defined by functions, counterexamples, completely regular, study, Lie

Quaestiones Mathematicae 24(2) 2001, 237-246
© 2001 NISC Pty Ltd, www.nisc.co.za 

Conjugacy Classes of Involutions in the Group of Homeomorphisms of the Rational Numbers

PV Bruyns

Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa.

e-mail: [email protected] 

Abstract: Let X be the set of real rational numbers $\mathbb{Q}$ with the topology induced by the usual topology on $\mathbb{R}$ and let AutX be the set of functions $\f : X \rightarrow X$ which are homeomorphisms with respect to this topology. The set AutX is a group with respect to composition of functions. Let $\f \in AutX$ be an involution whose fixed point set has derived set with n elements. It is shown that there are n + 2 conjugacy classes of such elements.

Mathematics Subject Classification (1991): 20B22

Keywords: conjugacy classes, rational numbers, involutions, fixed point sets, multiply transitive finite groups, composition, fixed point

Quaestiones Mathematicae 24(2) 2001, 247-259
© 2001 NISC Pty Ltd, www.nisc.co.za 

Bloch Constants of Order N

Philip R Brown

Department of Mathematics, Texas A&M University, College Station, TX77843-3368, USA.

e-mail: [email protected] 

Abstract: A variant of the classical Bloch's constant s studied: bounds are determined for the larger schlicht disc that must be present in the image of a holomorphic function on the unit disc, with the normalization $\f^{(n)}(0) = n!$

Mathematics Subject Classification (1991): 30E99

Keywords: holomorphic function