Manuel A. Morón

On the Monotonicity of the Compromise Set in Multicriteria Problems

This paper discusses the extension of results on monotonicity of the compromise set valid for bicriteria problems to general multicriteria problems under a very general condition, which is assumable in compromise programming problems coming from economics. Mainly, the problem that we treat is the following: find and describe the compromise set when the feasible set is a convex set in the positive cone, limited by a level hypersurface of a differentiable production–transformation function. This scenario is usual in many economic applications, chiefly in production analysis.

Generating well-behaved utility functions for compromise programming

The purpose of this paper is to seek utility functions satisfying a weak condition which guarantees that the utility optimum always belongs to the compromise set. This set is a special subset of the attainable or feasible set, which is generated through the application of the well-known operational research approach called compromise programming. It is shown that there are large families of utility functions satisfying this condition, thus reinforcing the value of compromise programming as a good surrogate of the traditional utility optimum.

Ultrametrics, Banach's fixed point theorem and the Riordan group

We interpret the reciprocation process in K[[x]] as a fixed point problem related to contractive functions for certain adequate ultrametric spaces. This allows us to give a dynamical interpretation of certain arithmetical triangles introduced herein. Later we recognize, as a special case of our construction, the so-called Riordan group which is a device used in combinatorics. In this manner we give a new and alternative way to construct the proper Riordan arrays. Our point of view allows us to give a natural metric on the Riordan group turning this group into a topological group. This construction allows us to recognize a countable descending chain of normal subgroups.

Shape as a Cantor completion process

Let X and Y be metric compacta, Y embedded in the Hilbert cube Q. For two maps f,g:X→Q the authors define F(f,g):=inf{e>0:f is homotopic to g in the e-neighborhood of Y}, and a sequence of maps fk :X→Q, k∈N, is said to be a Cauchy sequence provided for every e>0 there is a k0∈N such that F(fk,fk′)<e whenever k,k′≥k 0. Such sequences coincide with the approximative maps of K. Borsuk [Theory of shape, PWN, Warsaw, 1975] and represent shape morphisms from X to Y. The function F is not a pseudometric, but defining d(α,β):=lim k F(fk,gk), where the shape morphisms α,β∈Sh(X,Y) are represented by Cauchy sequences (fk),(gk), the authors prove that (Sh(X,Y),d) becomes a complete zero-dimensional ultrametric space, homeomorphic to a closed subset of the irrationals. Among other things, the authors prove that if two compacta X and Y are of the same shape, then for every compactum Z, the spaces Sh (X,Z) and Sh (Y,Z) are uniformly homeomorphic. In the last section, the authors show, for example, that for X compact and Y∈FANR , the space Sh (X,Y) is countable, give several characterizations of various kinds of movability, and translate their results to Z-sets in Q and sequences of proper maps between their complements.

Some duality properties of non-saddle sets

Abstract We show in this paper that the class of compacta that can be isolated non-saddle sets of flows in ANRs is precisely the class of compacta with polyhedral shape. We also prove—reinforcing the essential role played by shape theory in this setting—that the Conley index of a regular isolated non-saddle set is determined, in certain cases, by its shape. We finally introduce and study the notion of dual of a non-saddle set. Examples of compacta related by duality are attractor–repeller pairs. We use the complement theorems in shape theory to prove that the shape of the dual set is determined by the shape of the original non-saddle set.

Complementary Riordan arrays

Abstract Recently, the concept of the complementary array of a Riordan array (or recursive matrix) has been introduced. Here we generalize the concept and distinguish between dual and complementary arrays. We show a number of properties of these arrays, how they are computed and their relation with inversion. Finally, we use them to find explicit formulas for the elements of many recursive matrices.

A topology for the sets of shape morphisms

Abstract We introduce a topology on the set of shape morphisms between arbitrary topological spaces X , Y , Sn(X,Y) . These spaces allow us to extend, in a natural way, some classical concepts to the realm of topological spaces. Several applications are given to obtain relations between shape theory and N -compactness and shape-theoretic properties of the spaces of quasicomponents.

A note about the shape of attractors of discrete semidynamical systems

We state in a short way a result that improves one of the main theorems in a paper of M. Gobbino concerning the topological properties that the phase space induces in an attractor of a discrete dynamical system.

On weak shape equivalences

Abstract We prove that weak shape equivalences are monomorphisms in the shape category of uniformly pointed movable continua ShM. We use an example of Draper and Keesling to show that weak shape equivalences need not be monomorphisms in the shape category. We deduce that ShM is not balanced. We give a characterization of weak dominations in the shape category of pointed continua, in the sense of Dydak (1979). We introduce the class of pointed movable triples (X, F, Y), for a shape morphism F : X → Y, and we establish an infinite-dimensional Whitehead theorem in shape theory from which we obtain, as a corollary, that for every pointed movable pair of continua (Y, X) the embedding j : X → Y is a shape equivalence iff it is a weak shape equivalence.

Ultrametrics and infinite dimensional whitehead theorems in shape theory

We apply a Cantor completion process to construct a complete, non-Archimedean metric on the set of shape morphisms between pointed compacta. In the case of shape groups we obtain a canonical norm producing a complete, both left and right invariant ultrametric. On the other hand, we give a new characterization of movability and we use these spaces of shape morphisms and uniformly continuous maps between them, to prove an infinite-dimensional theorem from which we can show, in a short and elementary way, some known Whitehead type theorems in shape theory.