Fathi B. Saidi

Best Simultaneous Approximation in Lp(I,E)

Let G be a reflexive subspace of the Banach space E and let Lp(I,E) denote the space of all p-Bochner integrable functions on the interval I=0,1] with values in E, 1?p?∞. Given any norm N(?,?) on R2, N nondecreasing in each coordinate on the set R2+, we prove that Lp(I,G) is N-simultaneously proximinal in Lp(I,E). Other results are also obtained.

Uniqueness results in the representation of families of sets by fuzzy sets

The important concept of identifying a fuzzy subset of a set R with its corresponding chain of level sets is widely used in the literature. We investigate in this paper the problem of characterization of all fuzzy subsets of R that can be identified with a given arbitrary family C of subsets of R together with a given arbitrary subset S of [0,1]. Although our main focus is on the problem of uniqueness, we start by giving necessary and sufficient conditions for both existence and uniqueness of such fuzzy subsets. We continue by obtaining several algebraic and topological properties under which uniqueness is guaranteed. To further support our results, we present some examples which we hope will shed more light on the above mentioned problem of identification. We finish by raising a few natural questions that are left as open problems for further investigation.

Alternative characterizations for the representation of families of sets by fuzzy sets

We investigate the representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets. This concept of representation is widely used in the literature for the classification and study of fuzzy structures on the same underlying set. Negoita and Ralescu obtained two theorems, both giving necessary and sufficient conditions, one called the representation theorem and the second called the generalized representation theorem. Our aim in this paper is to take a closer look at both theorems, hence establishing some equivalent forms and several constructive remarks, including some directions for future investigations.

From fuzzy sets to the decompositions of non-rigid sets

The concept of identifying a fuzzy subset of a set R with its collection of level sets and its collection of membership values plays an important role in many real life applications. Such identification is possible only if the fuzzy set can be recaptured. This implies that uniqueness of the fuzzy set plays an important role. It has been recently shown that in order to have uniqueness it is necessary and sufficient that the collection S of membership values of the fuzzy set be a rigid set. With this in mind, the authors went on to show that if S has the min-max-property then S is a rigid set and left the converse as an open problem. We start this paper by constructing a counterexample that shows that rigid sets do not have to have the min-max-property. We then take a closer look at non-rigid sets by decomposing them into isomorphism-invariant components and into S-connected components in the hope of getting a characterization of non-rigid sets, which will lead to a characterization of rigid sets and hence of uniqueness of the fuzzy set. We also investigate, in the case of nonuniqueness, the number of fuzzy sets corresponding to a collection of level sets and a collection of membership values.

On the proximinality of the unit ball of proximinal subspaces in Banach spaces: A counterexample

A known, and easy to establish, fact in Best Approximation Theory is that, if the unit ball of a subspace G of a Banach space X is proximinal in X, then G itself is proximinal in X. We are concerned in this article with the reverse implication, as the knowledge of whether the unit ball is proximinal or not is useful in obtaining information about other problems. We show, by constructing a counterexample, that the answer is negative in general.

Uniqueness in the generalized representation by fuzzy sets

The concept of representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets is widely used in the literature for the classification and study of fuzzy algebraic structures on the same underlying set. The problem of existence of such fuzzy sets was investigated by various authors. As far as we know, the problem of uniqueness of such fuzzy sets was not studied before. We establish, among other things, necessary and sufficient conditions under which uniqueness is guaranteed in the cases of complete as well as partial representations of the level sets of the fuzzy set.

Existence and uniqueness of fuzzy ideals

Let R be a commutative ring. The idea of identifying a fuzzy ideal of R with its corresponding chain of level ideals and its range of values, although widely used in the literature for the study of fuzzy ideals, has never been investigated in a general setting before. We investigate in this paper the more general problem of characterization of all those fuzzy ideals that can be identified with a given arbitrary family C of subsets of R together with a given arbitrary subset S of [0,1]. Necessary and sufficient conditions for the existence and uniqueness of such fuzzy ideals are established. In particular, we obtain that uniqueness always holds true for Artinian rings. Moreover, examples are presented to further support our results and shed more light on the above-mentioned problem of identification. We finish by raising a natural question that is left as an open problem for further investigation.

Hadamard and Fejer-type inequalities

Abstract. We provide a generalization of the concept of r-convexity, thus obtaining a larger class of functions. Bounds for the averages of the functions in this class are established. Among other results, we obtain the upper Hadamards and Fejers inequalities as special cases of our inequalities.

Explicit characterizations of orthogonality in lSp(C)

Abstract In a previous paper, the author used a notion of orthogonality introduced in another article to establish characterizations for orthogonality in the spaces l S p ( C ), 1⩽ p l S 2 ( C ), via inner products. In this paper we make explicit these characterizations for some of the spaces l S p ( C ). We finish by presenting some remarks and open problems.

Generalized Deviations and Best Approximation Theory

ABSTRACT Given the importance of standard deviation in applications, we think that it is worthwhile considering other types of deviations. For this purpose, we introduce a generalization of the concept of deviation. We also introduce a new definition for the concept of generalized mean. We then highlight the connection that exists between arithmetic mean, standard deviation, and best approximation, and we establish necessary and sufficient conditions that would guarantee the existence of similar connections between generalized means, generalized deviations, and best approximation. We finish by presenting some open problems.