J. Bair - Academia.edu (original) (raw)

Papers by J. Bair

Results in Mathematics, 1984

Bulletin of the Belgian Mathematical Society - Simon Stevin, 1997

We give subtle, simple and precise results about the convergence or the divergence of the sequenc... more We give subtle, simple and precise results about the convergence or the divergence of the sequence (x n), where x j = f (x j−1) for every integer j, when the initial element x 0 is in the neighbourhood of a neutral fixed point, i.e. a point x * such that f (x *) = x * with |f (x *)| = 1 (where f is a C ∞ function defined on a subset of R).

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2008

Following the works of van Asch and van der Blij, we show that the horn angles introduced by Eucl... more Following the works of van Asch and van der Blij, we show that the horn angles introduced by Euclide can be measured by superreal numbers as Tall defined them. We deduce from this the possibility to estimate, on the one hand, the ratio of the measures of a horn angle and of the mixed angle formed by a spiral of Archimede and, on the other hand, the ratio of the measures of two horn angles. Résumé En nous appuyant sur des travaux de van Asch et van der Blij, nous montrons que les angles corniculaires introduits par Euclide peuvent se voir attribuer une mesure numérique donnée par un nombre superréel infiniment petit au sens de Tall. Nous en déduisons la possibilité d'estimer d'une part le rapport entre les mesures d'un angle corniculaire et d'un angle mixtiligne formé par une spirale d'Archimède, et d'autre part le rapport des mesures de deux angles corniculaires.

In this note, we give some characterizations of the cover of a convex polyhedra P, i.e. of the se... more In this note, we give some characterizations of the cover of a convex polyhedra P, i.e. of the set of points x such that the conical hull of P from x is closed ; for instance, we prove that the cover of P is the cover of the cover of P. We also apply these results in the theory of the strong separation of two convex polyhedra.

ABSTRACT We use the cover or the closure of the cover in order to determine when two disjoint clo... more ABSTRACT We use the cover or the closure of the cover in order to determine when two disjoint closed convex sets have parallel faces, one of them being bounded. Moreover, we show that the study of the closure of the cover gives interesting results about the cover itself.

Results in Mathematics, 1983

ABSTRACT

Optimization, 1988

ABSTRACT For a quasi-concave function f and a quasi-convex function g on . we study the two follo... more ABSTRACT For a quasi-concave function f and a quasi-convex function g on . we study the two following problems:we also establish a duality theorem. These results can be used to see the analogy between the classical economic theories of the consumer and of the firm.

Notices of the American Mathematical Society, 2013

We examine prevailing philosophical and historical views about the origin of infinitesimal mathem... more We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal mathematics in a new light. We also detail several procedures of the historical infinitesimal calculus that were only clarified and formalized with the advent of modern infinitesimals. These procedures include Fermat's adequality; Leibniz's law of continuity and the transcendental law of homogeneity; Euler's principle of cancellation and infinite integers with the associated infinite products; Cauchy's infinitesimal-based definition of continuity and "Dirac" delta function. Such procedures were interpreted and formalized in Robinson's framework in terms of concepts like microcontinuity (S-continuity), the standard part principle, the transfer principle, and hyperfinite products. We evaluate the critiques of historical and modern infinitesimals by their foes from Berkeley and Cantor to Bishop and Connes. We analyze the issue of the consistency, as distinct from the issue of the rigor, of historical infinitesimals, and contrast the methodologies of Leibniz and Nieuwentijt in this connection. Contents 1. The ABC's of the history of infinitesimal mathematics 2 2. Adequality to Chimeras 5 2.1. Adequality 5 2.2. Archimedean axiom 6 2.3. Berkeley, George 7 2.4. Berkeley's logical criticism 8 2.

Journal of Mathematical Analysis and Applications, 1975

Abstract In this note we study the separation of two convex sets in the straight line spaces intr... more Abstract In this note we study the separation of two convex sets in the straight line spaces introduced by J. Cantwell and in the convexity spaces defined by W. Prenowitz and investigated by V. W. Bryant and R. J. Webster: in the last spaces we particularly generalize the theorem of Hahn-Banach and characterize the elements of a finite convex partition.

Geometriae Dedicata, 1977

In this paper, we give a property of the linearly dosed non-linearly bounded Choquet simplices of... more In this paper, we give a property of the linearly dosed non-linearly bounded Choquet simplices of a real vector space which can be used to describe the dosed Choquet simplices of E ". Such a description can be obtained from a theorem of Gruber [3, Satz 2] using other techniques.

La Société Belge des Professeurs de Mathématique d'expression française a un rôle primordial... more La Société Belge des Professeurs de Mathématique d'expression française a un rôle primordial: l'amélioration de l'enseignement de la mathématique. Cela l'a amenée à publier elle-même de nombreux ouvrages. Qu'il me suffise de rappeler les documents de la série ...

Arkiv för matematik, 1978

In this note, we generalize two theorems of Klee [9] and a result of Bair-Jongmans [7] about the ... more In this note, we generalize two theorems of Klee [9] and a result of Bair-Jongmans [7] about the true separation of two convex cones; afterwards, we introduce the notion of true separation for n(n_~2) convex sets and we extend our three first statements for n convex cones. Nous nous placerons dans un espace vectoriel rrel, 6ventuellement muni d'une topologie vectorielle (ce qui sera prrcis6 dans chaque 6noncr) et adopterons les drfinitions et notations utilisres dans les ouvrages citrs dans la bibliographie. Signalons nranmoins que nous noterons iA l'internat, ~A l'enveloppe lindaire et mA la marge de A [5]; de plus, une cellule convexe drsignera un ensemble convexe d'internat non vide [4], tandis qu'un ensemble vrai sera un ensemble non vide et distinct de tout l'espace [3]. La notion de srparation vraie a 6t6 introduite dans [7]: deux sous-ensembles A, B d'un espace vectoriel sont vraiment sdpar~s par un hyperplan H lorsque A et B sont srparrs par H et qu'aucun de ces deux ensembles n'est contenu dans H. Le probl~me de la srparation vraie de deux convexes se ram~ne souvent h celui de la srparation vraie de deux crnes de m~me sommet [7]. Or, Klee [9] a tout d'abord obtenu un rrsultat intrressant sur la srparation vraie de deux cSnes convexes A, B de sommet O dans un espace localement convexe lorsque l'intersection de A et B se rrduit ~t {O}; cet ~nonc6 a 6t6 g~nrralis~ dans le cas oCa A n B est un sous-espace vectoriel quelconque, pour autant que ron restreigne ~t R" l'espace considrr6 [7; 4.5]: R" 6tait en effet le type d'espace le plus grnrral dans lequel le caract~re ferm6 de la diffrrence A-B 6tait assurr; rrcemment, un crit~re de fermeture vient d'etre donn6 dans [8], ce qui permet justement de lever cette restriction sur l'espace et d'obtenir ce rrsultat qui grnrralise ~t la lois les 6noncrs 4.5 de [7] et 2.5 de [9]:

Archiv der Mathematik, 1993

ABSTRACT 1. Introduction. In order to give a classification for non-linearly bounded convex sets ... more ABSTRACT 1. Introduction. In order to give a classification for non-linearly bounded convex sets in an arbitrary real vector space (paragraph 4), we introduce and study the notion of quasi-parabolic set (paragraph 3). The inner aperture cone, introduced by Larrnan [13] and Brondsted [6] and studied by several authors as Bait ([1], [2] or [3]), Jongmans [10], Sung-Tam [14], plays a central role in this matter; for example, it appears in Propositions 3.8 and 3.10, which are generalizations of results given by Bair in IR", and in Corollary 3.9 which is a general characterization of the quasi-parabolicity for non-linearly bounded convex sets. The examples in paragraph 5 and some ubiquituous convex sets (introduced by Klee [11] or [12]) and especially studied by Coquet and Dupin ([7], [8])) are useful to understand this classification; in this study, they also show the importance of th e intrinsic core of the convex set itself, but also the intrinsic core of the infinitude cone. 2. Notations and definitions. We work in a real vector space E; its dimension is finite or infinite, but greater than or equal to 2. The subset A of E is always supposed non-empty. As in [3], [4] or [9], we respectively denote by tA = U (~ c~iA'P ~ N, Z ~ = t), P,~i i=1 i=1 P SA ~ U (~, ~i A:p~, (~i E~) and CA = U (~. ~ ~ ~ O~ ~ ~i ~ 1)the p, Cti i=1 p, ai i=I i=i

... Language : English. Author, co-author : Bair, Jacques mailto [Université de Liège - ULg &... more ... Language : English. Author, co-author : Bair, Jacques mailto [Université de Liège - ULg > HEC-Ecole de gestion de l'ULg : UER > Mathématiques appliquées aux sc. ... économiques et de gestion >]. Justens, Daniel mailto [Haute Ecole Ferrer, Bruxelles > > > >]. ...

... Language : French. Author, co-author : Bair, Jacques mailto [Université de Liège - ULg &g... more ... Language : French. Author, co-author : Bair, Jacques mailto [Université de Liège - ULg > HEC-Ecole de gestion de l'ULg : UER > Mathématiques appliquées aux sc. économiques et de gestion >]. ... Justens, Daniel mailto [Haute Ecole Francisco Ferrer > > > >]. ...