Jacques Bair

Une modélisation d'un zoom au moyen de microscopes virtuels

In this paper, we explain how a computer works when “zooms” are made around a point on a planar curve. This modelisation leads to an easy and algorithmic method to find the (vertical or not vertical) tangents for the studied curve. Mots-cle: Zoom, microscope virtuel, tangente, point de non derivabilite, limite de courbes, convergence uniforme, didactique des mathematiques. ZDM Subject Classification: I10, P7, R20.

From Newton's Fluxions to Virtual Microscopes

The method of fluxions was originally given by Newton among others in order to determine the tangent to a curve. In this note, we will formulate this method by the light of some modern mathematical tools: using the concept of limit, but also with hyperreal numbers and their standard parts and with dual numbers; another way is the use of virtual microscopes both in the contexts of classical and non standard analysis.

Un problème de duaiité en programmation quasi-concave

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.

Separation geometrique de familles finies d'ensembles

In this paper, we introduce a geometric separation for an arbitrary finite family of sets in a real linear space; we give some conditions of geometric separation and we characterize the geometric separation for a family composed of three properly convex sets.

Fermat’s Dilemma: Why Did He Keep Mum on Infinitesimals? And the European Theological Context

The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. André Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve cannot be so treated and involve additional tools–leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat’s manuscript but it is Breger who tampers with Fermat’s procedure by moving all terms to the left-hand side so as to accord better with Breger’s own interpretation emphasizing the double root idea. We provide modern proxies for Fermat’s procedures in terms of relations of infinite proximity as well as the standard part function.

Cauchy, Infinitesimals and ghosts of departed quantifiers

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinsons frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibnizs distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinsons framework, while Leibnizs law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibnizs infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinsons framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Eulers own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinsons framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchys procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinsons framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Keywords: historiography; infinitesimal; Latin model; butterfly model

Séparation et polarité

To every subset of an arbitrary vector space we introduce a new cone in the dual, We show that there is a connection to the polar cone and to the well-known prepolar cone. Then we prove the relationship to the polarity at the separation:the characterization of all separating planes is possible.

Some characterizations of the asymptotic cone and the lineality space of a convex set

We give new characterizations of the asymptotic cone and the lineality space of a convex set; to this effect we use the notions of margin, extreme point, face and cone of admissible directions. For example we find this statement: if A. is a closed, line-free and finite-dimensional convex set, then wherepis the set of extreme points of A and W(A,α) is the witness cone of A from the point α.

On a property of extended sublinear functionals

In this note we consider this question: when does an extended sublinear functional on a real nonzero linear space majorize at least one linear functional which is not. identically zero.

Criteres de separation pour polyedres convexes

We give a few results about the separation of two convex polyhedra in an arbitrary real vector space; so we generalize theorems of Černikov, Klee, Rockafellar and Gallivan-Zaks.