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Concept# Définition

Résumé

Une définition est une proposition qui met en équivalence un élément définissant et un élément étant défini.
Une définition a pour but de clarifier, d'expliquer. Elle détermine les limites ou « un ensemble de traits qui circonscrivent un objet ».
Rhétorique
Selon les Définitions du pseudo-Platon, la définition est la . Aristote, dans le Topiques, définit le mot comme
Mathématiques
En mathématiques, on définit une notion à partir de notions antérieurement définies.
Les notions de bases étant les symboles non logiques du langage considéré, dont l'usage est défini par les axiomes de la théorie.
Se pose la question de la différence entre une définition et un axiome. Pour exemple, dans l'arithmétique de Peano, l'addition et la multiplication sont des symboles du langage et leur fonctionnement est régi par des axiomes. Mais on pourrait tout à fait réduire le langage de l’arithmétique en supprimant les symboles « + » et « * » et les définir à partir de 0 et de l

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In this thesis we address the problem of safe substitutability in mobile component-oriented formalisms. We try to give a solution for different definitions of "safe" through a new notion of subtyping based on flexibility and observation. The aim is to have a definition which is applicable to a large numbers of existing formalisms, and which itself may be seen as an abstraction of many existing definitions of subtyping. In a similar way, we also try to see the verification of such relations in a way that may be applicable to many formalisms, and which may be seen as "containing" many existing techniques. Our hope is that our work may link these subtype definitions or verification techniques to formalisms they were not originally designed for. The contribution of this thesis can be divided in several points. First of all, the full formal definition of a "descriptive" component-oriented formalism based on concurrency, non determinism, complex composition and mobility. Also, the definition of a general flexible notion of subtyping, as well as a more particular example. Verification techniques for these subtyping relations are also grouped under the concept of component transformation, which is shown to include many other works. Finally, to illustrate the main concepts of our work, we provide a case study with a model of a "mobile" drinks dispenser.

Krassimir Panajotov, Yiqi Zhang

The pioneering paper 'Optical rogue waves' by Solli et al (2007 Nature 450 1054) started the new subfield in optics. This work launched a great deal of activity on this novel subject. As a result, the initial concept has expanded and has been enriched by new ideas. Various approaches have been suggested since then. A fresh look at the older results and new discoveries has been undertaken, stimulated by the concept of 'optical rogue waves'. Presently, there may not by a unique view on how this new scientific term should be used and developed. There is nothing surprising when the opinion of the experts diverge in any new field of research. After all, rogue waves may appear for a multiplicity of reasons and not necessarily only in optical fibers and not only in the process of supercontinuum generation. We know by now that rogue waves may be generated by lasers, appear in wide aperture cavities, in plasmas and in a variety of other optical systems. Theorists, in turn, have suggested many other situations when rogue waves may be observed. The strict definition of a rogue wave is still an open question. For example, it has been suggested that it is defined as 'an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses'. This definition (as suggested by a peer reviewer) is clear at the intuitive level and can be easily extended to the case of spatial beams although additional clarifications are still needed. An extended definition has been presented earlier by N Akhmediev and E Pelinovsky (2010 Eur. Phys. J. Spec. Top. 185 1-4). Discussions along these lines are always useful and all new approaches stimulate research and encourage discoveries of new phenomena. Despite the potentially existing disagreements, the scientific terms 'optical rogue waves' and 'extreme events' do exist. Therefore coordination of our efforts in either unifying the concept or in introducing alternative definitions must be continued. From this point of view, a number of the scientists who work in this area of research have come together to present their research in a single review article that will greatly benefit all interested parties of this research direction. Whether the authors of this 'roadmap' have similar views or different from the original concept, the potential reader of the review will enrich their knowledge by encountering most of the existing views on the subject. Previously, a special issue on optical rogue waves (2013 J. Opt. 15 060201) was successful in achieving this goal but over two years have passed and more material has been published in this quickly emerging subject. Thus, it is time for a roadmap that may stimulate and encourage further research.

Let K be a field with char(K) ≠ 2. The Witt-Grothendieck ring (K) and the Witt ring W (K) of K are both quotients of the group ring ℤ[𝓖(K)], where 𝓖(K) := K*/(K*)2 is the square class group of K. Since ℤ[𝓖(K)] is integral, the same holds for (K) and W(K). The subject of this thesis is the study of annihilating polynomials for quadratic forms. More specifically, for a given quadratic form φ over K, we study polynomials P ∈ ℤ[X] such that P([φ]) = 0 or P({φ}) = 0. Here [φ] ∈ (K) denotes the isometry class and {φ} ∈ W(K) denotes the equivalence class of φ. The subset of ℤ[X] consisting of all annihilating polynomials for [φ], respectively {φ}, is an ideal, which we call the annihilating ideal of [φ], respectively {φ}. Chapter 1 is dedicated to the algebraic foundations for the study of annihilating polynomials for quadratic forms. First we study the general structure of ideals in ℤ[X], which later on allows us to efficiently determine complete sets of generators for annihilating ideals. Then we introduce a more natural setting for the study of annihilating polynomials for quadratic forms, i.e. we define Witt rings for groups of exponent 2. Both (K) and W(K) are Witt rings for the square class group 𝓖(K). Studying annihilating polynomials in this more general setting relieves us to a certain extent from having to distinguish between isometry and equivalence classes of quadratic forms. In Section 1.1 we study the structure of ideals in R[X], where R is a principal ideal domain. For an ideal I ⊂ R[X] there exist sets of generators, which can be obtained in a natural way by considering the leading coefficients of elements in I. These sets of generators are called convenient. By discarding super uous elements we obtain modest sets of generators, which under certain assumptions are minimal sets of generators for I. Let G be a group of exponent 2. In Section 1.2 we study annihilating polynomials for elements of ℤ[G]. With the help of the ring homomorphisms Hom(ℤ[G],ℤ) it is possible to completely classify annihilating polynomials for elements of ℤ[G]. Note that an annihilating polynomial for an element f ∈ ℤ[G] also annihilates the image of f in any quotient of ℤ[G]. In particular, Witt rings for G are quotients of ℤ[G]. In Section 1.3 we use the ring homomorphisms Hom(ℤ[G],ℤ) to describe the prime spectrum of ℤ[G]. The obtained results can then be employed for the characterisation of the prime spectrum of a Witt ring R for G. Section 1.4 is dedicated to proving the structure theorems for Witt rings. More precisely, we generalise the structure theorems for Witt rings of fields to the general setting of Witt rings for groups of exponent 2. Section 1.5 serves to summarise Chapter 1. If R is a Witt ring for G, then we use the structure theorems to determine, for an element x ∈ R, the specific shape of convenient and modest sets of generators for the annihilating ideal of x. In Chapter 2 we study annihilating polynomials for quadratic forms over fields. More specifically, we first consider fields K, over which quadratic forms can be classified with the help of the classical invariants. Calculations involving these invariants allow us to classify annihilating ideals for isometry and equivalence classes of quadratic forms over K. Then we apply methods from the theory of generic splitting to study annihilating polynomials for excellent quadratic forms. Throughout Chapter 2 we make heavy usage of the results obtained in Chapter 1. Let K be a field with char(K) ≠ 2. Section 2.1 constitutes an introduction to the algebraic theory of quadratic forms over fields. We introduce the Witt-Grothendieck ring (K) and the Witt ring W(K), and we show that these are indeed Witt rings for 𝓖(K). In addition we adapt the structure theorems to the specific setting of quadratic forms. In Section 2.2 we introduce Brauer groups and quaternion algebras, and in Section 2.3 we define the first three cohomological invariants of quadratic forms. In particular we use quaternion algebras to define the Clifford invariant. In Section 2.4 we begin our actual study of annihilating polynomials for quadratic forms. Henceforth it becomes necessary to distinguish between isometry and equivalence classes of quadratic forms. We start by classifying annihilating ideals for quadratic forms over fields K, for which (K) and W(K) have a particularly simple structure. Subsequently we use calculations involving the first three cohomological invariants to determine annihilating ideals for quadratic forms over a field K such that I3(K) = {0}, where I(K) ⊂ W(K) is the fundamental ideal. Local fields, which are a special class of such fields, are studied in Section 2.5. By applying the Hasse-Minkowski Theorem we can then determine annihilating ideals of quadratic forms over global fields. Section 2.6 serves as an introduction to the elementary theory of generic splitting. In particular we introduce Pfister neighbours and excellent quadratic forms, which are the subjects of study in Section 2.7. We use methods from generic splitting to study annihilating polynomials for Pfister neighbours. The obtained result can be applied inductively to obtain annihilating polynomials for excellent quadratic forms. We conclude the section by giving an alternative, elementary approach to the study of annihilating polynomials for excellent forms, which makes use of the fact that (K) and W(K) are quotients of ℤ[𝓖(K)].

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Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics as diverse as mathematical reasoning, combinatorics, discrete structures & algorithmic thinking.

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

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Dictionnaire

thumb|upright=1.2|Dictionnaire en latin constitué de plusieurs volumes, œuvre d'Egidio Forcellini (1771).
Un dictionnaire est un ouvrage de référence contenant un ensemble de mots d’une langue ou d

Mathématiques

thumb|upright|Raisonnement mathématique sur un tableau.
Les mathématiques (ou la mathématique) sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets

Ensemble

vignette|Ensemble de polygones dans un diagramme d'Euler
En mathématiques, un ensemble désigne intuitivement un rassemblement d’objets distincts (les éléments de l'ensemble), « une multitude qui peut