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This paper gives an elementary proof of a theorem by de Felipe and Teissier in the paper “Valuations and henselization” (arxiv.org/abs/1903.10793v1), to appear in Math. Annalen. The theorem compares two henselizations of a local domain dominated by a valuation domain. Our proofs are written in the constructive Bishop style.

International audience; In my paper at Inventiones 2009, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.; Dans [2], on détermine les groupes de cohomologie des espaces de Lubin-Tate par voie globale en calculant les fibres des faisceaux de cohomologie du faisceau pervers des cyclesévanescentscyclesévanescents Ψ d'une variété de Shimura de type Kottwitz-Harris-Taylor. L'ingrédient le plus complexe consistè a contrôler lesfì eches de deux suites spectrales calculant l'une les faisceaux de cohomologie des faisceaux pervers d'Harris-Taylor, et l'autre ceux de Ψ. Dans cet article, nous contournons ces difficultés en utilisant la théorie classique des représentations du groupe mirabolique ainsi qu'un argument géométrique simple. Abstract (Mirabolic group, ramified Newton stratification and cohomology of Lubin-Tate spaces) In [2], we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf coho-mology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.

Si la notion de validité fait l’objet de multiples recherches, notamment dans le champ de la psychométrie, elle n’a pas encore été beaucoup étudiée dans le cadre de la didactique des mathématiques. Nous proposons d’expliciter la façon dont les outils de ce champ de recherche, en particulier l’analyse a priori des tâches, peuvent être exploités pour apporter des preuves de validité complémentaires à celles de la psychométrie. Une première partie vise à proposer une méthodologie d’analyse et de conception du contenu d’une évaluation externe dans laquelle l’approche didactique de la validité est pensée en complémentarité de celle psychométrique. Nous illustrons, dans une deuxième partie, la mise en oeuvre de cette méthodologie sur l’étude de deux domaines mathématiques (l’arithmétique en fin d’école et l’algèbre en fin de collège) dans des évaluations menées nationalement en France à ces deux ordres d’enseignement., Even if many researches deal with validity, especially in psychometrics, this notion hasn’t been significantly discussed in the field of didactics of mathematics. This paper examines how didactics tools, especially a priori analysis of tasks, can be used to bring validity evidence in addition to those provided by psychometrics. A first part of the paper aims to expose a methodology, using a combination of didactic and psychometric approaches, to analyze and design external assessment tests. In a second part, we implement this methodology to study two national large scale assessments, in two mathematical domains at two levels of school (arithmetic at the end of primary school and algebra at the end of grade 9), in France., Embora o conceito de validade seja objeto de muitas investigações, particularmente no campo da psicometria, não foi ainda muito estudado no quadro da didática das matemáticas. Ora, o nosso propósito é explicar a maneira como as ferramentas deste campo de investigação, em particular a análise prévia de tarefas, podem ser exploradas para fornecer provas de validade complementares às da psicometria. A primeira parte visa propor uma metodologia de análise e conceção do conteúdo de uma avaliação externa na qual a abordagem didática da validade é pensada em complementaridade à da psicometria. Numa segunda parte, mostramos a implementação desta metodologia no estudo de dois domínios matemáticos (aritmética no final do 2.º ciclo e de álgebra no final do 3.º ciclo do ensino básico) nas avaliações realizadas a nível nacional em França nestes dois níveis de ensino.

Based on the framework of meta-didactic transposition analysis (Arzarello et al., 2014) and specifically the concepts of brokering and boundary object, this paper studies how knowledge to teach algebra is exchanged during a training program involving nine mathematics teachers and two researchers specialised in algebra teaching and learning. Organized in three half-day sessions, this program is based on a problem pointed out in the research literature as particularly rich to develop algebraic thinking. In addition, the materials used in training are come directly from the classes of the teachers participating in the program. In this sense, the program values knowledge that makes sense in both research and teaching practice. The analysis of interactions between researchers and teachers highlights three types of collaborative activities between the two groups and thus questions the potential of such a mechanism to foster integration of research results by teachers. En s’appuyant sur le cadre d’analyse de la transposition méta-didactique (Arzarello et al., 2014) et plus précisément sur les concepts de courtage en connaissances et d’objet frontière, cet article étudie la manière dont les connaissances pour enseigner l’algèbre sont échangées dans le cadre d’un programme de formation réunissant neuf enseignants de mathématiques et deux chercheurs spécialisés en didactique de l’algèbre. Organisé en trois séances d’une demi-journée, ce programme s’articule autour d’un problème pointé dans la littérature de recherche comme particulièrement riche pour développer la pensée algébrique des élèves. Les documents exploités en formation sont en outre directement issus des classes des enseignants participant au programme. En ce sens, le programme valorise des connaissances ayant du sens tant dans la recherche que dans la pratique enseignante. L’analyse des interactions entre les chercheurs et les enseignants met en évidence trois types d’activités de collaboration entre les deux groupes d’intervenants et questionnent le potentiel d’un tel dispositif pour favoriser une appropriation, par les enseignants, de résultats de recherches.

Ordinary Timed Event Graphs (TEG) have a linear transfer relation in some dioid algebras. These transfer relations can be described thanks to ultimately periodic formal series which have a graphicalZ 2 description. Firstly, this paper aims at carrying out a similar repre- sentation for Generalized TEG (GTEG), i.e. TEG with multipliers. The representation requires an additional dimension that reflects the gain of these systems, the latter being algebraically modelled by the introduction of new operators. This graphical representation is a natural ex- tension of the one used for ordinary TEG. Secondly, this paper shows that for GTEG having a property of periodicity, it is possible to consider the residuation theory so as to compute some control laws.

International audience; In the vast movement of appropriation, by Europe, of the sciences of Islamic countries, algebra holds an important position. In particular, al-Khwârizmî'’s Mukhtasar, a text published in Baghdad between 813 and 833 and considered the official birth of the discipline, was several times translated into Latin and vernacular languages. In this paper, we return briefly to the historiography of the Arab-Latin translations/versions of al-Khwârizmî'’s text to present and discuss both the content and the form (with paleographic and editorial data) of one of these versions: the Liber restauracionis [Book of Restoration]. We then attempt, in a mathematical and historical analysis, to highlight the elements of continuity and rupture between this text and the al-Khwarizmian algebraic tradition in order to present our hypotheses about the context of its composition. Our contribution ends with the first modern translation (in French) of the text we have studied.; Dans le vaste mouvement d’appropriation, par l’'Europe, des sciences des pays d’Islam, l’algèbre occupe une place non négligeable. En particulier, le Mukhtasar d’'al-Khwârizmî, texte publié à Bagdad entre 813 et 833 et reconnu comme acte de naissance ofciel de la discipline, est plusieurs fois traduit en latin puis en langue vernaculaire. Dans cette contribution, nous revenons brièvement sur l’historiographie des traductions/versions arabo-latines du texte d’'al-Khwârizmî pour présenter et commenter tant le contenu que la forme (avec des données paléographiques et éditoriales) de l’une de ces versions: le Liber restauracionis [Livre de la restauration]. Nous tentons ensuite, au cours d’une analyse mathématique et historique, de mettre en évidence les éléments de continuité et de rupture de ce texte avec la tradition algébrique al-khwarizmienne an de livrer nos hypothèses sur son contexte de rédaction. Enfin, notre contribution s’achève sur la première traduction moderne du texte étudié (en français).

We study the geometry and the cohomology of the tamely ramified cover of Drinfeld’s p-adic symmetric space over a p-adic field K. For this tame level, we prove, in a purely local way, most of a conjecture of Harris on the form of the $$\ell $$ -adic cohomologies, as well as the torsion freeness of the integral cohomology. In this paper, we also compute the $$\ell $$ -adic cohomology of Coxeter Deligne–Lusztig variety associated to $$\mathrm{GL}_d,$$ and some results of independent interest on the coefficient systems over the Bruhat–Tits building associated to $$\mathrm{GL}_d(K)$$ have been established.

International audience; After having pointed out that the difficulty in reading contemporary formalisms lies in the fact that they combine algebraic and geometric dimensions in their written form, this paper – which aims to question the merits of a number of dichotomies (visual versus oral reading, doing vs. saying ...) which are usually referred to in describing the specificity of formal writings – first focuses on the practice of mathematicians, which was at the origin of the creation of this type of writing, in order to highlight different facets of mathematical doing in the fields of activity where a formalism occurs, and argues that these doings are usually interlaced with a saying. Examining the conceptual upheaval, resulting from logicians’ and computer scientists’ use of the proof-program correspondence (or Curry-Howard isomorphism), is then undertaken and this study offers us the opportunity to propose keys to understanding the main contemporary formalisms used for the analysis of natural languages.; Après avoir mis en évidence que la difficulté de lecture des formalismes contemporains réside dans le fait qu’ils conjuguent dans leur graphie les dimensions algébrique et géométrique, cet article – qui entend questionner le bien-fondé d’un certain nombre de dichotomies (lecture visuelle versus orale, faire et dire, …) usuellement mobilisées pour qualifier la spécificité des écritures formelles – se penche tout d’abord sur la pratique des mathématiciens, qui fut à l’origine de la création de ce type d’écriture, pour dégager, dans les champs d’activité où le formalisme intervient, les différents facettes du faire mathématique et argumenter qu’il se retrouve le plus souvent entrelacé à un dire. L’examen du bouleversement conceptuel résultant de l’exploitation par les logiciens et les informaticiens de la correspondance preuve-programme (ou isomorphisme de Curry-Howard) est ensuite entrepris et cette étude offre alors la possibilité de proposer des clés de lecture des principaux formalismes contemporains utilisés pour l’analyse des langues naturelles.

Euclid’s lemma can be derived from the algebraic gcd property, but it is not at all apparent that Euclid himself does this. We would be quite surprised if he didn’t use this property because he points it out early on and because we expect him to make use of the Euclidean algorithm in some significant way. In this paper, we explore the question of just how the algebraic gcd property enters into Euclid’s proof, if indeed it does. Central to Euclid’s development is the idea of four numbers being proportional: a is to b as c is to d. Euclid gives two different definitions of proportionality, one in Book VII for numbers (“Pythagorean proportionality”) and one in Book V for general magnitudes (“Eudoxean proportionality”). We will discover that it is essential to keep in mind the difference between these two definitions and that many authorities, possibly including Euclid himself, have fallen into the trap of believing that Eudoxean proportionality for numbers is easily seen to be the same as Pythagorean proportionality. Finally, we will suggest a way to make Euclid’s proof good after 2300 years.

La Fédération Wallonie-Bruxelles met à la disposition des enseignants des outils d’évaluation susceptibles de diagnostiquer la capacité des élèves à mobiliser et à combiner, en situation inédite, un ensemble de procédures apprises. Si plusieurs chercheurs se sont investis dans la réflexion scientifique autour de ces outils et de leur pouvoir diagnostique, la question de l’impact des contenus spécifiques évalués par ceux-ci est peu envisagée. En analysant un outil centré sur l’algèbre élémentaire, cet article a pour ambition de rendre compte de la plus-value apportée par un regard alimenté par les recherches menées dans la discipline ciblée. Dépassant les constats généraux posés par la passation de l’outil telle que prévue par les auteurs dans une logique de compétences, les résultats mettent en perspective les démarches mises en oeuvre par les élèves devant les différentes tâches proposées. Au-delà des constats spécifiques à la tâche analysée, cet article montre l’importance de regards croisés sur la question de l’évaluation des compétences., The Wallonia-Brussels Federation offers teachers assessment tools meant to measure students’ abilities to select and combine, in unusual situations, learnt-in-class procedures. Even though many researchers were involved into scientific reflections on these tools and their diagnostic value, the impact of the contents specifically assessed has not been really considered. The purpose of this paper is to highlight the importance of this aspect through the analysis of a tool related to elementary algebra. While extending the general findings arising from the tool as intended by the authors in the competency-based approach, the results confront the strategies adopted by students when dealing with the different tasks proposed. Beside the specific results to tested task, the paper intends to show the added value of this integrated approach to the competence assessment issue., A Federação Valónia-Bruxelas coloca à disposição dos professores instrumentos de avaliação sucetíveis de diagnosticar a capacidade dos alunos para mobilizar e combinar, em situação inédita, um conjunto de procedimentos aprendidos. Apesar da reflexão científica de muitos investigadores em torno destes instrumentos e do seu poder de diagnóstico, a questão do impacto dos conteúdos específicos avaliados tem sido pouco considerada. Analisando um instrumento centrado na álgebra elementar, este artigo tem como propósito dar conta da mais-valia que resulta de um olhar alimentado pelas investigações realizadas nesta disciplina em particular. Ultrapassando as constatações gerais colocadas pela aplicação do instrumento numa lógica de competências de acordo com a intenção dos autores, os resultados permitem perspetivar as estratégias adotadas pelos alunos perante as tarefas propostas. Para além das constatações específicas da tarefa analisada, este artigo mostra os olhares cruzados sobre a problemática da avaliação de competências.

We give a proof and extension of two formulas of Frobenius and Stickelberger of Differential Calculus that they used in a fundamental paper concerning elliptic functions theory. Our main ingredient is the introduction of a bilinear form which is a vast generalization of Leibniz formula. This bilinear form is also connected to the Lagrange product formula. As a corollary, we give a different proof of a theorem of P.J. Olver., 26 pages, in French

In this short paper we try to describe the fundamental contribution of Quillenin the development of abstract homotopy theory and we explain how he uses this theory to lay the foundations of rational homotopy theory.

This paper introduces the notions of inaugural texts and inaugural statements and applied them to La Géometrie of Descartes. Some consequences of the existence of this kind of texts and statements are given for mathematics and their history., Cet article introduit les notions de textes et d'énoncés inauguraux et les applique à La Géométrie de Descartes. Quelques conséquences sur les mathématiques et leur histoire sont ensuite tirées de l'existence de tels textes et de tels énoncés.

International audience; In this paper, we describe constructions of majority logic decodable codes which stem out description of finite groups. To this end, we generalize some previous studies of Tonchev, Key and Moori on codes from finite groups. We also extend some results obtained by Rudolph for majority logic decodable codes. Finally, we describe applications of these codes in the area of soft-decoding techniques.

Let S be a finite set of primes, p in S, and Q_S a maximal algebraic extension of Q unramified outside S and infinity. Assume that |S|>=2. We show that the natural maps Gal(Q_p^bar/Q_p) --> Gal(Q_S/Q) are injective. Much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of GL(2n,A_Q) with prescribed properties at all places, that we study via the twisted trace formula of J. Arthur. The techniques we develop shed also some lights on the orthogonal/symplectic alternative for selfdual representations of GL(2n)., 50 pages, french. Section 4.18 has been extended : let F be a totally real field and \pi a selfdual cuspidal automorphic representation of GL(2n,A_F) which is cohomological at all the archimedean places and discrete at a finite place at least, we show that for each place v the L-parameter of \pi_v preserves a non-degenerate symplectic pairing

What did "algebra" mean before the development of the algebraic theories of the 20th century ? This paper stresses the identities taken by the algebraic practices developped during the century long discussion around the equation around the equation of secular inequalities (1766- 1874). In 1874, a strong controversy on the theory of bilinear and quadratic forms opposed Camille Jordan and Leopold Kronecker. The arithmetical ideal of Kronecker faced Jordan's claim for the simplicity of his algebraic canonical form. As the controversy combined mathematical and historical arguments, it gave rise to the writing of a history of the methods used by Lagrange, Laplace and Weierstrass in a century long mathematical discussion around the "equation of secular inequalities"., Comment: Il s'agit d'une version pr\'e publication ant\'erieure de plusieurs mois \`a la version publi\'ee (r\'ef\'erence jointe). Les commentaires du comit\'e de lecture et le travail d'\'edition ont amen\'e une \'evolution importante du contenu de cet article dans la version finale. Se reporter \`a la version publi\'ee

This paper answers a question of Clozel and Ullmo, showing that certain sequences of adelically-defined probability measures defined on an adelic quotient of a solvable group converge to the uniform measure on that quotient. This turns out to depend on any non-trivial estimate for classical Kloosterman sums. At the end, a “horizontal” analogue of the problem is stated and solved using a result of Duke, Friedlander and Iwaniec.

In this paper, we propose a method of specific Correspondence Analysis which allows to treat questionnaires when some responses are missing, and thus to free oneself from the yoke of complete disjunctive encoding. The method of specific analysis is presented within the general framework of Geometric Data Analysis for a Euclidean cloud, then particularized to multinumerical protocols and to questionnaires. We show that, in this approach, beweighted Principal Component Analysis (PCA) is privileged and that Multiples Correspondence Analysis (MCA) is equivalent to a biweighted PCA on indicator variables. Finally, we compare the specific analysis to the conventional one by writing inequalities between eigenvalues and studying the rotation of principal subspaces when one goes from the global analysis to the specific one. Dans cet article, on propose une méthode d'analyse des correspondances spécifique qui permet de traiter des questionnaires où manquent certaines réponses, et ainsi de s'affranchir du carcan du codage disjonctif complet. La méthode d'analyse spécifique est présentée dans le cadre général de l'analyse géométrique des données pour un nuage euclidien, puis particularisée à un protocole multinumérique et à un questionnaire. En particulier, on montre que l'analyse en composantes principales (ACP) bipondérée est privilégiée dans cette approche, et que l'analyse des correspondances multiples (ACM) est équivalente à une ACP bipondérée sur variables indicatrices. Enfin, on compare analyse spécifique et analyse usuelle, en donnant des inégalités sur les valeurs propres et en étudiant la rotation des sous-espaces principaux lorsque l'on passe de l'analyse globale à l'analyse spécifique.

We show that the main part of the genus of the preimages of a projective line by a generic holomorphic endomorphism of CP^2 goes to the support of the Green measure., 16 pages, paper in french. Final version. To appear in Math. Ann