Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.     

Geometry of polarised varieties

Publications mathématiques de l'IHÉS - In this paper, we investigate the geometry of projective varieties polarised by ample and more generally nef and big Weil divisors. First we...     

On the intellectual heritage of Henri Poincaré

Poincaré's understanding of how mathematics grows over time was informed by the theory of evolution by natural selection and Mach's economy of thought. According to Poincaré, mathematics is neither created nor discovered but cultivated as part of our intellectual inheritance. This paper is based on a lecture delivered at the joint meeting of the BSHM and CSHPM at Trinity College, Dublin, 2011. The title echoes The mathematical heritage of Henri Poincaré (1983) a collection of essays edited by Felix Browder.     

Two new multivariable generating relations

A multidimensional extension of Bailey's transform is utilised to deduce two new generating relations of quite a general character. These expressions are then specialised to give more practical formulae in terms of Karlsson's generalised Kampé de Fériet functions which embody very many generating relations. A number of interesting special cases are given in an appendix which includes results involving Lauricella polynomials, generalised hypergeometric polynomials and the polynomials of Meixner, Charlier and Laguerre.     

Galois sections for abelianized fundamental groups

Given a smooth projective curve X of genus at least 2 over a number field k, Grothendieck’s Section Conjecture predicts that the canonical projection from the étale fundamental group of X onto the absolute Galois group of k has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of k but not over k. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the étale fundamental group ‘with abelianized geometric part’ onto the Galois group. We also point out the relation to the elementary obstruction of Colliot-Thélène and Sansuc. This article has an appendix by E. V. Flynn.     

Invariants de classes : exemples de non-annulation en dimension supérieure

The so-called class-invariant homomorphism ψ measures the Galois module structure of torsors—under a finite flat group scheme G—which lie in the image of a coboundary map associated to an isogeny between (Néron models of) abelian varieties with kernel G. When the varieties are elliptic curves with semi-stable reduction and the order of G is coprime to 6, it is known that the homomorphism ψ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number p > 2, of an abelian variety A of dimension p endowed with an isogeny (with kernel μ _p ) whose coboundary map is surjective. In the case when A has rank zero and the p-part of the Picard group of the base is non-trivial, we obtain examples where ψ does not vanish on torsion points.

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Résumé  Le class-invariant homomorphism permet de mesurer la structure galoisienne des torseurs—sous un schéma en groupes fini et plat G—qui sont dans l’image du cobord associé à une isogénie, de noyau G, entre des (modèles de Néron de) variétés abéliennes. Quand les variétés sont des courbes elliptiques à réduction semi-stable et que l’ordre de G est premier à 6, on sait que cet homomorphisme s’annule sur les points de torsion. Dans cet article, en nous servant de restrictions de Weil de courbes elliptiques, nous construisons, pour tout nombre premier p > 2, une variété abélienne A de dimension p munie d’une isogénie (de noyau μ _p ) dont le cobord est surjectif. Si A est de rang nul, et si la p-partie du groupe de Picard de la base est non triviale, nous obtenons ainsi un exemple où le class-invariant homomorphism ne s’annule pas sur les points de torsion.

     

Optimal (4up, 5, 1) optical orthogonal codes

By a (ν, k, 1)‐OOC we mean an optical orthogonal code. In this paper, it is proved that an optimal (4p, 5, 1)‐OOC exists for prime p ≡ 1 (mod 10), and that an optimal (4up, 5, 1)‐OOC exists for u = 2, 3 and prime p ≡ 11 (mod 20). These results are obtained by applying Weil's theorem. © 2004 Wiley Periodicals, Inc.     

Mikhail Nikolaevich Lagutinskii (1871–1915): Un Mathématicien Méconnu

Dans cette note biographique nous rapportons, à quelques détails près, tout ce qui est connu de la vie de M. N. Lagutinskii. (Certains faits supplémentaires concernant M. N. Lagutinskii seront publiés dans la version russe de ce travail (en préparation).) Ce mathématicien russe, dans une série de travaux importants mais complètement méconnus, a développé substantiellement la méthode de Darboux de recherche, en termes finis, des intégrales premières de systèmes d'équations différentielles ordinaires polynomiales. Il a aussi développé la théorie de l'intégrabilité en termes finis de tels systèmes d'équations. Nous fournissons sa bibliographie complète. L'analyse détaillée du contenu mathématique de son œuvre est le sujet de la seconde partie ([30], en préparation) de ce travail. Elle paraı̂tra ultérieurement.Copyright 1998 Academic Press.In this biographical note, we report virtually all that is known about the life of the Russian mathematician, M. N. Lagutinskii. (Some additional facts concerning Lagutinskii will be published in the Russian version of this paper (to appear).) In a series of important but completely unrecognised works, Lagutinskii developed the Darboux method of determination in finite terms of the first integrals of systems of polynomial, ordinary differential equations. He also developed the theory of integrability in finite terms of such systems of equations. We provide his complete bibliography. A detailed analysis of the mathematical contents of his works is the subject of the second part [30] of the present study. It will appear at a later date.Copyright 1998 Academic Press.[formula]Copyright 1998 Academic Press.AMS 1991 subject classifications: 01A60, 01A70, 01A73, 34–03     

On the classification of exceptional planar functions over \mathbb {F}_{p}

We will present many strong partial results towards a classification of exceptional planar/PN monomial functions on finite fields. The techniques we use are the Weil bound, Bézout’s theorem, and Bertini’s theorem.     

Poincaré polynomial of the moduli spaces of parabolic bundles

In this paper we use Weil conjectures (Deligne’s theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. The quasi parabolic analogue of the Siegel formula, together with the method of HarderNarasimhan filtration gives us a recursive formula for the Poincaré polynomials of the moduli. We solve the recursive formula by the method of Zagier, to give the Poincaré polynomial in a closed form. We also give explicit tables of Betti numbers in small rank, and genera.     

An investigation of pedagogical techniques in Descartes' La géométrie

Much research has been conducted about the philosophy and mathematical writings of René Descartes, but that which focuses on pedagogy does so in a holistic manner. The present study uses a systematic approach to identify pedagogical techniques within each sentence of Descartes' La géométrie. Next, the study provides an analysis of La géométrie based on the techniques identified, their frequencies, and patterns of use within the text. The results of this analysis indicate that Descartes placed a high value on the use of demonstration, particularly in conjunction with deductive reasoning and multiple representations; that Descartes believed his method of approaching mathematical problems was superior to other methods; and that Descartes was in fact concerned with whether his readers understood his ideas or not.     

From National to International Society: The London Mathematical Society, 1867–1900

The London Mathematical Society had been founded in 1865 as little more than a college club. Thanks to the support of prominent members from the British mathematical community, it had quickly grown in size and stature during its first two years; yet, while firmly established at home, it had still to secure an academic reputation overseas. This paper, a sequel to [35], examines the principal developments that occurred in the period from 1867 until the turn of the century, during which time the Society would consolidate its position as a prestigious learned body in both the national and international mathematical arenas.Copyright 1998 Academic Press.The London Mathematical Society a été fondée en 1865 et a débuté en tant que club de collège. Grâce au soutien de membres éminents de la communauté mathématique britannique, le nombre d'adhérents et la stature de cette Société se sont rapidement accrus pendant ses deux premières années; cependant, bien que résolument établie en Grande Bretagne elle devait encore établir sa renommée à l'étranger. Cet exposé qui fait suite à [35], examine les développements importants qui se sont produits entre 1867 et la fin du dix-neuvième siècle, une période pendant laquelle la Société a consolidé sa position de Société prestigieuse dans les arènes nationales et internationales.Copyright 1998 Academic Press.Als die London Mathematical Society 1865 gegründet wurde, war sie kaum mehr als ein Universitätsverein. Weil prominente Mathematiker aus ganz Großbritannien sie unterstützten, nahm sie in den ersten zwei Jahren schnell an Größe und Gestalt zu; obwohl sie sich im eigenen Lande beständig entwickelte, mußte sie sich im Ausland zunächst noch um einen akademischen Ruf bemühen. In dieser Arbeit werden (im Anschluß an [35]), die wesentlichen Entwicklungen in der Zeit von 1867 bis zur Jahrhundertwende untersucht, also in der Zeit, in der die “Society” ihre Stellung als anerkannte gelehrte Vereinigung sowohl auf nationaler als auch auf internationaler Ebene gefestigt hat.Copyright 1998 Academic Press.MSC 1991 subject classification: 01A55     

PariTOP: A goal programming-based software for real estate assessment

In Quebec's municipal taxation system, the tax burden is allocated according to the market value of properties. Determining the market values for housing inventories containing tens of thousands of properties is obviously a complex task with enormous implications in terms of the resources required and the quality of the resulting estimates. This article presents PariTOP, a decision-support system that uses mathematical programming in general and goal programming in particular to estimate the market value of large numbers of residential properties. PariTOP is the result of a partnership between the Communauté urbaine de Québec (CUQ), an administrative unit that comprises 12 cities, and Modellium, a Canadian company that develops decision-support systems.     

Les Pré-(a b)-algèbres à Homotopie Près

Dans cet article on étudie le concept d'algèbre à homotopie près pour une structure définie par deux opérations ? et ?. Des exemples importants d'une telle structure sont ceux des algèbres pré-Gerstenhaber et pré-Poisson graduées. Etant donnée une structure d'algèbre pré-commutative et pré-Lie graduée pour deux décalages des degrés donnés par a et b, on définit la structure d'une pré-(a, b)-algèbre graduée et on donne une construction explicite de l'algèbre à homotopie près associée.

We study in this article the concept of algebra up to homotopy for a structure defined by two operations, ? and ?. Important examples of such structure are those of graded pre-Gerstenhaber and pre-Poisson algebras.

Given a structure of pre-commutative and pre-Lie algebra for two shifts of degree given by a and b, we define the structure of a graded pre-(a, b)-algebra, and we give an explicit construction of the associated algebra up to homotopy     

Linear liftings of skew symmetric tensor fields of type (1, 2) to Weil bundles

The paper contains a classification of linear liftings of skew symmetric tensor fields of type (1, 2) on n-dimensional manifolds to tensor fields of type (1, 2) on Weil bundles under the condition that n ⩾ 3. It complements author’s paper “Linear liftings of symmetric tensor fields of type (1, 2) to Weil bundles” (Ann. Polon. Math. 92, 2007, pp. 13–27), where similar liftings of symmetric tensor fields were studied. We apply this result to generalize that of author’s paper “Affine liftings of torsion-free connections to Weil bundles” (Colloq. Math. 114, 2009, pp. 1–8) and get a classification of affine liftings of all linear connections to Weil bundles.     

A twisted theorem of Chebotarev

The classical function field version of Chebotarev's Theorem follows from the Lang–Weil estimate and an ‘untwisting’ trick. We obtain an analogue in the framework of difference schemes, using Hrushovski's twisted Lang–Weil estimate. To cite this article: I. Toma?i?, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

We compute a local linearization for the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as a solution of the nonlinear, dynamic, elastic wave equation, where the first Piola–Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense, the mathematical model of the direct problem is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article, we summarize some continuity results for this operator and deduce its Fréchet derivative as well as the adjoint of this derivative. Because the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton‐type methods, the knowledge of the Fréchet derivative and its adjoint is of utmost importance. Copyright © 2016 John Wiley & Sons, Ltd.     

Dévissages des F-complexes de \mathcal{D}-modules arithmétiques en F-isocristaux surconvergents

Résumé Nous construisons, avec le langage des -modules arithmétiques de Berthelot, ?les F-isocristaux surcohérents?. Cette catégorie, sur certains schémas séparés et lisses, est équivalente à celle des F-isocristaux surconvergents. Nous en déduisons une notion de ?F-complexes de -modules arithmétiques dévissables en F-isocristaux surconvergents?. Nous prouvons que les F-complexes surholonomes se dévissent en F-isocristaux surconvergents. On établit ensuite une formule cohomologique, étendant celle d’étesse et Le Stum, des fonctions L associées aux duaux des F-complexes de -modules arithmétiques dévissables en F-isocristaux surconvergents. Puis, nous obtenons un analogue p-adique de Weil II généralisant celui de Kedlaya.     

Of Our Own Nation: John Wallis's Account of Mathematical Learning in Medieval England

In A treatise of algebra both historical and practical (London 1685), John Wallis wrote the first survey of the state of mathematical learning in medieval England, and discussed with particular care the arrival and significance of the Hindu–Arabic numeral system. This paper offers a detailed commentary on Wallis's account in relation to the sources he used and the 17th-century Oxford context in which he wrote. The paper also supplements Wallis's treatment where possible with some of the findings of modern scholarship. It therefore provides on the one hand an overview of the spread of mathematical learning into medieval England, and on the other an insight into late 17th-century historiography. Wallis pioneered several new historiographical methods and can perhaps be claimed as the first modern historian of mathematics. Copyright 2001 Academic Press.Dans A treatise of algebra both historical and practical (Londres 1685) John Wallis a été le premier à écrire sur l'etat des études de mathématiques dans l'Angleterre du Moyen Age, et il a discuté avec un soin particulier de l'arrivée et du sens du système numérique hindou–arabe. Cet exposé offre un commentaire détaillé sur l'essai de Wallis en ce qui concerne les sources utilisées et le contexte d'Oxford au dix-septième siècle, époque à laquelle il écrivit. Cet exposé ajoute en plus au traitement de Wallis certaines découvertes d'erudition moderne. Il fournit donc d'une part un aperçu général de l'étendue des études de mathématiques dans l'Angleterre moyenâgeuse et d'autre part un aperçu dans l'historiographie à la fin du dix-septième siècle. Wallis à été le pionier de plusiers méthodes d'historiographie et pourrait être acclamé comme le premier historien moderne de mathématiques. Copyright 2001 Academic Press.MSC classifications: 01A35, 01A45, 01A85.     

Constructions of optimal optical orthogonal codes with weight five

Several direct constructions via skew starters and Weil's theorem on character sum estimates are given in this paper for optimal (gv, 5, 1) optical orthogonal codes (OOCs) where 60 ≤ g ≤ 180 satisfying g ≡ 0 (mod 20) and v is a product of primes greater than 5. These improve the known existence results on optimal OOCs. Especially, we provide an optimal (v, 5, 1)‐OOC for any integer v ≡ 60, 420, 660, 780, 1020, 1140, 1380, 1740 (mod 1800). © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 54–69, 2005.