Algebra in Roth,Faulhaber, and Descartes

Descartes' “multiplicative” theory of equations in the Géométrie (1637) systematically treats equations as polynomials set equal to zero, bringing out relations between equations, roots, and polynomial factors. We here consider this theory as a response to Peter Roth's suggestions in Arithmetica Philosophica (1608), notably in his “seventh-degree” problem set. These specimens of arithmetic-masterly problem design develop skills with multiplicative and other degree-independent techniques. The challenges were fine-tuned by introducing errors disguised as printing errors. During Descartes' visit to Germany in 1619–1622, he probably worked with Johann Faulhaber (1580–1635) on these problems; they are discussed in Faulhaber's Miracula Arithmetica (1622), which also looks forward to fuller publication, probably by Descartes.     

A remark on Carathéodory balls

Member of URA CNRS D 1322 Groupes de Lie et Géométrie     

What Descartes knew of mathematics in 1628

The aim of this paper is to give an account of Descartes’ mathematical achievements in 1628–1629 using, as far as is possible, only contemporary documents, and in particular Beeckman’s Journal for October 1628. In the first part of the paper, I study the content of these documents, bringing to light the mathematical weaknesses they display. In the second part, I argue for the significance of these documents by comparing them with other independent sources, such as Descartes’ Regulae ad directionem ingenii. Finally, I outline the main consequences of this study for understanding the mathematical development of Descartes before and after 1629.     

Cohomology of semi 1-coronae and extension of analytic subsets

We deal with the cohomology of semi 1-coronae. Semi 1-coronae are domains whose boundary is the union of a Levi flat part, a 1-pseudoconvex part and a 1-pseudoconcave part. Using the main result in [C. Laurent-Thiébaut, J. Leiterer, Uniform estimates for the Cauchy-Riemann equation on q-concave wedges, in: Colloque d'Analyse Complexe et Géométrie, Marseille, 1992, Astérisque 217 (7) (1993) 151-182], we prove a bump lemma for compact semi 1-coronae in Cn and then, applying Andreotti-Grauert theory, we get a cohomology finiteness theorem for coherent sheaves whose depth is at least 3. As an application we get an extension theorem for coherent sheaves and analytic subsets.     

Quadriques partielles d'indice deux

Résumé L'origine de ce travail réside dans l'observation que le groupe de Higman-Sims possède une géométrie très proche de celle d'une quadrique d'indice de Witt deux, constituée de 100 points, de droites de 2 points et de cercles de 6 points. Notre but est de décrire un système d'axiomes qui caractérise simultanément la géométrie des droites et des cercles des quadriques finies d'indice deux et la quadrique de Higman-Sims.Le premier auteur a disposé des subventions T1306 et A3485 du Conseil de Recherche en Sciences Naturelles et Génie du Canada. Elle étend ses remerciements les plus profonds à l'Université Libre de Bruxelles et au Service de Géométrie de celle-ci, pour leur hospitalité durant l'année académique 1979–1980.     

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.     

On multiple roots in Descartes’ Rule and their distance to roots of higher derivatives

If an open interval I contains a k  -fold root α$\alpha$ of a real polynomial f, then, after transforming I   to (0,∞)$(0,\infty )$, Descartes’ Rule of Signs counts exactly k roots of f in I, provided I is such that Descartes’ Rule counts no roots of the kth derivative of f. We give a simple proof using the Bernstein basis.     

A generalization of Descartes’ rule of signs and fundamental theorem of algebra

Descartes’ rule of signs yields an upper bound for the number of positive and negative real roots of a given polynomial. The fundamental theorem of algebra implies a similar property; every real polynomial of degree n ? 1 has at most n real zeroes. In this paper, we describe axiomatically function families possessing one or another of these properties. The resulting families include, at least, all polynomial functions and sums of exponential functions. As an application of our approach, we consider, among other things, a method for identifying certain type of bases for the Euclidean space.     

The dramatic episode of Sundman

In 1912 the Finnish mathematical astronomer Karl Sundman published a remarkable solution to the three-body problem, of a type that mathematicians such as Poincaré had believed impossible to achieve. Although lauded at the time, the result dimmed from view as the 20th century progressed and its significance was often overlooked. This article traces Sundman’s career and the path to his achievement, bringing to light the involvement of Ernst Lindelöf and Gösta Mittag-Leffler in Sundman’s research and professional development, and including an examination of the reception over time of Sundman’s result. A broader perspective on Sundman’s research is provided by short discussions of two of Sundman’s later papers: his contribution to Klein’s Encyklopädie and his design for a calculating machine for astronomy.     

General Weissenberg effect in free surface rheometry part I: analytical considerations

Summary The steady free surface motion of a layered medium of viscoelastic fluids driven by a vertical rotating rod and the free surface motion of a viscoelastic fluid between vertical, eccentric, rotating cylinders is solved by means of domain perturbations. The flow field, the shape of the free surfaces and the interface and the climb at the rod are determined by expanding the stress in a series of Rivlin-Ericksen tensors and developing the theory as a perturbation of the rest state in powers of the angular frequency of the rod or of the cylinders. It is shown that rheometrical measurements in the rotating rod geometry are reliably reproducible whilst concentric cylinder geometry presents important practical problems in measuring rheological constants.

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Résumé Les problèmes des écoulements stationnaires à surface libre de plusieurs couches superposées de fluides viscoélastiques mis en mouvement par une barre verticale en révolution et d'un fluide viscoélastique entre des cylindres verticaux, excentriques et tournants sont résolus par le moyen des perturbations de domaine. Le champ des vitesses, les formes de la surface libre et de l'interface sont déterminées en développant la contrainte en une série de tenseurs Rivlin-Ericksen. La théorie est basée sur la perturbation de l'état statique et les variables sont exprimées en une série de puissances de la vitesse de rotation de la barre ou des cylindres. On a démontré que les mesurements rhéométriques dans la géométrie de barre tournante sont sûrement reproductibles tandis que l'utilisation de la géométrie de cylindres concentriques présente d'importants problèmes pratiques en mesurant les constantes rhéologiques.

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Parts of this paper presented at the 9th Canadian Congress of Applied Mechanics, University of Saskatchewan, Saskatoon, Canada, May 29–June 3, 1983 and at the XVIth International Congress of Theoretical and Applied Mechanics, The Technical University of Denmark, Lyngby, Denmark, August 19–25, 1984.     

Metrizability of spaces of holomorphic functions with the Nachbin topology

In this paper we prove, among other things, that the space of all holomorphic functions on an open subset U of a complex metrizable space E, endowed with the Nachbin ported topology, is metrizable only if E has finite dimension. This answers a question by Mujica in [J. Mujica, Gérmenes holomorfos y funciones holomorfas en espacios de Fréchet, Publicaciones del Departamento de Teoría de Funciones, Universidad de Santiago, Spain, 1978].     

On the back-scatter cross-sections of ice spheres

Résumé La section droite de diffusion arrière de sphères de grand diamètre constituées par de la glace ou une matière (Stycast) ayant le même indice de réfraction est envisagée d'une manière à la fois théorique et expérimentale. La section droite est évaluée théoriquement par l'application d'une méthode d'optique géométrique simplifiée dans laquelle on considère seulement les réflexions sur les faces avant et arrière de la sphère. Dans le domaine du rapport diamètre/longueur d'onde de l'ordre de 1,3 à 6,7, les résultats obtenus par cette méthode s'accordent bien avec le calcul rigoureux de la section droite obtenu par la solution de l'équation demie. Pour des diamètres plus grands, la solution approchée sous-estime la section droite demie par un facteur qui est cependant plus petit que 3. La théorie simplifiée est étendue au cas de sphères diélectriques ayant leur surface partiellement couverte par une couche réfléchissante.Des données expérimentales sur la diffusion arrière, à une longuer d'onde de 3,22 cm, par des sphères en Stycast recouvertes d'une calotte sphérique de petite dimension, sont présentées dans le but d'illustrer la théorie précédente. En effet, les résultats obtenus, pour toutes les positions de la calotte sphérique au cours de sa rotation autour d'un axe vertical, sont en plein accord, dans la limite des erreurs expérimentales, avec la théorie faisant intervenir la composition de 3 parcours optiques principaux.

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A summary of this work was presented at the Tenth Weather Radar Conference, Washington, D. C., April 22–25, 1963.     

A priori reduction method for solving the two-dimensional Burgers’ equations

The two-dimensional Burgers’ equations are solved here using the A Priori Reduction method. This method is based on an iterative procedure which consists in building a basis for the solution where at each iteration the basis is improved. The method is called a priori because it does not need any prior knowledge of the solution, which is not the case if the standard Karhunen-Loéve decomposition is used. The accuracy of the APR method is compared with the standard Newton-Raphson scheme and with results from the literature. The APR basis is also compared with the Karhunen-Loéve basis.     

Pliages de Géométries minces

Résumé Nous montrons que tout pliage d'une géométrie (au sens de Buekenhout [1] et Tits [3]) mince fortement connexe est associé à une réflexion; il n'en est plus de même si la connexité de la géométrie est quelque peu affaiblie.

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Summary We prove that every folding of a thin geometry (in the sense of Buekenhout [1] and Tits [3]) is associated to a reflection, whenever the geometry is strongly connected; this is false, if the connexity of the geometry is somewhat weaker.

     

Rethinking geometrical exactness

A crucial concern of early modern geometry was fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed into it. According to Bos, this is the exactness concern. I argue that Descartes’s way of responding to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In Section 2, I outline the exactness concern as, I think, it appeared to Descartes. In Section 3, I account for Descartes’s views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in which sense his geometry can be conceived as a conservative extension of EPG. I conclude by briefly discussing some structural similarities and differences between Descartes’s geometry and EPG.     

Normal integral bases for cyclic Kummer extensions

Let K/F be a Kummer cyclic extension of number fields. In the case when the degree is a prime number, Gómez Ayala gave an explicit criterion for the existence of a normal integral basis. More recently Ichimura proposed a generalization of that result for cyclic extensions of arbitrary degree, but we have found that Ichimura’s result is incorrect. In this paper we present a counter-example to Ichimura’s result as well as the correct generalization of Gómez Ayala’s result.     

Preservers of eigenvalue inclusion sets of matrix products

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the union of Cassini ovals, and the Ostrowski’s set. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)Φ(B))=S(AB) for all matrices A and B.     

Large deviations of bootstrapped U -statistics

We develop large deviation results with Cramér’s series and the best possible remainder term for bootstrapped U-statistics with non-degenerate bounded kernels. The method of the proof is based on the contraction technique of Keener, Robinson and Weber [R.W. Keener, J. Robinson, N.C. Weber, Tail probability approximations for U-statistics, Statist. Probab. Lett. 37 (1) (1998) 59-65], which is a natural generalization of the classical conjugate distribution technique due to Cramér [H. Cramér, Sur un nouveau théoréme-limite de la theorie des probabilites, Actual. Sci. Indust. 736 (1938) 5-23].     

On the deformation of elastic plates

Summary The purpose of the paper is to indicate some elementary applications of a method for the resolution of elliptic problems. The method consists in fitting solutions obtained in several subdomains and depending on arbitrary constants. One of the interests of the method is that it seems particularly efficient in the case of singularities. An application to the bending of triangular and parallelogramic plates is given.

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Résumé Les auteurs se proposent d'exposer sur quelques cas simples une méthode de résolution pour des problèmes elliptiques. La méthode, qui consiste à raccorder en certains points choisis diverses solutions obtenues dans des sous-domaines, semble particulièrement efficace pour traiter des problèmes présentant des singularités. La méthode est appliquée à titre d'exemple à la flexion de plaques triangulaires ou en forme de paralléllogrammes.

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Dédié au Professeur E. Stiefel, avec notre entière reconnaissance     

Gâteaux differentiability of the dual gap function of a variational inequality

In terms of the mapping involved in a variational inequality, we characterize the Gâteaux differentiability of the dual gap function G and present several sufficient conditions for its directional derivative expression, including one weaker than that of Danskin [J.M. Danskin, The theory of max–min, with applications, SIAM Journal on Applied Mathematics 14 (1966) 641–664]. When the solution set of a variational inequality problem is contained in that of its dual problem, the Gâteaux differentiability of G on the latter turns out to be equivalent to the conditions appearing in the authors’ recent results about the weakly sharp solutions of the variational inequality problem.