Les Récréations Mathématiques d'Édouard Lucas: quelques éclairages

The article is devoted to Edouard Lucas's contribution to the development of mathematical recreations in the France of the post 1870 war period. Lucas's name is associated to four volumes of Récréations mathématiques published between 1882 and 1894 (the last two having been published posthumously) and to a posthumous volume L'Arithmétique amusante, which appeared in 1895. The author analyzes the context of reform of science education in relation to which mathematical recreations appeared as a means of attracting a wider public to scientific activities and inspiring young people to study science. The article brings to light how the milieu of new associations which took shape to promote science (Association Française pour l'Avancement des Sciences, Société Mathématique de France) allowed the constitution of social groups internationally connected and quite active in the promotion and development of mathematical recreations. Lastly, the article suggests that this type of mathematical activity allowed the cultivation of fields that at the time the French academic milieu perceived as marginal such as number theory and analysis situs as well as their applications.     

Some results concerning maximum Rényi entropy distributions

We consider the Student-t and Student-r distributions, which maximise Rényi entropy under a covariance condition. We show that they have information-theoretic properties which mirror those of the Gaussian distributions, which maximise Shannon entropy under the same condition. We introduce a convolution which preserves the Rényi maximising family, and show that the Rényi maximisers are the case of equality in a version of the Entropy Power Inequality. Further, we show that the Rényi maximisers satisfy a version of the heat equation, motivating the definition of a generalised Fisher information.     

q-Analogs of symmetric function operators

For any homomorphism V on the space of symmetric functions, we introduce an operation that creates a q-analog of V. By giving several examples we demonstrate that this quantization occurs naturally within the theory of symmetric functions. In particular, we show that the Hall-Littlewood symmetric functions are formed by taking this q-analog of the Schur symmetric functions and the Macdonald symmetric functions appear by taking the q-analog of the Hall-Littlewood symmetric functions in the parameter t. This relation is then used to derive recurrences on the Macdonald q,t-Kostka coefficients.RésuméPour un homomorphisme V sur l'espace des fonctions symétriques, nous présentons une opération qui crée un q-analogue de V. En donnant plusieurs exemples nous démontrons que cette quantization se produit naturellement dans la théorie de fonctions symétriques. En particulier, nous prouvons que les fonctions symétriques de Hall-Littlewood sont constituées en prenant ce q-analogue des fonctions symétriques de Schur et les fonctions symétriques de Macdonald apparaissent en prenant le q-analogue des fonctions symétriques de Hall-Littlewood dans le paramètre t. Cette relation est alors employée pour dériver des récurrence sur les coefficients Macdonald q,t-Kostka.     

Random quantum channels II: Entanglement of random subspaces, Rényi entropy estimates and additivity problems

In this paper we obtain new bounds for the minimum output entropies of random quantum channels. These bounds rely on random matrix techniques arising from free probability theory. We then revisit the counterexamples developed by Hayden and Winter to get violations of the additivity equalities for minimum output Rényi entropies. We show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the p-Rényi entropy, for all p>1. For some sequences of random quantum channels, we compute almost surely the limit of their Schatten S₁→Sp norms.     

Lamé operators with finite monodromy—a combinatorial approach

We are describing Lamé differential operators with a full set of algebraic solutions. For each finite group G, we are describing the possible values of the degree parameter n such that the Lamé operator L_n has the projective monodromy group G. The main technical tool is the combinatorics associated to Belyi functions, ideas that we already used in (Rend. Sem. Mat. Univ. Padova 107 (2002) 191-208) for describing the case n=1. We also supply proofs to some finiteness properties conjectured by Baldassarri and by Dwork, and we work out an explicit formula for the number of essentially different Lamé equations when n=2. This approach can be generalized for arbitrary degree n (see (Counting Integral Lamé Equations by Means of Dessins d'Enfants, arXiv:math.CA/0311510) for n integer).     

Maximum distance separable codes and arcs in projective spaces

Given any linear code C over a finite field GF(q) we show how C can be described in a transparent and geometrical way by using the associated Bruen-Silverman code.Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes. Our proofs make use of the connection between the work of Rédei [L. Rédei, Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973. Translated from the German by I. Földes. [18]] and the Rédei blocking sets that was first pointed out over thirty years ago in [A.A. Bruen, B. Levinger, A theorem on permutations of a finite field, Canad. J. Math. 25 (1973) 1060-1065]. The main results of this paper significantly strengthen those in [A. Blokhuis, A.A. Bruen, J.A. Thas, Arcs in PG(n,q), MDS-codes and three fundamental problems of B. Segre—Some extensions, Geom. Dedicata 35 (1-3) (1990) 1-11; A.A. Bruen, J.A. Thas, A.Blokhuis, On M.D.S. codes, arcs in PG(n,q) with q even, and a solution of three fundamental problems of B. Segre, Invent. Math. 92 (3) (1988) 441-459].     

On locking materials

This work is devoted to the study of variational problems encountered in the theory of locking materials introduced by Prager. In a first step, we are interested in the dual principles governing the equilibrium state of an elastic-locking material. We establish the inf-sup equality by means of a penalty method. In a second step, we introduce and discuss the locking limit analysis problem. This problem allows one to determine which displacements can be imposed on a locking body before complete locking. We show the equivalence of the statical and the kinematical methods. Two appendices close the paper. The first one focuses the attention on stress concentrations in a problem of torsion, and the second is devoted to materials with limited compressibility which lead to simpler mathematical problems.

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Résumé Ce travail est consacré à l'étude des problèmes variationnels posés par les matériaux à blocage introduits par Prager. Dans un premier temps nous nous intéressons aux principes duaux qui gouvernent l'équilibre d'un milieu élastique à blocage. Nous établissons l'égalité Inf-Sup par une méthode de pénalisation. Dans un deuxième temps nous introduisons et discutons l'analyse limite de blocage qui permet de déterminer les déplacements susceptibles d'être imposés au corps avant son blocage complet. On montre que les approches statiques et cinématiques conduisent à la même valeur de la charge limite de blocage. Le travail se termine par deux annexes: l'une met l'accent sur la possibilité de concentration de contraintes (qui sont alors des mesures) dans un problème de torsion; l'autre étudie les matériaux à compressibilité limitée pour lesquels les problèmes mathémqtiques se simplifient notablement.

     

Multivariate Frobenius–Padé approximants: Properties and algorithms

The aim of this paper is to construct rational approximants for multivariate functions given by their expansion in an orthogonal polynomial system. This will be done by generalizing the concept of multivariate Padé approximation. After defining the multivariate Frobenius–Padé approximants, we will be interested in the two following problems: the first one is to develop recursive algorithms for the computation of the value of a sequence of approximants at a given point. The second one is to compute the coefficients of the numerator and denominator of the approximants by solving a linear system. For some particular cases we will obtain a displacement rank structure for the matrix of the system we have to solve. The case of a Tchebyshev expansion is considered in more detail.     

Carathéodory interpolation on the non-commutative polydisk

The Carathéodory problem in the N-variable non-commutative Herglotz-Agler class and the Carathéodory-Fejér problem in the N-variable non-commutative Schur-Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory-Fejér) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set Λ) takes operator values with positive semidefinite real part (resp., contractive operator values) on N-tuples of Λ-jointly nilpotent contractive n×n matrices, for all n∈N.     

Lévy group and density measures

We will deal with finitely additive measures on integers extending the asymptotic density. We will study their relation to the Lévy group G of permutations of N. Using a new characterization of the Lévy group G we will prove that a finitely additive measure extends density if and only if it is G-invariant.     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

Optimal vector quantization in terms of Wasserstein distance

The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the Rényi-α-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained (α=1) and memory-size constrained (α=0) quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by György and Linder (2002, 2003) [11] and [12].     

The ovals in the Excerpta Mathematica and the origins of Descartes’ method of normals

The only occurrence of Descartes’ method of normals before La Géométrie (1637) is to be found in the Excerpta Mathematica. These mathematical fragments, published posthumously among others works in 1701, and dated by Tannery before 1629, deal with curves used in dioptrics which Descartes called ovals. I study in detail two of the texts on ovals together with the related texts in La Géométrie in order to shed light on the geometrical origins of Descartes’ method of normals.     

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.     

Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids

Any étale Lie groupoid G is completely determined by its associated convolution algebra C_c^∞(G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c^∞(G)-C_c^∞(H)-bimodule C_c^∞(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c^∞gives an equivalence between the Morita category of étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.     

Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany

Remainder problems have a long tradition and were widely disseminated in books on calculation, algebra, and recreational mathematics from the 13th century until the 18th century. Many singular solution methods for particular cases were known, but Bachet de Méziriac was the first to see how these methods connected with the Euclidean algorithm and with Diophantine analysis (1624). His general solution method contributed to the theory of equations in France, but went largely unnoticed elsewhere. Later Euler independently rediscovered similar methods, while von Clausberg generalized and systematized methods that used the greatest common divisor procedure. These were followed by Euler's and Lagrange's continued fraction solution methods and Hindenburg's combinatorial solution. Shortly afterwards, Gauss, in the Disquisitiones Arithmeticae, proposed a new formalism based on his method of congruences and created the modular arithmetic framework in which these problems are posed today.     

The introduction of logarithms into Spain

During the first half of the 17th century, logarithms were taught by some professors in Spain, but knowledge of this subject remained scanty until the publication of Architectura civil by Juan Caramuel (1678) and especially of Trigonometria española by José Zaragoza (1672). Logarithms were considered only as an aid for computation up to the second half of the 18th century. Only when the infinitesimal calculus became more widely spread in Spanish mathematics, analytical interpretations of logarithms were also taken into account in books such as Elementos de matemáticas by Benito Bails (1776).     

Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas dissymétrique

This paper generalizes the first author's preceding works concerning admissible functions on certain Fano manifolds [A. Ben Abdesselem, Lower bound of admissible functions on sphere, Bull. Sci. Math. 126 (2002) 675-680 [2]; A. Ben Abdesselem, Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas symétrique, Bull. Sci. Math. 130 (2006) 341-353 [3]]. Here, we study a larger class of functions which can be less symmetric than the ones studied before. When the sup of these functions is null, we prove that they admit a lower bound, giving precisely Tian invariant [G. Tian, On Kähler-Einstein metrics on certain Kähler manifolds with C₁(M)>0, Invent. Math. 89 (1987) 225-246 [7]] (see also [T. Aubin, Réduction du cas positif de l'équation de Monge-Ampère sur les variétés Kählériennes à la démonstration d'une inégalité, J. Funct. Anal. 57 (1984) 143-153 [1]]) on these manifolds.     

A variant of the hypergraph removal lemma

Recent work of Gowers [T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001) 465-588] and Nagle, Rödl, Schacht, and Skokan [B. Nagle, V. Rödl, M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Applications of the regularity lemma for uniform hypergraphs, preprint] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975) 299-345], and Furstenberg and Katznelson [H. Furstenberg, Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Anal. Math. 34 (1978) 275-291] concerning one-dimensional and multidimensional arithmetic progressions, respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [T. Tao, The Gaussian primes contain arbitrarily shaped constellations, preprint] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes.