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Maurice Janet’S Algorithms on Systems of Linear Partial Differential Equations
Maurice Janet’s algorithms on systems of linear partial differential equations Kenji Iohara Philippe Malbos Abstract – This article describes the emergence of formal methods in theory of partial differential equations (PDE) in the French school of mathematics through Janet’s work in the period 1913-1930. In his thesis and in a series of articles published during this period, Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the uniqueness of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the 20th century in various algebraic contexts. Keywords – Linear PDE systems, formal methods, Janet’s bases. M.S.C. 2010 – Primary: 01-08, 01A60. Secondary: 13P10, 12H05, 35A25. 1 Introduction 2 2 Historical context of Janet’s work 6 3 Emergence of formal methods for linear PDE systems 8 4 Algebraisation of monomial PDE systems 12 5 Janet’s completion procedure 17 6 Initial value problem 19 7 Janet’s monomial order on derivatives 23 1. -
The “Geometrical Equations:” Forgotten Premises of Felix Klein's
The “Geometrical Equations:” Forgotten Premises of Felix Klein’s Erlanger Programm François Lê October 2013 Abstract Felix Klein’s Erlanger Programm (1872) has been extensively studied by historians. If the early geometrical works in Klein’s career are now well-known, his links to the theory of algebraic equations before 1872 remain only evoked in the historiography. The aim of this paper is precisely to study this algebraic background, centered around particular equations arising from geometry, and participating on the elaboration of the Erlanger Programm. An- other result of the investigation is to complete the historiography of the theory of general algebraic equations, in which those “geometrical equations” do not appear. 1 Introduction Felix Klein’s Erlanger Programm1 is one of those famous mathematical works that are often invoked by current mathematicians, most of the time in a condensed and modernized form: doing geometry comes down to studying (transformation) group actions on various sets, while taking particular care of invariant objects.2 As rough and anachronistic this description may be, it does represent Klein’s motto to praise the importance of transformation groups and their invariants to unify and classify geometries. The Erlanger Programm has been abundantly studied by historians.3 In particular, [Hawkins 1984] questioned the commonly believed revolutionary character of the Programm, giving evi- dence of both a relative ignorance of it by mathematicians from its date of publication (1872) to 1890 and the importance of Sophus Lie’s ideas4 about continuous transformations for its elabora- tion. These ideas are only a part of the many facets of the Programm’s geometrical background of which other parts (non-Euclidean geometries, line geometry) have been studied in detail in [Rowe 1989b]. -
Université Du Québec À Montréal Edwin B. Wilson Aux Origines De L
UNIVERSITÉ DU QUÉBEC À MONTRÉAL EDWIN B. WILSON AUX ORIGINES DE L’ÉCONOMIE MATHÉMATIQUE DE PAUL SAMUELSON: ESSAIS SUR L’HISTOIRE ENTREMÊLÉE DE LA SCIENCE ÉCONOMIQUE, DES MATHÉMATIQUES ET DES STATISTIQUES AUX ÉTATS-UNIS, 1900-1940 THÈSE PRÉSENTÉE COMME EXIGENCE PARTIELLE DU DOCTORAT EN ÉCONOMIQUE PAR CARVAJALINO FLOREZ JUAN GUILLERMO DÉCEMBRE 2016 UNIVERSITÉ DU QUÉBEC À MONTRÉAL EDWIN B. WILSON AT THE ORIGIN OF PAUL SAMUELSON'S MATHEMATICAL ECONOMICS: ESSAYS ON THE INTERWOVEN HISTORY OF ECONOMICS, MATHEMATICS AND STATISTICS IN THE U.S., 1900-1940 THESIS PRESENTED AS PARTIAL REQUIREMENT OF DOCTOR OF PHILOSPHY IN ECONOMIC BY CARVAJALINO FLOREZ JUAN GUILLERMO DECEMBER 2016 ACKNOWLEDGMENTS For the financial aid that made possible this project, I am thankful to the Université du Québec À Montréal (UQAM), the Fonds de Recherche du Québec Société et Culture (FRQSC), the Social Science and Humanities Research Council of Canada (SSHRCC) as well as the Centre Interuniversitaire de Recherche sur la Science et la Technologie (CIRST). I am also grateful to archivists for access to various collections and papers at the Harvard University Archives (Edwin Bidwell Wilson), David M. Rubenstein Rare Book & Manuscript Library, Duke University (Paul A. Samuelson and Lloyd Metzler), Library of Congress, Washington, D.C. (John von Neumann and Oswald Veblen) and Yale University Library (James Tobin). I received encouraging and intellectual support from many. I particularly express my gratitude to the HOPE Center, at Duke University, where I spent the last semester of my Ph.D. and where the last chapter of this thesis was written. In particular, I thank Roy Weintraub for all his personal support, for significantly engaging with my work and for his valuable research advice. -
The Case of Sicily, 1880–1920 Rossana Tazzioli
Interplay between local and international journals: The case of Sicily, 1880–1920 Rossana Tazzioli To cite this version: Rossana Tazzioli. Interplay between local and international journals: The case of Sicily, 1880–1920. Historia Mathematica, Elsevier, 2018, 45 (4), pp.334-353. 10.1016/j.hm.2018.10.006. hal-02265916 HAL Id: hal-02265916 https://hal.archives-ouvertes.fr/hal-02265916 Submitted on 12 Aug 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Interplay between local and international journals: The case of Sicily, 1880–1920 Rossana Tazzioli To cite this version: Rossana Tazzioli. Interplay between local and international journals: The case of Sicily, 1880–1920. Historia Mathematica, Elsevier, 2018, 45 (4), pp.334-353. 10.1016/j.hm.2018.10.006. hal-02265916 HAL Id: hal-02265916 https://hal.archives-ouvertes.fr/hal-02265916 Submitted on 12 Aug 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. -
Arxiv:Math.HO/0507203 V1 11 Jul 2005
SAUNDERS MAC LANE, THE KNIGHT OF MATHEMATICS S. S. KUTATELADZE Abstract. This is a short obituary of Saunders Mac Lane (1909–2005). San Francisco and April 14, 2005 form the terminal place and date of the mar- vellous almost centennial life of the prominent American mathematician Saunders Mac Lane who shared with Samuel Eilenberg (1913–1998) the honor of creation of category theory which ranks among the most brilliant, controversial, ambitious, and heroic mathematical achievements of the 20th century. Category theory, alongside set theory, serves as a universal language of modern mathematics. Categories, functors, and natural transformations are widely used in all areas of mathematics, allowing us to look uniformly and consistently on vari- ous constructions and formulate the general properties of diverse structures. The impact of category theory is irreducible to the narrow frameworks of its great ex- pressive conveniences. This theory has drastically changed our general outlook on the foundations of mathematics and widened the room of free thinking in mathe- matics. Set theory, a great and ingenious creation of Georg Cantor, occupies in the common opinion of the 20th century the place of the sole solid base of modern mathematics. Mathematics becomes sinking into a section of the Cantorian set theory. Most active mathematicians, teachers, and philosophers consider as obvi- ous and undisputable the thesis that mathematics cannot be grounded on anything but set theory. The set-theoretic stance transforms paradoxically into an ironclad -
A History of Galois Fields
A HISTORY OF GALOIS FIELDS * Frédéric B RECHENMACHER Université d’Artois Laboratoire de mathématiques de Lens (EA 2462) & École polytechnique Département humanités et sciences sociales 91128 Palaiseau Cedex, France. ABSTRACT — This paper stresses a specific line of development of the notion of finite field, from Éva- riste Galois’s 1830 “Note sur la théorie des nombres,” and Camille Jordan’s 1870 Traité des substitutions et des équations algébriques, to Leonard Dickson’s 1901 Linear groups with an exposition of the Galois theory. This line of development highlights the key role played by some specific algebraic procedures. These in- trinsically interlaced the indexations provided by Galois’s number-theoretic imaginaries with decom- positions of the analytic representations of linear substitutions. Moreover, these procedures shed light on a key aspect of Galois’s works that had received little attention until now. The methodology of the present paper is based on investigations of intertextual references for identifying some specific collective dimensions of mathematics. We shall take as a starting point a coherent network of texts that were published mostly in France and in the U.S.A. from 1893 to 1907 (the “Galois fields network,” for short). The main shared references in this corpus were some texts published in France over the course of the 19th century, especially by Galois, Hermite, Mathieu, Serret, and Jordan. The issue of the collective dimensions underlying this network is thus especially intriguing. Indeed, the historiography of algebra has often put to the fore some specific approaches developed in Germany, with little attention to works published in France. -
Landau's Problems on Primes
János PINTZ Landau’s problems on primes Tome 21, no 2 (2009), p. 357-404. <http://jtnb.cedram.org/item?id=JTNB_2009__21_2_357_0> © Université Bordeaux 1, 2009, tous droits réservés. L’accès aux articles de la revue « Journal de Théorie des Nom- bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb.cedram. org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Journal de Théorie des Nombres de Bordeaux 21 (2009), 357-404 Landau’s problems on primes par János PINTZ Résumé. Au congrès international de Cambridge en 1912, Lau- dau dressa la liste de quatre problèmes de base sur les nombres pre- miers. Ces problèmes furent caractérisés dans son discours comme “inaccessibles en l’état actuel de la science”. Ces problèmes sont les suivants : (1) Existe-t-il une infinité de nombres premiers de la forme n2 + 1 ? (2) La conjecture (binaire) de Goldbach, que chaque nombre pair supérieur à 2 est somme de deux nombres premiers. (3) La conjecture des nombres premiers jumeaux. (4) Existe-t-il toujours un nombre premier entre deux carrés consécutifs ? Tous ces problèmes sont encore ouverts. Le travail présenté ici est un exposé des résultats partiels aux problèmes (2)–(4), avec une attention particuliere concernant les résultats récents de D. -
The Role of GH Hardy and the London Mathematical Society
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematica 30 (2003) 173–194 www.elsevier.com/locate/hm The rise of British analysis in the early 20th century: the role of G.H. Hardy and the London Mathematical Society Adrian C. Rice a and Robin J. Wilson b,∗ a Department of Mathematics, Randolph-Macon College, Ashland, VA 23005-5505, USA b Department of Pure Mathematics, The Open University, Milton Keynes MK7 6AA, UK Abstract It has often been observed that the early years of the 20th century witnessed a significant and noticeable rise in both the quantity and quality of British analysis. Invariably in these accounts, the name of G.H. Hardy (1877–1947) features most prominently as the driving force behind this development. But how accurate is this interpretation? This paper attempts to reevaluate Hardy’s influence on the British mathematical research community and its analysis during the early 20th century, with particular reference to his relationship with the London Mathematical Society. 2003 Elsevier Inc. All rights reserved. Résumé On a souvent remarqué que les premières années du 20ème siècle ont été témoins d’une augmentation significative et perceptible dans la quantité et aussi la qualité des travaux d’analyse en Grande-Bretagne. Dans ce contexte, le nom de G.H. Hardy (1877–1947) est toujours indiqué comme celui de l’instigateur principal qui était derrière ce développement. Mais, est-ce-que cette interprétation est exacte ? Cet article se propose d’analyser à nouveau l’influence d’Hardy sur la communauté britannique sur la communauté des mathématiciens et des analystes britanniques au début du 20ème siècle, en tenant compte en particulier de son rapport avec la Société Mathématique de Londres. -
Arxiv:Math/0402085V1 [Math.NT] 5 Feb 2004 Eafntosi Omlsfrvolumes
VOLUMES OF SYMMETRIC SPACES VIA LATTICE POINTS HENRI GILLET AND DANIEL R. GRAYSON Introduction In this paper we show how to use elementary methods to prove that the vol- ume of Slk R/ Slk Z is ζ(2)ζ(3) ζ(k)/k; see Corollary 3.16. Using a version of reduction theory presented in this··· paper, we can compute the volumes of certain unbounded regions in Euclidean space by counting lattice points and then appeal to the machinery of Dirichlet series to get estimates of the growth rate of the number of lattice points appearing in the region as the lattice spacing decreases. In section 4 we present a proof of the closely related result that the Tamagawa number of Slk,Q is 1 that is somewhat simpler and more arithmetic than Weil’s in [34]. His proof proceeds by induction on k and appeals to the Poisson summation formula, whereas the proof here brings to the forefront local versions (5) of the formula, one for each prime p, which help to illuminate the appearance of values of zeta functions in formulas for volumes. The volume computation above is known; see, for example, [24], and formula (24) in [27]. The methods used in the computation of the volume of Slk R/ Slk Z in the book [28, Lecture XV] have a different flavor from ours and do not involve counting lattice points. One positive point about the proof there is that it proceeds by induction on k, making clear how the factor ζ(k) enters in at k-th stage. -
A Calendar of Mathematical Dates January
A CALENDAR OF MATHEMATICAL DATES V. Frederick Rickey Department of Mathematical Sciences United States Military Academy West Point, NY 10996-1786 USA Email: fred-rickey @ usma.edu JANUARY 1 January 4713 B.C. This is Julian day 1 and begins at noon Greenwich or Universal Time (U.T.). It provides a convenient way to keep track of the number of days between events. Noon, January 1, 1984, begins Julian Day 2,445,336. For the use of the Chinese remainder theorem in determining this date, see American Journal of Physics, 49(1981), 658{661. 46 B.C. The first day of the first year of the Julian calendar. It remained in effect until October 4, 1582. The previous year, \the last year of confusion," was the longest year on record|it contained 445 days. [Encyclopedia Brittanica, 13th edition, vol. 4, p. 990] 1618 La Salle's expedition reached the present site of Peoria, Illinois, birthplace of the author of this calendar. 1800 Cauchy's father was elected Secretary of the Senate in France. The young Cauchy used a corner of his father's office in Luxembourg Palace for his own desk. LaGrange and Laplace frequently stopped in on business and so took an interest in the boys mathematical talent. One day, in the presence of numerous dignitaries, Lagrange pointed to the young Cauchy and said \You see that little young man? Well! He will supplant all of us in so far as we are mathematicians." [E. T. Bell, Men of Mathematics, p. 274] 1801 Giuseppe Piazzi (1746{1826) discovered the first asteroid, Ceres, but lost it in the sun 41 days later, after only a few observations. -
From the Ergodic Hypothesis in Physics to the Ergodic Axiom in Economics Prepared for the 7
From the Ergodic Hypothesis in Physics to the Ergodic Axiom in Economics Prepared for the 7. Wintertagung des ICAE Linz Mark Kirstein∗ Department of Economics, TU Dresden October 12, 2015 Abstract The direction of the mathematisation of predominant economics is unthinkable without the tacit underlying assumption of ergodicity. Despite its foundational charac- ter, the assumption of ergodicity is largely overlooked while discussing the intellectual 5 history of the discipline. This is the cause why ergodicity is absent from the cur- riculum – although intimately intertwined with the equilibrium concept. Contrasted by such popular assumptions like rational expectations formation, representative agent, efficient markets, perfect competition, etc., that every student is aware of. Nevertheless, ergodicity is more fundamental than all of the mentioned assumptions 10 taken together. Ergodicity is a property of a mathematical system and originated from statistical mechanics, invoked by L. Boltzmann, who literally called it a ‘trick’ to simplify the needed mathematics. Ergodicity is fulfilled, if the time average of a system or process equals its ensemble average. However, if the time average of a system is 15 different from its ensemble average, the system or process is called non-ergodic. The time average is the average of one observed trajectory or realisation of a process (one time series). The ensemble average is the average over every possible state of a system. Non-ergodicity is a necessary property of a mathematical model, if the model is supposed to describe occurences of endogenous novelties and change. The 20 non-ergodic case is the more general, whereas the ergodic case is much easier to handle mathematically. -
What Groups Were: a Study of the Development of the Axiomatics of Group Theory
BULL. AUSTRAL. MATH. SOC. 01A55, 01A60, 20-03, 20A05 VOL. 60 (1999) [285-301] WHAT GROUPS WERE: A STUDY OF THE DEVELOPMENT OF THE AXIOMATICS OF GROUP THEORY PETER M. NEUMANN For my father for his ninetieth birthday: 15 October 1999 This paper is devoted to a historical study of axioms for group theory. It begins with the emergence of groups in the work of Galois and Cauchy, treats two lines of development discernible in the latter half of the nineteenth century, and concludes with a note about some twentieth century ideas. One of those nineteenth century lines involved Cayley, Dyck and Burnside; the other involved Kronecker, Weber (very strongly), Holder and Probenius. INTRODUCTION As is well known, my father has long had an interest in axiomatics, especially the axiomatics of group theory (see, for example, [21, 34, 35]). My purpose in this essay is to trace the origins and development of currently familiar axiom systems for groups. A far more general study of the origins of the concept of abstract group has been undertaken by Wussing (see [42, Chapter 3, Section 4]) who, however, treats the axiomatics rather differently. George Abram Miller has also written on the subject but perhaps a little erratically. Statements like It should perhaps be noted in this connection that an abstract goup is a set of distinct elements which obey the associative law when they are com- bined and is closed with respect to the unique solutions of linear equations in the special form ax = b. This special form appears already in the Rhind Mathematical Papyrus.