What Groups Were: a Study of the Development of the Axiomatics of Group Theory
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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]. -
Appendix a Historical Notes
Appendix A Historical notes A.I Determining all permutation groups of a given degree For this section only, we adopt the following conventions. 1. The early researchers in this area adopted the convention that the degree of a permutation group is the number of points in the set on which it acts minus the number of orbits of length 1. For example, the copy of S3 in S4 (a point stabiliser) was counted as a group of degree 3, not 4. 2. When we say. 'all' permutation groups, we mean up to permutational isomorphism, with the definition of degree as given above. 3. We use the word 'group' to mean 'permutation group'. 4. Whenever a range of degrees is given, it is an inclusive one. 5. We make no claims on the accuracy of others' results unless explicitly stated. How ever, we adopt four benchmarks with which to compare all other results: (a) for all groups up to degree 7 we take the enumeration produced by the CAYLEY function lattice; (b) for the imprimitive groups up to degree 11 we take the list of Butler and McKay (1983); (c) for the imprimitive groups of degree 12 we take the list of Royle (1987); (d) for the primitive groups up to degree 50 we take the list of Sims, which first appeared in full in version 3.5 of CAYLEY, released in 1987. Tables containing these figures appear at the end of the section. 122 Appendix A. Historical notes The first work in this area is probably due to Ruffini (1799), who gives the possible orders of the groups of degree 5. -
Regular Permutation Groups and Cayley Graphs Cheryl E Praeger University of Western Australia
Regular permutation groups and Cayley graphs Cheryl E Praeger University of Western Australia 1 What are Cayley graphs? Group : G with generating set S = s, t, u, . 1 { } Group elements: ‘words in S’ stu− s etc vertices: group elements Cayley graph: Cay(G, S) edges: multiplication from S 2 What are Cayley graphs? If S inverse closed: s S s 1 S ∈ ⇐⇒ − ∈ Cayley graph Cay(G, S): may use undirected edges 3 Some reasonable questions Where: do they arise in mathematics today? Where: did they originate? What: kinds of groups G give interesting Cayley graphs Cay(G, S)? Which graphs: arise as Cayley graphs? Does it matter: what S we choose? Are: Cayley graphs important and why? And: what about regular permutation groups? Let’s see how I go with answers! 4 In Topology: Embedding maps in surfaces Thanks to Ethan Hein: flickr.com 5 Computer networks; experimental layouts (statistics) Thanks to Ethan Hein: flickr.com and Jason Twamley 6 Random walks on Cayley graphs Applications: from percolation theory to group computation How large n: g ‘approximately random’ in G? Independence? Method: for random selection in groups – underpins randomised algorithms for group computation (Babai, 1991) 7 Fundamental importance for group actions in combinatorics and geometry Iwilldescribe: Regular permutation groups Origins of Cayley graphs Links with group theory Some recent work and open problems on primitive Cayley graphs 8 Permutation groups Permutation : of set Ω, bijection g :Ω Ω Symmetric group of all permutations→ of Ω group Sym (Ω): under composition, for -
Regular Permutation Groups and Cayley Graphs Cheryl E Praeger University of Western Australia
Regular permutation groups and Cayley graphs Cheryl E Praeger University of Western Australia 1 Permutation groups Permutation : of set Ω, bijection g :Ω ! Ω Symmetric group of all permutations of Ω group Sym (Ω): under composition, for example g = (1; 2) followed by h = (2; 3) yields gh = (1; 3; 2) g = (1; 2; 3) has inverse g−1 = (3; 2; 1) = (1; 3; 2) Permutation G ≤ Sym (Ω), that is, subset group on Ω: closed under inverses and products (compositions) Example: G = h(0; 1; 2; 3; 4)i < Sym (Ω) on Ω = f0; 1; 2; 3; 4g 2 Interesting permutation groups occur in: Graph Theory: Automorphism groups (edge-preserving perm's) Geometry: Collineations (line-preserving permutations) Number Theory and Cryptography: Galois groups, elliptic curves Differential equations: Measure symmetry - affects nature of so- lutions Many applications: basic measure of symmetry 3 Regular permutation groups Permutation group: G ≤ Sym (Ω) G transitive: all points of Ω equivalent under elements of G G regular: `smallest possible transitive' that is only the identity element of G fixes a point Example: G = h(0; 1; 2; 3; 4)i on Ω = f0; 1; 2; 3; 4g Alternative view: G = Z5 on Ω = f0; 1; 2; 3; 4g by addition 0 4 1 3 2 4 View of regular permutation groups Take any: group G, set Ω := G Define action: ρg : x ! xg for g 2 G; x 2 Ω(ρg is bijection) Form permutation group: GR = fρgjg 2 Gg ≤ Sym (Ω) ∼ GR = G and GR is regular 5 Visualise regular permutation groups as graphs Given generating set S: G = hSi with s 2 S () s−1 2 S Define graph: vertex set Ω = G, edges fg; sgg for g 2 G; s 2 S Example: G = Z5, S = f1; 4g, obtain Γ = C5, Aut (Γ) = D10. -
View This Volume's Front and Back Matter
Titles in This Series Volume 8 Kare n Hunger Parshall and David £. Rowe The emergenc e o f th e America n mathematica l researc h community , 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moore 1994 7 Hen k J. M. Bos Lectures in the history of mathematic s 1993 6 Smilk a Zdravkovska and Peter L. Duren, Editors Golden years of Moscow mathematic s 1993 5 Georg e W. Mackey The scop e an d histor y o f commutativ e an d noncommutativ e harmoni c analysis 1992 4 Charle s W. McArthur Operations analysis in the U.S. Army Eighth Air Force in World War II 1990 3 Pete r L. Duren, editor, et al. A century of mathematics in America, part III 1989 2 Pete r L. Duren, editor, et al. A century of mathematics in America, part II 1989 1 Pete r L. Duren, editor, et al. A century of mathematics in America, part I 1988 This page intentionally left blank https://doi.org/10.1090/hmath/008 History of Mathematics Volume 8 The Emergence o f the American Mathematical Research Community, 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moor e Karen Hunger Parshall David E. Rowe American Mathematical Societ y London Mathematical Societ y 1991 Mathematics Subject Classification. Primary 01A55 , 01A72, 01A73; Secondary 01A60 , 01A74, 01A80. Photographs o n th e cove r ar e (clockwis e fro m right ) th e Gottinge n Mathematisch e Ges - selschafft, Feli x Klein, J. J. Sylvester, and E. H. Moore. -
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. -
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. -
Making Higher Mathematics Accessible: G.A
Making Higher Mathematics Accessible: G.A. Miller’s Project for a Mathematical Dictionary Making Higher Mathematics Accessible: G.A. Miller’s Project for a Mathematical Dictionary Laura E. Turner Department of Mathematics Monmouth University CIRMATH-Americas University of Virginia, 27-30 May 2018 Making Higher Mathematics Accessible: G.A. Miller’s Project for a Mathematical Dictionary American mathematicians have good reason to protest against such a butchery of their subject in a popular work of reference. Mathematical Definitions in the New Standard Dictionary, 1913 George Abram Miller (1863–1951) I have heard the question raised as to where or when our American LaGranges are to appear. A Decade of American Mathematics, 1921 Oliver Dimon Kellogg (1878–1932) [. ] mathematicians are still in the shadow of the Tower of Babel [. ] A History of Mathematical Notations: Volume II, 1929 Florian Cajori (1859–1930) Making Higher Mathematics Accessible: G.A. Miller’s Project for a Mathematical Dictionary Introduction Conference Goals: I To provide a venue in which European CIRMATH members can intensively interact with historians of mathematics from the Americas working on related historical questions. I To provide researchers the opportunity to explore the significant archival and library holdings at both the University of Virginia and the nearby Library of Congress. I To provide opportunities for American and European researchers at earlier stages in their careers to develop significant and enduring research collaborations with established scholars in the history of mathematics. I To trace the impact of journals on the circulation of mathematics within the Americas and between the Americas and Europe from 1850 to 1950. -
Jarník's Notes of the Lecture Course Allgemeine Idealtheorie by B. L
Jarník’s Notes of the Lecture Course Allgemeine Idealtheorie by B. L. van der Waerden (Göttingen 1927/1928) Monographs and Textbooks in Algebra at the Turn of the 20th Century In: Jindřich Bečvář (author); Martina Bečvářová (author): Jarník’s Notes of the Lecture Course Allgemeine Idealtheorie by B. L. van der Waerden (Göttingen 1927/1928). (English). Praha: Matfyzpress, 2020. pp. 61–110. Persistent URL: http://dml.cz/dmlcz/404381 Terms of use: Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz 61 MONOGRAPHS AND TEXTBOOKS IN ALGEBRA at the turn of the 20th century 1 Introduction Algebra as a mathematical discipline was initially concerned with equation solving. It originated in ancient Egypt and Mesopotamia four thousand years ago, and somewhat later also in ancient China and India. At that time, equations in the present sense, i.e., formal expressions based on a certain, perhaps very primitive notation, did not exist yet. The ancient arithmeticians were able to solve word problems leading to equations or their systems by means of meticulously memorized procedures, which can be nowadays aptly designated as algorithms. They have successfully tackled a number of problems, which often correspond to present-day problems of school mathematics, sometimes being much more difficult. Their methods of calculation largely correspond to our procedures used for solving equations or their systems. -
Regular Permutation Groups and Cayley Graphs Cheryl E Praeger University of Western Australia
Regular permutation groups and Cayley graphs Cheryl E Praeger University of Western Australia 1 What are Cayley graphs? Group : G with generating set S = fs; t; u; : : : g Group elements: `words in S' stu−1s etc vertices: group elements Cayley graph: Cay(G; S) edges: multiplication from S 2 What are Cayley graphs? If S inverse closed: s 2 S () s−1 2 S Cayley graph Cay(G; S): may use undirected edges 3 Some reasonable questions Where: do they arise in mathematics today? Where: did they originate? What: kinds of groups G give interesting Cayley graphs Cay(G; S)? Which graphs: arise as Cayley graphs? Does it matter: what S we choose? Are: Cayley graphs important and why? And: what about regular permutation groups? Let's see how I go with answers! 4 In Topology: Embedding maps in surfaces Thanks to Ethan Hein: flickr.com 5 Computer networks; experimental layouts (statistics) Thanks to Ethan Hein: flickr.com and Jason Twamley 6 Random walks on Cayley graphs Applications: from percolation theory to group computation How large n: g `approximately random' in G? Independence? Method: for random selection in groups { underpins randomised algorithms for group computation (Babai, 1991) 7 Fundamental importance for group actions in combinatorics and geometry I will describe: Regular permutation groups Origins of Cayley graphs Links with group theory Some recent work and open problems on primitive Cayley graphs 8 Permutation groups Permutation : of set Ω, bijection g :Ω ! Ω Symmetric group of all permutations of Ω group Sym (Ω): under composition, for -
A Bibliography of Collected Works and Correspondence of Mathematicians Steven W
A Bibliography of Collected Works and Correspondence of Mathematicians Steven W. Rockey Mathematics Librarian Cornell University [email protected] December 14, 2017 This bibliography was developed over many years of work as the Mathemat- ics Librarian at Cornell University. In the early 1970’s I became interested in enhancing our library’s holdings of collected works and endeavored to comprehensively acquire all the collected works of mathematicians that I could identify. To do this I built up a file of all the titles we held and another file for desiderata, many of which were out of print. In 1991 the merged files became a printed bibliography that was distributed for free. The scope of the bibliography includes the collected works and correspon- dence of mathematicians that were published as monographs and made available for sale. It does not include binder’s collections, reprint collec- tions or manuscript collections that were put together by individuals, de- partments or libraries. In the beginning I tried to include all editions, but the distinction between editions, printings, reprints, translations and now e-books was never easy or clear. In this latest version I have supplied the OCLC identification number, which is used by libraries around the world, since OCLC does a better job connecting the user with the various ver- sion than I possibly could. In some cases I have included a title that has an author’s mathematical works but have not included another title that may have their complete works. Conversely, if I believed that a complete works title contained significant mathematics among other subjects, Ihave included it.