Rubel's Problem: from Hayman's List to the Chabauty Method

Total Page:16

File Type:pdf, Size:1020Kb

Rubel's Problem: from Hayman's List to the Chabauty Method i “NLMS_492” — 2020/12/21 — 10:40 — page 24 — #24 i i i 24 FEATURES Rubel’s Problem: from Hayman’s List to the Chabauty Method EDWARD CRANE AND GENE S. KOPP Walter Hayman’s list Research Problems in Function Theory has been updated for its 50th anniversary. Problem 4.27, posed by Lee Rubel, was related to the famous Prouhet-Tarry-Escott problem. Some modern tools of computational number theory can be used to nd explicit counterexamples. Walter Hayman’s list a problem in function theory but rather a problem in number theory. It was posed by the American Research Problems in Function Theory [9] is a analyst Lee Rubel (1927–1995), whose work spanned collection of problems collated by Walter Hayman in dierential equations, approximation theory, and the 1967 and fondly known as Hayman’s list. It gave a big theory of analog computing. Among many other impetus to the development of geometric function surprising results, Rubel showed that there is a single theory, the part of complex analysis concerned with entire function whose derivatives are dense in the the geometric properties of analytic functions. space of entire functions with respect to the topology of locally uniform convergence. Here is problem 4.27, Walter loved to as it appears in [9]: encourage younger mathematicians. He and his wife Margaret 4.27 (Lee Rubel). Let f ¹xº be a real polynomial founded the British of degree n in the real variable x such that Mathematical Olympiad, f ¹xº = 0 has n distinct (real) rational roots. and the mathematics Does there necessarily exist a (real) non-zero genealogy website lists number t such that f ¹xº − t = 0 has n distinct his 20 PhD students (real) rational roots? and 133 mathematical (I can prove this for n = 1,2,3.) descendants. His own research was You might like to try the case n = 3 for yourself. groundbreaking, particul- Walter Hayman. arly his work on the Photo credit: MFO The phrasing suggests Rubel hoped for an armative Bieberbach conjecture answer, extrapolating from the low-degree cases. and in the area of Nevanlinna theory, which relates the growth and covering properties of meromorphic In a straw poll of our local number theorists at the functions. He was awarded the LMS De Morgan medal University of Bristol, all had the opposite intuition to in 1995. Fifty years after the publication of the list, Rubel’s. We’ll give a calculation-free proof below that ≥ Walter Hayman teamed up with Eleanor Lingham the answer is no, at least for even degrees n 8. (the editor of this newsletter) to gather the state What is more interesting is to try to exhibit explicit ≥ of the art on each of the problems. The resulting counterexamples. For each degree n 4 we pose Fiftieth Anniversary Edition [10] was published in 2019. the following challenge: We were sorry to learn that Walter Hayman passed away on January 1st 2020. Problem Rubel¹nº: Exhibit f 2 Q»x¼ of degree n such that f − t has n distinct rational roots if and only if t = 0. Rubel’s problem If f is a polynomial of degree n, t < 0, and both The fourth chapter of Hayman’s list is about f and f − t have n distinct rational roots, then we polynomials. Among the problems there, number can rescale the roots of f and f − t to obtain an 4.27 stands out for the strange reason that it is not ideal solution of the Prouhet-Tarry-Escott problem (see i i i i i “NLMS_492” — 2020/12/21 — 10:40 — page 25 — #25 i i i FEATURES 25 inset). Those are hard to nd, but this doesn’t mean rst case f − t cannot have two distinct real roots, it is easy to prove that any particular polynomial f and in the second it cannot have two distinct 2-adic is a solution of Rubel¹nº. roots. Ruling out repeated roots stops Rubel’s problem from Prouhet–Tarry–Escott problem being trivial. For example f = x ¹x − 1º2 ¹x ¸ 1º2 has a local minimum at 1 and a local maximum at −1, so Given k,n 2 N, nd two distinct sets of f − t has at most three real roots if t < 0. integers A and B, both of size n, such that The constraint that f has n distinct rational roots Õ i Õ i makes the problem a lot more fun. It implies that a = b for i = 1,...,k. for K = R or K = Q , every suciently small a 2A b 2B p perturbation f˜ of f still factors completely over K . A simple appplication of the pigeonhole This is because each root of f can be perturbed to principle shows that solutions exist whenever a nearby root of f˜ in K . This is shown in the p-adic n ¡ k ¹k ¸1º/2. A solution is called ideal when case by Hensel’s lemma. So we cannot hope to solve n = k ¸ 1. In that case we have Rubel¹nº by nding a local obstruction. Ö Ö ¹x − aº − ¹x − bº = c a 2A b 2B for some non-zero constant c. The largest k A non-constructive solution of Rubel’s problem for which an ideal solution is known is 11. This solution was found in 1999 by Nuutti Kuosa, We will answer Rubel’s problem by showing that for Jean-Charles Meyrignac and Chen Shuwen: each n ≥ 4 , there exists a solution to Rubel¹2nº. A = f±35,±47,±94,±121,±146,±148g, Choose rationals 0 < a1 < ··· < an such that the polynomial f = În ¹x −a2º has n−1 distinct critical B = f±22,±61,±86,±127,±140,±151g . i=1 i values. This condition holds generically in An ¹Qº. For this example, we have Case 1: f is a solution of Rubel¹nº. 2 c = 67440294559676054016000 Then F := f ¹x º is a solution of Rubel¹2nº, since it has the 2n distinct roots ±a ,...,±a . However, if = 212.39.53.72.112.132.17.19.23.29.31 . 1 n F −t has 2n distinct rational roots for some nonzero − If you can prove that there is no ideal t, then f t must have n distinct rational roots, solution of Prouhet-Tarry-Escott with k ≥ 12, contrary to the hypothesis. then you will also have proved that every Case 2: f is not a solution of Rubel¹nº. ≥ polynomial of degree n 13 with n distinct Dene G ¹x,wº = ¹f ¹xº − f ¹wºº/¹x − wº. Thinking Rubel¹ º rational roots is a solution of n . See of G as a polynomial in x, its coecients are Borwein [5] for more information about the polynomials in w, so its discriminant h belongs to Prouhet-Tarry-Escott problem. Q»w¼. The roots of h in C are the n − 2 non-critical preimages under f of each of the n − 1 critical values of f . Each of these is a simple root of h. So h has degree d = ¹n − 1º¹n − 2º and has no No local obstructions repeated complex root. In particular, d ≥ 6, and the discriminant of h is non-zero. It is easy to write down an f 2 Q»x¼ of degree n ≥ 3 for which there is no t 2 Q such that f − t If f − t has rational roots q1 < ··· < qn, we have 2n has n distinct roots. For instance, this is true for rational points on the hyperelliptic curve Y 2 = h¹X º, f = x3 ¸ x, because it maps R to R injectively. given by For another example, f = x3 − 2x is an injective mapping from Q to Q. If x < y, but f ¹xº = f ¹yº, then x2 ¸ xy ¸ y 2 = 2, but for x,y 2 Q, not both © Ö ª ­q , ± ¹q − q º® , k = 1,...,n . zero, the valuation of x2 ¸ xy ¸ y 2 at 2 is always even. ­ k i j ® ­ i < j ® These are both examples of local obstructions. In the « i,j <k ¬ i i i i i “NLMS_492” — 2020/12/21 — 10:40 — page 26 — #26 i i i 26 FEATURES Since disc¹hº < 0, this is a smooth ane curve over Q of genus g = d¹d − 2º/2e = ¹n − 1º¹n − 2º/2 − 1. Faltings’ theorem Although its closure in the projective plane is not smooth, it can be embedded in a smooth curve In 1922 Louis Mordell proved the seminal result in projective 3-space, by a map that takes rational that the abelian group of rational points on points to rational points. Faltings’ theorem says that any elliptic curve dened over Q is nitely any smooth projective curve of genus at least 2 has generated. At the end of same paper, he only nitely many rational points (see inset). It follows conjectured a restricted version of what that the curve Y 2 = h¹X º has only nitely many became known as the Mordell conjecture, that rational points. any smooth algebraic curve dened over Q with genus at least 2 has only nitely many We deduce that there are only nitely many rational rational points. − values of t for which f t has n distinct rational The Mordell conjecture was proved by Gerd < ··· < roots. Let t1 tm be the complete sorted list Faltings in 1983, and he won a Fields Medal of these rational values. Since f is not a solution of in 1986 for this work. A few years later, Rubel¹ º ≥ n , m 2. Consider the degree 2n polynomial Paul Vojta gave a very dierent proof using Diophantine approximation and Arakelov F := ¹f − t1º¹f − t2º .
Recommended publications
  • Condenser Capacity and Hyperbolic Perimeter
    CONDENSER CAPACITY AND HYPERBOLIC PERIMETER MOHAMED M. S. NASSER, OONA RAINIO, AND MATTI VUORINEN Abstract. We apply domain functionals to study the conformal capacity of condensers (G; E) where G is a simply connected domain in the complex plane and E is a compact subset of G. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of E. Novel computational algorithms based on implementations of the fast multipole method are combined with analytic techniques. Computational experiments are used throughout to, for instance, demonstrate sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed. 1. Introduction A condenser is a pair (G; E), where G ⊂ Rn is a domain and E is a compact non-empty subset of G. The conformal capacity of this condenser is defined as [14, 17, 20] Z (1.1) cap(G; E) = inf jrujndm; u2A G 1 where A is the class of C0 (G) functions u : G ! [0; 1) with u(x) ≥ 1 for all x 2 E and dm is the n-dimensional Lebesgue measure. The conformal capacity of a condenser is one of the key notions in potential theory of elliptic partial differential equations [20, 29] and it has numerous applications to geometric function theory, both in the plane and in higher dimensions, [10, 14, 15, 17]. The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length, or, perimeter. Constrained extremal problems of this type, where the constraints involve geometric or physical quantities, have been studied in several thousands of papers, see e.g.
    [Show full text]
  • Harmonic and Complex Analysis and Its Applications
    Trends in Mathematics Alexander Vasil’ev Editor Harmonic and Complex Analysis and its Applications Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. For further volumes: http://www.springer.com/series/4961 Harmonic and Complex Analysis and its Applications Alexander Vasil’ev Editor Editor Alexander Vasil’ev Department of Mathematics University of Bergen Bergen Norway ISBN 978-3-319-01805-8 ISBN 978-3-319-01806-5 (eBook) DOI 10.1007/978-3-319-01806-5 Springer Cham Heidelberg New York Dordrecht London Mathematics Subject Classification (2010): 13P15, 17B68, 17B80, 30C35, 30E05, 31A05, 31B05, 42C40, 46E15, 70H06, 76D27, 81R10 c Springer International Publishing Switzerland 2014 This work is subject to copyright.
    [Show full text]
  • Complex Numbers and Colors
    Complex Numbers and Colors For the sixth year, “Complex Beauties” provides you with a look into the wonderful world of complex functions and the life and work of mathematicians who contributed to our understanding of this field. As always, we intend to reach a diverse audience: While most explanations require some mathemati- cal background on the part of the reader, we hope non-mathematicians will find our “phase portraits” exciting and will catch a glimpse of the richness and beauty of complex functions. We would particularly like to thank our guest authors: Jonathan Borwein and Armin Straub wrote on random walks and corresponding moment functions and Jorn¨ Steuding contributed two articles, one on polygamma functions and the second on almost periodic functions. The suggestion to present a Belyi function and the possibility for the numerical calculations came from Donald Marshall; the November title page would not have been possible without Hrothgar’s numerical solution of the Bla- sius equation. The construction of the phase portraits is based on the interpretation of complex numbers z as points in the Gaussian plane. The horizontal coordinate x of the point representing z is called the real part of z (Re z) and the vertical coordinate y of the point representing z is called the imaginary part of z (Im z); we write z = x + iy. Alternatively, the point representing z can also be given by its distance from the origin (jzj, the modulus of z) and an angle (arg z, the argument of z). The phase portrait of a complex function f (appearing in the picture on the left) arises when all points z of the domain of f are colored according to the argument (or “phase”) of the value w = f (z).
    [Show full text]
  • Nevanlinna Theory of the Askey-Wilson Divided Difference
    NEVANLINNA THEORY OF THE ASKEY-WILSON DIVIDED DIFFERENCE OPERATOR YIK-MAN CHIANG AND SHAO-JI FENG Abstract. This paper establishes a version of Nevanlinna theory based on Askey-Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane C. A second main theorem that we have derived allows us to define an Askey-Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole- scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey-Wilosn type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey-Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey-Wilson. This paper concludes with an application to difference equations generalising the Askey-Wilson second-order divided difference equation. Contents 1. Introduction 2 2. Askey-Wilson operator and Nevanlinna characteristic 6 3. Askey-Wilson type Nevanlinna theory – Part I: Preliminaries 8 4. Logarithmic difference estimates and proofs of Theorem 3.2 and 3.1 10 5. Askey-Wilson type counting functions and proof of Theorem 3.3 22 6. ProofoftheSecondMaintheorem3.5 25 7. Askey-Wilson type Second Main theorem – Part II: Truncations 27 8. Askey-Wilson-Type Nevanlinna Defect Relation 29 9. Askey-Wilson type Nevanlinna deficient values 31 arXiv:1502.02238v4 [math.CV] 3 Feb 2018 10. The Askey-Wilson kernel and theta functions 33 11.
    [Show full text]
  • Multiplicities in Hayman's Alternative
    J. Aust. Math. Soc. 78 (2005), 37-57 MULTIPLICITIES IN HAYMAN'S ALTERNATIVE WALTER BERGWEILER and J. K. LANGLEY (Received 3 September 2002; revised 28 August 2003) Communicated by P. C. Fenton Abstract In 1959 Hayman proved an inequality from which it follows that if/ is transcendental and meromorphic in the plane then either/ takes every finite complex value infinitely often or each derivative /(k), k > 1, takes every finite non-zero value infinitely often. We investigate the extent to which these values may be ramified, and we establish a generalization of Hayman's inequality in which multiplicities are not taken into account. 2000 Mathematics subject classification: primary 3OD35. 1. Introduction Our starting point is the following result of Hayman [8, 9]. THEOREM 1.1. Let k e H and let a, b e C, b ^ 0. Let f be nonconstant and meromorphic in the plane. Then at least one of f — a and f(Ar) — b has at least one zero, and infinitely many iff is transcendental. Hayman's proof of Theorem 1.1 is based on Nevanlinna theory, the result deduced from the following inequality. Here the notation is that of [9], with S(r,f) denoting any term which is o(T(r,f)) as r -> oo, possibly outside a set of finite Lebesgue measure. THEOREM 1.2 ([8, 9]). Let ieN and let f be transcendental and meromorphic in the plane. Then, as r -» oo, (1) r(rJ) © 2005 Australian Mathematical Society 1446-7887/05 $A2.00 + 0.00 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • Math Times Is Published Twice a Year by Teaching, and Their Leadership in the Profession
    Department of Mathematics Spring/Summer 2014 MathDepartment of Mathematics Times — Spring 2019 Architects hired to create designs for $100 million plan to modernize and expand learning spaces The University of Illinois has hired a Chicago design firm to “We are very pleased that the trustees approved CannonDesign begin planning the $100 million Altgeld Hall and Illini Hall for the Altgeld Hall and Illini Hall project,” said Derek Fultz, project to modernize learning spaces and increase capacity in director of facilities for the College of LAS who also served data science and other mathematical sciences. on the selection committee that recommended the firm. “We During their March meeting, the Board of Trustees approved fully expect the buildings to display the same, high levels of CannonDesign to conceptualize and create schematic designs creativity and practicality that CannonDesign has provided in for the project, which will include the construction of a new the past.” building on the site of Illini Campus is still raising money Hall, and the renovation for the Altgeld Hall and Illini of Altgeld Hall. Campus Hall project. The new building officials expect the new will be constructed on the building to be constructed site of Illini Hall, which is by 2022, and for the scheduled to be razed at the renovation of Altgeld to be corner of South Wright and complete by 2024. East John streets beginning Once the schematic in fall 2020. It will be replaced designs are complete— with a larger, world-class which will provide facility for learning and basically an overall discovery, including a data visualization of the science center that does not project—they will be currently exist on campus.
    [Show full text]
  • Walter Kurt Hayman (6.1.1926 – 1.1.2020)
    Walter Kurt Hayman (6.1.1926 – 1.1.2020) Am Neujahrstag 2020 verstarb kurz vor seinem 94. Geburtstag Walter Kurt Hayman. Er war seit 1982 korrespondierendes Mitglied der Bayerischen Akademie der Wissenschaften. Seine Eltern waren der Jura-Professor Franz Haymann und Ruth Therese Haymann, Tochter des bedeutenden Mathematikers Kurt Hensel. Beide Eltern hatten judische¨ Vorfahren; der Vater verlor deshalb 1935 seine Professur in Koln.¨ Ein Jahr zuvor war ein Vetter der Mutter (Kurt Hahn, einer der Grunder¨ des Internats Schloss Salem) nach Schottland emigriert, wo er die Gordonstoun School grundete.¨ Walter wurde von seinen Eltern 1938 alleine nach Schottland geschickt, um in Gordonstoun seine Schulbildung fortzusetzen. Es gelang ihm, mithilfe seiner dortigen Forderer¨ seinen Eltern die Emigration nach Großbritannien zu ermoglichen,¨ sodass sie der Verfolgung durch das NS-Regime entgingen. Das Studium der Mathematik in Cambridge schloss er in Rekordzeit mit seiner Promotion bei Mary Cartwright1 ab und nahm 1947 eine Stelle als Lecturer an der University of Newcastle an. Noch im selben Jahr wurde er Fellow des St. John’s College in Cambridge und spater¨ Lecturer in Exeter. Das hohere¨ Gehalt ermoglichte¨ die Heirat mit Margaret Riley. Er blieb bis 1956 in Exeter (seit 1953 als Reader) und wurde dann als erster Professor fur¨ Reine Mathematik ans Imperial College London berufen, wo er bis 1985 tatig¨ war. Ebenfalls 1956 wurde er zum damals jungsten¨ Mitglied der Royal Society gewahlt.¨ Nach seiner Emeritierung am Imperial College nahm er bis 1993 eine Teilzeit-Professur an der University of York wahr und ging dann 1995 als Senior Research Fellow ans Imperial College zuruck.¨ Fur¨ seine Forschungsarbeiten in der Funktionentheorie (das Studium von komplex differen- zierbaren Funktionen einer komplexen Variablen) hat Walter Hayman neben anderen Preisen sowohl den Junior (1955) als auch den Senior Berwick Prize (1964) und schließlich 1995 die de-Morgan-Medaille der London Mathematical Society erhalten.
    [Show full text]
  • 30388 OID RS Annual Review
    Review of the Year 2005/06 >> President’s foreword In the period covered by this review*, the Royal Society has continued and extended its activities over a wide front. There has, in particular, been an expansion in our international contacts and our engagement with global scientific issues. The joint statements on climate change and science in Africa, published in June 2005 by the science academies of the G8 nations, made a significant impact on the discussion before and at the Gleneagles summit. Following the success of these unprecedented statements, both of which were initiated by the Society, representatives of the science academies met at our premises in September 2005 to discuss how they might provide further independent advice to the governments of the G8. A key outcome of the meeting was an We have devoted increasing effort to nurturing agreement to prepare joint statements on the development of science academies overseas, energy security and infectious diseases ahead particularly in sub-Saharan Africa, and are of the St Petersburg summit in July 2006. building initiatives with academies in African The production of these statements, led by the countries through the Network of African Russian science academy, was a further Science Academies (NASAC). This is indicative illustration of the value of science academies of the long-term commitment we have made to working together to tackle issues of help African nations build their capacity in international importance. science, technology, engineering and medicine, particularly in universities and colleges. In 2004, the Society published, jointly with the Royal Academy of Engineering, a widely Much of the progress we have has made in acclaimed report on the potential health, recent years on the international stage has been environmental and social impacts of achieved through the tireless work of Professor nanotechnologies.
    [Show full text]
  • NEWSLETTER Issue: 492 - January 2021
    i “NLMS_492” — 2020/12/21 — 10:40 — page 1 — #1 i i i NEWSLETTER Issue: 492 - January 2021 RUBEL’S MATHEMATICS FOUR PROBLEM AND DECADES INDEPENDENCE ON i i i i i “NLMS_492” — 2020/12/21 — 10:40 — page 2 — #2 i i i EDITOR-IN-CHIEF COPYRIGHT NOTICE Eleanor Lingham (Sheeld Hallam University) News items and notices in the Newsletter may [email protected] be freely used elsewhere unless otherwise stated, although attribution is requested when EDITORIAL BOARD reproducing whole articles. Contributions to the Newsletter are made under a non-exclusive June Barrow-Green (Open University) licence; please contact the author or David Chillingworth (University of Southampton) photographer for the rights to reproduce. Jessica Enright (University of Glasgow) The LMS cannot accept responsibility for the Jonathan Fraser (University of St Andrews) accuracy of information in the Newsletter. Views Jelena Grbic´ (University of Southampton) expressed do not necessarily represent the Cathy Hobbs (UWE) views or policy of the Editorial Team or London Christopher Hollings (Oxford) Mathematical Society. Robb McDonald (University College London) Adam Johansen (University of Warwick) Susan Oakes (London Mathematical Society) ISSN: 2516-3841 (Print) Andrew Wade (Durham University) ISSN: 2516-385X (Online) Mike Whittaker (University of Glasgow) DOI: 10.1112/NLMS Andrew Wilson (University of Glasgow) Early Career Content Editor: Jelena Grbic´ NEWSLETTER WEBSITE News Editor: Susan Oakes Reviews Editor: Christopher Hollings The Newsletter is freely available electronically at lms.ac.uk/publications/lms-newsletter. CORRESPONDENTS AND STAFF LMS/EMS Correspondent: David Chillingworth MEMBERSHIP Policy Digest: John Johnston Joining the LMS is a straightforward process. For Production: Katherine Wright membership details see lms.ac.uk/membership.
    [Show full text]
  • Visiting Mathematicians Jon Barwise, in Setting the Tone for His New Column, Has Incorporated Three Articles Into This Month's Offering
    OTICES OF THE AMERICAN MATHEMATICAL SOCIETY The Growth of the American Mathematical Society page 781 Everett Pitcher ~~ Centennial Celebration (August 8-12) page 831 JULY/AUGUST 1988, VOLUME 35, NUMBER 6 Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences This calendar lists all meetings which have been approved prior to Mathematical Society in the issue corresponding to that of the Notices the date this issue of Notices was sent to the press. The summer which contains the program of the meeting. Abstracts should be sub­ and annual meetings are joint meetings of the Mathematical Associ­ mitted on special forms which are available in many departments of ation of America and the American Mathematical Society. The meet­ mathematics and from the headquarters office of the Society. Ab­ ing dates which fall rather far in the future are subject to change; this stracts of papers to be presented at the meeting must be received is particularly true of meetings to which no numbers have been as­ at the headquarters of the Society in Providence, Rhode Island, on signed. Programs of the meetings will appear in the issues indicated or before the deadline given below for the meeting. Note that the below. First and supplementary announcements of the meetings will deadline for abstracts for consideration for presentation at special have appeared in earlier issues. sessions is usually three weeks earlier than that specified below. For Abstracts of papers presented at a meeting of the Society are pub­ additional information, consult the meeting announcements and the lished in the journal Abstracts of papers presented to the American list of organizers of special sessions.
    [Show full text]
  • On the Kronecker Nachlass 423
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector HISTORIA MATHEMATICA 5 (1978), 419-426 ON THE KRONECKERNACHLASS BY HAROLD M , EDWARDS t COURANT INST, OF MATH, SCI,, NYU SUMMARIES This article is an account of my unsuccessful efforts to find the Nachlass of Leopold Kronecker (1823-1891), and of my reasons for believing that these papers survived intact until 1945 but then were accidentally destroyed. In the same accident, I believe, the papers of Kurt Hensel (1861-1941) and many other valuable documents were lost. Cet article est le r&it de mes efforts infructueux pour trouver les Nachlass, de Leopold Kronecker (1823-1891). Nous y elaborons aussi les raisons qui nous font croire que ces papiers furent conserves intacts jusqu'en 1945, alors qu'ils furent accidentellement detruits. Le m&me accident serait, a notre avis, la cause de la perte des papiers de Kurt Hensel (1861-1941) et de beaucoup d'autres documents de valeur. Initially my interest in Kronecker's scientific manuscripts or Nachlass was stimulated by my quest for Kummer's papers [Edwards 1975, X9-2361. In particular, I hoped that it might be possible to find the originals of the letters from Kummer to Kronecker which Hensel excerpted in his Kummer memorial volume [Hensel 19101. This led me naturally to Hensel and to Hensells Lieblingsschiiler Helmut Hasse (1898- ). In the meantime, I had taken note of Pierre Dugac's remark in this journal [Historia Mathematics 1976, S-191 about the lack of any Kronecker Nachlass, and this stimulated my interest in finding any of Kronecker's papers, not merely Kummer's letters to him.
    [Show full text]
  • EUROPEAN WOMEN in MATHEMATICS NEWSLETTER NUMBER 7 March 2000
    EUROPEAN WOMEN IN MATHEMATICS NEWSLETTER NUMBER 7 March 2000 Editors: Nadia Larsen Department of Mathematics, University of Copenhagen DK-2100 Copenhagen Ø, Denmark e-mail: [email protected] Maren Riemenschneider Arbeitsgruppe 2 – Geometrie und Algebra Fachbereich Mathematik Technische Universit¨at Darmstadt D-64289 Darmstadt, Germany e-mail: [email protected] This is the seventh EWM Newsletter, mainly to be distributed via the EWM e-mail network. Contents: 1 Editorial 2 Information and useful e-mail addresses 3 Report on the 9th General Meeting of EWM 4 Reports on other meetings held in 1999 5 Announcements of forthcoming meetings 6 Other News 7 Questionnaires 1. EDITORIAL Irene Sciriha, convenor Mentoring in 2000 Means Future Women Mathematicians With apologies to US Awise The year 2000 has been designated as the World Mathematical Year, a tribute to the importance of our subject as the backbone of a true scientific approach. It is also the 150th anniversary of the birth of Sonia Kovalevskaya, a pioneer of women mathematicians. The situation for women mathematicians has improved considerably and yet in most civilized countries the opportunities for women are too restricted. To alleviate these difficulties for members of EWM who work in isolation or who are embarking on a mathematical career, we are setting up a mentor ring. Members are urged to fill in the questionnaire that is included in this issue. It is meant to provide us with information to set up a data base that should serve as a talent bank which we can access when a member needs mathematical help.
    [Show full text]