DOCSLIB.ORG

Pi Mu Epsilon Journal the Official Publication of the Honorary Mathematical Fraternity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Pi Mu Epsilon Journal the Official Publication of the Honorary Mathematical Fraternity PI MU EPSILON JOURNAL THE OFFICIAL PUBLICATION OF THE HONORARY MATHEMATICAL FRATERNITY VOLUME 3 NUMBER 3 CONTENTS Plane Geometry Without Postulates - Leonard M. Blumenthal 101 An Appreciation of Giuseppe Peano - Hubert C. Kennedy . 107 On The Division of One Polynomial by Another - Michael C. Mackey . 114 Problem Department . 118 Book Reviews - . 122 Richard Kao, Franz E. Hohn, W. J. Poppelbaum, F. M. Woodworth, M. K. Fort, Jr., Gian-Carlo Rota, A. S. Householder, Murray S. Klamkin, Jane Robertson, Paul T. Bateman, W. A. Vezeau, Horace W. Norton Books Received for Review. 135 Operations Unlimited . 136 News and Notices . 146 Department Devoted to Chapter Activities . 153 , Initiates.. 160 FALL 19601 f .. Copyright 1960 by Pi Mu Epsilon Fraternity, Inc. PLANE GEOMETRY WITHOUT POSTULATESa LEONARD M. BLUMENTHAL Missouri Alpha, 1937 PI MU EPSILON JOURNAL Introduction. In his famous Elements, Euclid made a distinc- THE OFFICIAL PUBLICATIOJN tion, usually neglected today, between certain assumptions that OFTHEHONORARY MATHEMATICALFRATERNITY he called axioms, and others that he referred to as postulates. The former characterized those basic statements that expressed Francis Regan, Editor common notions (e.g., If equals are added to equals, the sums are equal) while the latter term was used for those assumptions that are peculiar to geometry (e.g., To describe a circle with any ASSOCIATE EDITORS center and radius). - The title of this paper is to be understood in the light of that Josephine Chanler Franz E. Hohn ancient discrimination. The basis for euclidean plane geometry Mary Cummings H. T. Kames presented here contains no postulates whatever (and consequent- M. S. Klarnkin ly is free of primitive or undefined terms), but it is not devoid of common notions. The sole such notion that will be used is that John J. Andrews, Business Manager of real number. This notion occurs in nearly all systems of postu- GENERAL OFFICERS OF THE FRATERNITY lates for geometry, usually without attention being called to its status as an axiom. Director General: J. S. Frame, Michigan State University It was recently observed that "according to modern standards Vice-Director General: R. H. Bing, University of Wisconsin of rigor, each branch of pure mathematics must be founded in Secretary-Treasurer General: R. V. Andree, University of Oklahoma one of two ways: (1) either its basic concepts must be defined in terms of the concepts of some prior branch of mathematics (in Councilors General: which case its theorems are deduced from those of the prior Angus E. Taylor, University of California, Los Angeles branch of mathematics with the aid of these definitions) or else Ivan Niven, University of Oregon (2) its basic concepts are taken as undefined and its theorems James C. Eaves, University of Kentucky are deduced from a set of postulates involving these undefined Marion K. Fort, Jr., University of Georgia term^."^ This foundation of plane geometry utilizes the first way; its basic concepts are defined in terms of that prior branch Chapter reports, books for review, problems for solution and solutions of mathematics - the real number system. to problems, and news items should be mailed directly to the special editors found in this issue under the various sections. Editorial cor- respondence, including manuscripts should be mailed to THE EDITOR three positive real numbers (the reason for selecting this nota- OF THE PI MU EPSILON JOURNAL, Department of Mathematics, St. tion for the three numbers will be made clear later) such that the Louis University, 221 North Grand Blvd., St. Louis 3, Mo. determinant PI MU EPSILON JOURNAL is published semi-annually at St. Louis University. SUBSCRIPTION PRICE: To Individual Members, $1.50 for 2 years; to Non-Members and Libraries, $2.00 for 2 years. Subscriptions, orders for back numbers and correspondence concerning subscriptions and advertising should be addressed to the PI MU EPSILON JOUR- NAL, Department of Mathematics, St. Louis University, 221 North is negative. There are many such triples of numbers; indeed, any Grand Blvd., St. Louis 3, Mo. three positive numbers, each one of which is less than the sum of the other two, may be selected. veceived by the editors June 23, 1960 seefootnotes. 102 PI MU EPSILON JOURNAL PLANE GEOMETRY WITHOUT POSTULATES Definition. A point is any ordered triple of non-negative real numbers (rpl,rp2,rp3) such that 0 1 1 1 1 D(P~,P~~P~^)= 1 0 pip; P^P~^4 p1r2 1 p2p; 0 p2e; p2r2 1 pap; p3pJ 0 par2 1 rp; rp; rp; 0 where rp, = pir, (i = 1,2,3). If r denotes the point (rp1,rp2,rp3) and s the point (spvsp2,sp3), we define r = s if and only if p,r = pis, (i = 1,2,3). This definition must be justified by showing that the equation has a unique non-negative real root. Let P denote the set of all points. Justification of distanc e definition. Since D(p ,p2,p3,p) = 0 it follows (as in the proof of Theorem 1) that D(pl,p2,p) $ 0, - Theorem 1. lfr is the point (0,rp2rp3), then rp2 = pip2 D(p ^p3,p) 5 0, and D(p2,p3,p) < 0. Now the equality sign can- and re= PIP. - - not hold in all three relations for if it did, then Proof. The determinant D(pl,p2,r) may be written in factored D(p ,p) = D(p2 ,p) = D(p3,p) = 0, and expansions of these form D(pl1p2,r) =-(P~P~+~P~+~~~)(P~P~+~Pz-~~~)(PIPZ-~P~+~PI) vanishing bordered third-order determinants yields ppl = pp2 = (-p lp2 +re2+rp1), and since rp = 0, pp3 = 0. But this is impossible, for then the determinant D(P,,P~,~)= (P~P~+~~~)~(P~P~-^)~S 0. D(p llp2,p3.p)reduces to -2.plpJ.p2p~.plp~, which does not But since D(pl,p2,p3,r) = 0, it follows that vanish. Hence we may assume D(pl,p2,p)< 0. D(P~,P~,P~).D(P~,P~,~)- crp; :I2 = 01 where crpjldenotes the minor of rpi in D(plp2,p3,r).3 This equality, together with D(p ,p2,p3) 4 0, implies that D(pl,p2,r) 2 0, and consequently D(P ,p2 ,r) = 0. Hence rp2 = pip2, and in a similar manner it is shown that 'P3 = Pip3* Denote the point (Ol~lp21~l~3)by pi, the point (P~P,~O,P~P~) by p2, and the point (p3pl,p3p2,0) by p3.(It is easily shown that Consider the function (O,P~P~,P1~3)1 etc- are points). 2. Distances. We are now in position to define the important notion - distance of two points. Definition. If P = (ppVpp2.pp3) and q = (qq vqq2,qq3) are two points of R, their distance is the unique non-negative real root of the equation D(pl,p2,p3,p,q;x) = 0, where 104 PI MU EPSILON JOURNAL PLANE GEOMETRY WITHOUT POSTULATES Its graph is a parabola whose axis is perpendicular to the Proof. Since P is a metric space, the plane contains three x-axis, and which opens upward. It crosses the x-axis at the points Pi,P$,Pi such that Pip$' = P1P2a Pfi = P2P3, Pip; = points x = (pp1 - qp1l2 and x = bpi + qp1l2. If pq2 be sub- pip3. We symbolize this by writing stituted for x in the function, the corresponding functional value Pl1P2,P3 -à Pl1P;'P; 1 d y = D(pl,p,q) 2 0, and it follows that read "p1,p2,p3 are congruent to pi,p^p;". L& p~ denote any point of E2, and consider the three non- 0 2 (pip - 5 pq2 5 (pip + pid2. Hence pq2 2 0. negative numbers pp = p*pi, pp2 = pSp;, pp3 = plpi, where plp; denotes the distance in the plane of the points p*and pi, (i = 1,2,3). Clearly D(P,,P~,P~,P)= D~~,P;,P;,P~, and since (as is well-known) the D-determinant vanishes for each quadruple of points of the plane, then D(pllp2,p3,p) = 0 and so p = (pp,,pp2,pp3) is a point of P. Hence, with each point pwf E there is associated a unique point of P. It is easily seen that each point q of P is the associate o exactly one point q*of E2. For since p ,p2,p ,q are a metri quadruple and D(pl,p2,p3,q) = 0, it follows t%at the plane 1 IP; qp; qpj x2 I contains four points contains four points Fl,T2,y3,qcongruent to that quadruple; This equation obviously has at most one non-negativ that is, root, and the above argument shows that x = pq > 0 is such a root. p1,p2,p3,q S PllP21P31~-4 Remarks. The distance of points p = (0,plp2,plp3) and p2 Then TllT2,& à PllP21~3à PisPisPi , and there is a = (P~P~,OÈP~P~is pip2; of p2 and p3 = (P~P~,P~P~,O)is p2p3; congruence of the plane with itself that carries -pll~21&into and of p and p3 is p ,p3. This explains why the notation pi,p;,p; , respectively. This congruence is unique (for since pip2, p2p3, pip3 was adopted for the three positive numbers JXP;,P~,P~)= D(p1,p2,p3) < 01 the points P~JP~~P;are not selected at the outset. They turn out to be the mutual distances collinear) and carries "q into a point qO.Consequently of the points pp2p3. The distances of a point p = (ppl,pp2,pp3) from points PlsP2sP3,q % Pi~P;*P;~l1 pl,p2,p3 are ppl,pp2,pp3, respectively. Hence we have a tri- andsoq = (qpl1qp2,qp3) = (q~pi,q~p;,qtp~);thatis,qis polar coordinate system in P.
Load more

Recommended publications
  • Journal Symbolic Logic
    THT, E JOURNAL : OF SYMBOLIC LOGIC •' . X , , ' V. ;'••*• • EDITED B\Y \ ' ALONZO CHURCH S. C. KLEENE ALICE A. LAZEROWITZ Managing Editor'. ALFONS BOROERS Consulting Editors'. i W. ACKERMANN ROBERT FEYS ANDRZEJ MOSTOWSKJ G. A. BAYLIS FREDERIC B. FITCH R6ZSA PETER PAUL BERNAYS CARL G. HEMPEL BARKLEY ROSSER G. D. W. BERRY LEON HENKIN THORALF SKOLEM MARTIN DAVIS JOHN G. KEMENY A. R. TURQUETTE VOLUME 19 NUMBER 4 DECEMBER 1954 1 s PUBLISHED QUARTERLY BY THE ASSOCIATION FOR SYMBOLIC LOGIC, INC. WITH THE AID OF SUBVENTIONS FROM EDWARD C. HEGELER TRUST FUND HARVARD UNIVERSITY RUTGERS UNIVERSITY INSTITUTE FOR ADVANCED STUDY SMITH COLLEGE UNIVERSITY OF MICHIGAN 1_—; , Copyright igss by the Association for Symbolic Logic, Inc. Reproduction by photostat, photo-print, microfilm, or like process by permission only r • A Downloaded from https://www.cambridge.org/core. IP address: 170.106.51.11, on 05 Oct 2021 at 08:48:31, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022481200086618 TABLE OF CONTENTS The formalization of mathematics. By HAO WANG 24.1 The recursive irrationality of n. By R. L. GOODSTEIN 267 Distributivity and an axiom of choice. By GEORGE E. COLLINS . 275 Reviews . ! t 278 List of officers and members of the Association for Symbolic Logic . 305 The JOURNAL OF SYMBOLIC LOCIC is the official organ of the Association for Symbolic Logic, Inc., published quarterly, in the months of March, June, September, and December. The JOURNAL is published for the Association by N.V. Erven P. Noordhoff, Publishers, Gronin- gen, The Netherlands.
    [Show full text]
  • Some History of the Conjugate Gradient and Lanczos Algorithms: 1948-1976
    Some History of the Conjugate Gradient and Lanczos Algorithms: 1948-1976 Gene H. Golub; Dianne P. O'Leary SIAM Review, Vol. 31, No. 1. (Mar., 1989), pp. 50-102. Stable URL: http://links.jstor.org/sici?sici=0036-1445%28198903%2931%3A1%3C50%3ASHOTCG%3E2.0.CO%3B2-S SIAM Review is currently published by Society for Industrial and Applied Mathematics. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/siam.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Fri Aug 17 11:22:18 2007 SIAM REVIEW 0 1989 Society for Industrial and Applied Mathematics Vol.
    [Show full text]
  • Alfred Tarski and a Watershed Meeting in Logic: Cornell, 1957 Solomon Feferman1
    Alfred Tarski and a watershed meeting in logic: Cornell, 1957 Solomon Feferman1 For Jan Wolenski, on the occasion of his 60th birthday2 In the summer of 1957 at Cornell University the first of a cavalcade of large-scale meetings partially or completely devoted to logic took place--the five-week long Summer Institute for Symbolic Logic. That meeting turned out to be a watershed event in the development of logic: it was unique in bringing together for such an extended period researchers at every level in all parts of the subject, and the synergetic connections established there would thenceforth change the face of mathematical logic both qualitatively and quantitatively. Prior to the Cornell meeting there had been nothing remotely like it for logicians. Previously, with the growing importance in the twentieth century of their subject both in mathematics and philosophy, it had been natural for many of the broadly representative meetings of mathematicians and of philosophers to include lectures by logicians or even have special sections devoted to logic. Only with the establishment of the Association for Symbolic Logic in 1936 did logicians begin to meet regularly by themselves, but until the 1950s these occasions were usually relatively short in duration, never more than a day or two. Alfred Tarski was one of the principal organizers of the Cornell institute and of some of the major meetings to follow on its heels. Before the outbreak of World War II, outside of Poland Tarski had primarily been involved in several Unity of Science Congresses, including the first, in Paris in 1935, and the fifth, at Harvard in September, 1939.
    [Show full text]
  • A Bibliography of Publications in American Mathematical Monthly: 1990–1999
    A Bibliography of Publications in American Mathematical Monthly: 1990{1999 Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 14 October 2017 Version 1.19 Title word cross-reference NF [737]. !(n) [82]. p [619, 149, 694, 412]. P2 [357]. p ≡ 1 (mod 4) [34]. φ [674]. φ(30n + 1) [947]. Φpq(X) [618]. π 0 y − z 2 [105]. 1 [21]. (logx N) [333]. (x +1) x = 1 [845]. 0 [740, 684, 693, 950, 489]. π 2 3 [495, 1]. 1 [495, 1]. 1=p [511]. 10 [140]. 168 Qc(x)=x + c [399]. R [35, 226].p R [357].p n n R [62, 588]. S6 [315]. σ [19]. −1 [995]. 2 [538]. 2 [335]. $29.50 [792]. 2 :n! [1003]. p p p P1 n × 2 × 2 [26]. 3 [828, 583]. 4 − 2 [748]. A [435]. [473]. 2 3= 6 [257]. n=0 1=n! [619]. A∗A = B∗B [607]. AB [620]. BA [620]. 2n tan(k) z [160]. } [512]. x [859]. x=(sin x) n 0 [260]. mod5 [982]. C1 [832]. cos(2π/n) [322]. [755]. x = f(x ) [832]. x2 + ym = z2n [7]. d [844]. dy=dx [449]. ex [859]. f(x; y) [469]. x2 + ym = z2n [65]. x5 + ax + b [465]. xn =1 − [235]. Z3 [975].
    [Show full text]
  • The Legacy of Norbert Wiener: a Centennial Symposium
    http://dx.doi.org/10.1090/pspum/060 Selected Titles in This Series 60 David Jerison, I. M. Singer, and Daniel W. Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994) 59 William Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994) 58 Bill Jacob and Alex Rosenberg, Editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992) 57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993) 56 William J. Haboush and Brian J. Parshall, Editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, Editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, Editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, Editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, Editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, Editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, Editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, Editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R.
    [Show full text]
  • Report for the Academic Year 1999
    l'gEgasag^a3;•*a^oggMaBgaBK>ry^vg^.g^._--r^J3^JBgig^^gqt«a»J^:^^^^^ Institute /or ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1998-99 PRINCETON • NEW JERSEY HISTORICAL STUDIES^SOCIAl SC^JCE LIBRARY INSTITUTE FOR ADVANCED STUDY PRINCETON, NEW JERSEY 08540 Institute /or ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1 998 - 99 OLDEN LANE PRINCETON • NEW JERSEY • 08540-0631 609-734-8000 609-924-8399 (Fax) http://www.ias.edu Extract from the letter addressed by the Institute's Founders, Louis Bamberger and Mrs. FeUx Fuld, to the Board of Trustees, dated June 4, 1930. Newark, New Jersey. It is fundamental m our purpose, and our express desire, that in the appointments to the staff and faculty, as well as in the admission of workers and students, no account shall be taken, directly or indirectly, of race, religion, or sex. We feel strongly that the spirit characteristic of America at its noblest, above all the pursuit of higher learning, cannot admit of any conditions as to personnel other than those designed to promote the objects for which this institution is established, and particularly with no regard whatever to accidents of race, creed, or sex. ni' TABLE OF CONTENTS 4 • BACKGROUND AND PURPOSE 7 • FOUNDERS, TRUSTEES AND OFFICERS OF THE BOARD AND OF THE CORPORATION 10 • ADMINISTRATION 12 • PRESENT AND PAST DIRECTORS AND FACULTY 15 REPORT OF THE CHAIRMAN 18 • REPORT OF THE DIRECTOR 22 • OFFICE OF THE DIRECTOR - RECORD OF EVENTS 27 ACKNOWLEDGMENTS 41 • REPORT OF THE SCHOOL OF HISTORICAL STUDIES FACULTY ACADEMIC ACTIVITIES MEMBERS, VISITORS,
    [Show full text]
  • Theorem Proving in Higher Order Logics
    NASA/CP-2002-211736 Theorem Proving in Higher Order Logics Edited by Victor A. Carre_o Langley Research Center, Hampton, Virginia Cksar A. Mugoz Institute[or Computer- Applications in Science and Engineering Langley Research Center, Hampton, Virginia So/lOne Tahar Concordia UniversiO; Montreal, Canada August 2002 The NASA STI Program Office... in Profile Since its founding, NASA has been dedicated to the CONFERENCE PUBLICATION. advancement of aeronautics and space science. The Collected papers from scientific and NASA Scientific and Technical Information (STI) technical conferences, symposia, Program Office plays a key part in helping NASA seminars, or other meetings sponsored or maintain this important role. co-sponsored by NASA. The NASA STI Prograrn Office is operated by SPECIAL PUBLICATION. Scientific, Langley Research Center, the lead center for NASA's technical, or historical information from scientific and technical information. The NASA STI NASA programs, projects, and missions, Program Office provides access to the NASA STI often concerned with subjects having Database, the largest collection of aeronautical and substantial public interest. space science STI in the world. The Program Office is also NASA's institutional mechanism for TECHNICAL TRANSLATION. English- disseminating the results of its research and language translations of foreign scientific development activities. These results are and technical material pertinent to published by NASA in the NASA STI Report NASA's mission. Series, which includes the following report types: Specialized services that complement the STI Program Office's diverse offerings include TECHNICAL PUBLICATION. Reports of creating custom thesauri, building customized completed research or a major significant databases, organizing and publishing phase of research that present the results research results.., even providing videos.
    [Show full text]
  • William Karush and the KKT Theorem
    Documenta Math. 255 William Karush and the KKT Theorem Richard W. Cottle 2010 Mathematics Subject Classification: 01, 90, 49 Keywords and Phrases: Biography, nonlinear programming, calculus of variations, optimality conditions 1 Prologue This chapter is mainly about William Karush and his role in the Karush-Kuhn- Tucker theorem of nonlinear programming. It tells the story of fundamental optimization results that he obtained in his master’s thesis: results that he neither published nor advertised and that were later independently rediscov- ered and published by Harold W. Kuhn and Albert W. Tucker. The principal result – which concerns necessary conditions of optimality in the problem of minimizing a function of several variables constrained by inequalities – first became known as the Kuhn–Tucker theorem. Years later, when awareness of Karush’s pioneering work spread, his name was adjoined to the name of the theorem where it remains to this day. Still, the recognition of Karush’s discov- ery of this key result left two questions unanswered: why was the thesis not published? and why did he remain silent on the priority issue? After learning of the thesis work, Harold Kuhn wrote to Karush stating his intention to set the record straight on the matter of priority, and he did so soon thereafter. In his letter to Karush, Kuhn posed these two questions, and Karush answered them in his reply. These two letters are quoted below. Although there had long been optimization problems calling for the maxi- mization or minimization of functions of several variables subject to constraints, it took the advent of linear programming to inspire the name “nonlinear pro- gramming.” This term was first used as the title of a paper [30] by Harold W.
    [Show full text]
  • RM Calendar 2019
    Rudi Mathematici x3 – 6’141 x2 + 12’569’843 x – 8’575’752’975 = 0 www.rudimathematici.com 1 T (1803) Guglielmo Libri Carucci dalla Sommaja RM132 (1878) Agner Krarup Erlang Rudi Mathematici (1894) Satyendranath Bose RM168 (1912) Boris Gnedenko 2 W (1822) Rudolf Julius Emmanuel Clausius (1905) Lev Genrichovich Shnirelman (1938) Anatoly Samoilenko 3 T (1917) Yuri Alexeievich Mitropolsky January 4 F (1643) Isaac Newton RM071 5 S (1723) Nicole-Reine Étable de Labrière Lepaute (1838) Marie Ennemond Camille Jordan Putnam 2004, A1 (1871) Federigo Enriques RM084 Basketball star Shanille O’Keal’s team statistician (1871) Gino Fano keeps track of the number, S( N), of successful free 6 S (1807) Jozeph Mitza Petzval throws she has made in her first N attempts of the (1841) Rudolf Sturm season. Early in the season, S( N) was less than 80% of 2 7 M (1871) Felix Edouard Justin Émile Borel N, but by the end of the season, S( N) was more than (1907) Raymond Edward Alan Christopher Paley 80% of N. Was there necessarily a moment in between 8 T (1888) Richard Courant RM156 when S( N) was exactly 80% of N? (1924) Paul Moritz Cohn (1942) Stephen William Hawking Vintage computer definitions 9 W (1864) Vladimir Adreievich Steklov Advanced User : A person who has managed to remove a (1915) Mollie Orshansky computer from its packing materials. 10 T (1875) Issai Schur (1905) Ruth Moufang Mathematical Jokes 11 F (1545) Guidobaldo del Monte RM120 In modern mathematics, algebra has become so (1707) Vincenzo Riccati important that numbers will soon only have symbolic (1734) Achille Pierre Dionis du Sejour meaning.
    [Show full text]
  • An Interview with Martin Davis
    Notices of the American Mathematical Society ISSN 0002-9920 ABCD springer.com New and Noteworthy from Springer Geometry Ramanujan‘s Lost Notebook An Introduction to Mathematical of the American Mathematical Society Selected Topics in Plane and Solid Part II Cryptography May 2008 Volume 55, Number 5 Geometry G. E. Andrews, Penn State University, University J. Hoffstein, J. Pipher, J. Silverman, Brown J. Aarts, Delft University of Technology, Park, PA, USA; B. C. Berndt, University of Illinois University, Providence, RI, USA Mediamatics, The Netherlands at Urbana, IL, USA This self-contained introduction to modern This is a book on Euclidean geometry that covers The “lost notebook” contains considerable cryptography emphasizes the mathematics the standard material in a completely new way, material on mock theta functions—undoubtedly behind the theory of public key cryptosystems while also introducing a number of new topics emanating from the last year of Ramanujan’s life. and digital signature schemes. The book focuses Interview with Martin Davis that would be suitable as a junior-senior level It should be emphasized that the material on on these key topics while developing the undergraduate textbook. The author does not mock theta functions is perhaps Ramanujan’s mathematical tools needed for the construction page 560 begin in the traditional manner with abstract deepest work more than half of the material in and security analysis of diverse cryptosystems. geometric axioms. Instead, he assumes the real the book is on q- series, including mock theta Only basic linear algebra is required of the numbers, and begins his treatment by functions; the remaining part deals with theta reader; techniques from algebra, number theory, introducing such modern concepts as a metric function identities, modular equations, and probability are introduced and developed as space, vector space notation, and groups, and incomplete elliptic integrals of the first kind and required.
    [Show full text]
  • Belairglassch2-1997
    Interdisciplinary Applied Mathematics Volume 25 Editors S.S.Antman J.E. Marsden L. Sirovich S. Wiggins Geophysics and Planetary Sciences Mathematical Biology L. Glass, J.D. Murray Mechanics and Materials R.V.Kohn Systems and Control S.S. Sastry, P.S. Krishnaprasad Problems in engineering, computational science, and the physical and biologica! sciences are using increasingly sophisticated mathematical techniques. Thus, the bridge between the mathematical sciences and other disciplines is heavily traveled. The correspondingly increased dialog between the disciplines bas led to the establishment ofthe series: Interdisciplinary Applied Mathematics. The purpose of this series is to meet the current and future needs for the interaction between various science and technology areas on the one hand and mathematics on the other. This is done, frrstly, by encouraging the ways that that mathematics may be applied in traditional areas, as well as point towards new and innovative areas of applications; and, secondly, by encouraging other scientific disciplines to engage in a dialog with mathernaticians outlining their problems to both access new methods and suggest innovative developments within mathernatics itself. The series will consist of monographs and high-level texts from researchers working on the interplay between mathematics and other fields of science and technology. Interdisciplinary Applied Mathematics Volumes published are listed at the end of the book. Springer Science+Business Media, LLC Anne Beuter Leon Glass Michael C. Mackey Michele S. Titcombe Editors Nonlinear Dynamics in Physiology and Medicine With 162 Illustrations ~Springer Anne Beuter Leon Glass Institut de Biologie Department of Physiology Universite de Montpellier 1 McGill University 4, Boulevard Henri IV Montreal, Quebec H3G 1Y6 Montpellier Cedex 1, 34060 Canada France [email protected] [email protected] Michael C.
    [Show full text]
  • NBS-INA-The Institute for Numerical Analysis
    t PUBUCATiONS fl^ United States Department of Commerce I I^^^V" I ^1 I National Institute of Standards and Tectinology NAT L INST. OF STAND 4 TECH R.I.C. A111D3 733115 NIST Special Publication 730 NBS-INA — The Institute for Numerical Analysis - UCLA 1947-1954 Magnus R, Hestenes and John Todd -QC- 100 .U57 #730 1991 C.2 i I NIST Special Publication 730 NBS-INA — The Institute for Numerical Analysis - UCLA 1947-1954 Magnus R. Hestenes John Todd Department of Mathematics Department of Mathematics University of California California Institute of Technology at Los Angeles Pasadena, CA 91109 Los Angeles, CA 90078 Sponsored in part by: The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, DC 20036 August 1991 U.S. Department of Commerce Robert A. Mosbacher, Secretary National Institute of Standards and Technology John W. Lyons, Director National Institute of Standards U.S. Government Printing Office For sale by the Superintendent and Technology Washington: 1991 of Documents Special Publication 730 U.S. Government Printing Office Natl. Inst. Stand. Technol. Washington, DC 20402 Spec. Publ. 730 182 pages (Aug. 1991) CODEN: NSPUE2 ABSTRACT This is a history of the Institute for Numerical Analysis (INA) with special emphasis on its research program during the period 1947 to 1956. The Institute for Numerical Analysis was located on the campus of the University of California, Los Angeles. It was a section of the National Applied Mathematics Laboratories, which formed the Applied Mathematics Division of the National Bureau of Standards (now the National Institute of Standards and Technology), under the U.S.
    [Show full text]