DOCSLIB.ORG

This Page Intentionally Left Blank [50] Develop Computer Programs for Simplifying Sums That Involve Binomial CoeCients

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
This Page Intentionally Left Blank [50] Develop Computer Programs for Simplifying Sums That Involve Binomial Coe�Cients This page intentionally left blank [50] Develop computer programs for simplifying sums that involve binomial coe±cients. Exercise 1.2.6.63 in The Art of Computer Programming, Volume 1: Fundamental Algorithms by Donald E. Knuth, Addison Wesley, Reading, Massachusetts, 1968. A=B Marko Petkov·sek Herbert S. Wilf University of Ljubljana University of Pennsylvania Ljubljana, Slovenia Philadelphia, PA, USA Doron Zeilberger Temple University Philadelphia, PA, USA April 27, 1997 ii Contents Foreword vii AQuickStart::: ix I Background 1 1 Proof Machines 3 1.1Evolutionoftheprovinceofhumanthought.............. 3 1.2Canonicalandnormalforms....................... 7 1.3Polynomialidentities........................... 8 1.4Proofsbyexample?............................ 9 1.5Trigonometricidentities......................... 11 1.6Fibonacciidentities............................ 12 1.7Symmetricfunctionidentities...................... 12 1.8 Elliptic function identities ........................ 13 2 Tightening the Target 17 2.1Introduction................................ 17 2.2Identities.................................. 21 2.3Humanandcomputerproofs;anexample................ 23 2.4AMathematicasession.......................... 27 2.5AMaplesession.............................. 29 2.6Whereweareandwhathappensnext.................. 30 2.7Exercises.................................. 31 3 The Hypergeometric Database 33 3.1Introduction................................ 33 3.2Hypergeometricseries........................... 34 3.3Howtoidentifyaseriesashypergeometric............... 35 3.4Softwarethatidenti¯eshypergeometricseries.............. 39 iv CONTENTS 3.5Someentriesinthehypergeometricdatabase.............. 42 3.6Usingthedatabase............................ 44 3.7Istherereallyahypergeometricdatabase?............... 48 3.8Exercises.................................. 50 II The Five Basic Algorithms 53 4 Sister Celine's Method 55 4.1Introduction................................ 55 4.2SisterMaryCelineFasenmyer...................... 57 4.3SisterCeline'sgeneralalgorithm..................... 58 4.4 The Fundamental Theorem ....................... 64 4.5 Multivariate and \q"generalizations.................. 70 4.6Exercises.................................. 72 5 Gosper's Algorithm 73 5.1Introduction................................ 73 5.2Hypergeometricstorationalstopolynomials.............. 75 5.3Thefullalgorithm:Step2........................ 79 5.4Thefullalgorithm:Step3........................ 84 5.5Moreexamples.............................. 86 5.6Similarityamonghypergeometricterms................. 91 5.7Exercises.................................. 95 6 Zeilberger's Algorithm 101 6.1Introduction................................ 101 6.2Existenceofthetelescopedrecurrence.................. 104 6.3Howthealgorithmworks......................... 106 6.4Examples................................. 109 6.5Useoftheprograms........................... 112 6.6Exercises.................................. 118 7 The WZ Phenomenon 121 7.1Introduction................................ 121 7.2WZproofsofthehypergeometricdatabase............... 126 7.3Spino®sfromtheWZmethod...................... 127 7.4Discoveringnewhypergeometricidentities............... 135 7.5SoftwarefortheWZmethod....................... 137 7.6Exercises.................................. 140 CONTENTS v 8 Algorithm Hyper 141 8.1Introduction................................ 141 8.2Theringofsequences........................... 144 8.3Polynomialsolutions........................... 148 8.4Hypergeometricsolutions......................... 151 8.5AMathematicasession.......................... 156 8.6Findingallhypergeometricsolutions.................. 157 8.7Findingallclosedformsolutions..................... 158 8.8Somefamoussequencesthatdonothaveclosedform......... 159 8.9Inhomogeneousrecurrences........................ 161 8.10Factorizationofoperators........................ 162 8.11Exercises.................................. 164 III Epilogue 169 9 An Operator Algebra Viewpoint 171 9.1Earlyhistory............................... 171 9.2Lineardi®erenceoperators........................ 172 9.3Eliminationintwovariables....................... 177 9.4Modi¯edeliminationproblem...................... 180 9.5Discreteholonomicfunctions....................... 184 9.6Eliminationintheringofoperators................... 185 9.7Beyondtheholonomicparadigm..................... 185 9.8Bi-basicequations............................. 187 9.9Creativeanti-symmetrizing........................ 188 9.10Wavelets.................................. 190 9.11Abel-typeidentities............................ 191 9.12Anothersemi-holonomicidentity.................... 193 9.13Theart.................................. 193 9.14Exercises.................................. 195 A The WWW sites and the software 197 A.1 The Maple packages EKHAD and qEKHAD ................. 198 A.2Mathematicaprograms.......................... 199 Bibliography 201 Index 208 vi CONTENTS Foreword Science is what we understand well enough to explain to a computer. Art is everything else we do. During the past several years an important part of mathematics has been transformed from an Art to a Science: No longer do we need to get a brilliant insight in order to evaluate sums of binomial coe±cients, and many similar formulas that arise frequently in practice; we can now follow a mechanical procedure and discover the answers quite systematically. I fell in love with these procedures as soon as I learned them, because they worked for me immediately. Not only did they dispose of sums that I had wrestled with long and hard in the past, they also knocked o® two new problems that I was working on at the time I ¯rst tried them. The success rate was astonishing. In fact, like a child with a new toy, I can't resist mentioning how I used the new P ³ ´³ ´ 2n¡2k 2k methods just yesterday. Long ago I had run into the sum k n¡k k ,whichtakes the values 1, 4, 16, 64 for n =0,1,2,3soitmustbe4n. Eventually I learned a tricky way to prove that it is, indeed, 4n; but if I had known the methods in this book I could have proved the identity immediately. Yesterday I was working on a harder problem P ³ ´ ³ ´ 2n¡2k 2 2k 2 whose answer was Sn = k n¡k k . I didn't recognize any pattern in the ¯rst values 1, 8, 88, 1088, so I computed away with the Gosper-Zeilberger algorithm. In 3 ¡ 1 2 ¡ ¡ ¡ 3 afewminutesIlearnedthatn Sn =16(n 2 )(2n 2n +1)Sn¡1 256(n 1) Sn¡2. Notice that the algorithm doesn't just verify a conjectured identity \A = B". It also answers the question \What is A?", when we haven't been able to formulate a decent conjecture. The answer in the example just considered is a nonobvious recurrence from which it is possible to rule out any simple form for Sn. I'm especially pleased to see the appearance of this book, because its authors have not only played key roles in the new developments, they are also master expositors of mathematics. It is always a treat to read their publications, especially when they are discussing really important stu®. Science advances whenever an Art becomes a Science. And the state of the Art ad- vances too, because people always leap into new territory once they have understood more about the old. This book will help you reach new frontiers. Donald E. Knuth Stanford University 20 May 1995 viii CONTENTS A Quick Start ::: You've been up all night working on your new theory, you found the answer, and it's in the form of a sum that involves factorials, binomial coe±cients, and so on, such as à !à ! Xn ¡ ¡ ¡ k x k +1 x 2k f(n)= ( 1) ¡ : k=0 k n k You know that many sums like this one have simple evaluations and you would like to know, quite de¯nitively, if this one does, or does not. Here's what to do. 1. Let F (n; k)beyoursummand, i.e., the function1 that is being summed. Your ¯rst task is to ¯nd the recurrence that F satis¯es. 2. If you are using Mathematica, go to step 4 below. If you are using Maple, then get the package EKHAD either from the included diskette or from the World- WideWeb site given on page 197. Read in EKHAD,andtype zeil(F(n; k); k; n; N); in which your summand is typed, as an expression, in place of \F(n,k)". So in the example above you might type f:=(n,k)->(-1)^k*binomial(x-k+1,k)*binomial(x-2*k,n-k); zeil(f(n,k),k,n,N); Then zeil will print out the recurrence that your summand satis¯es (it does satisfy one; see theorems 4.4.1 on page 65 and 6.2.1 on page 105). The output recurrence will look like eq. (6.1.3) on page 102. In this example zeil prints out the recurrence ((n +2)(n ¡ x) ¡ (n +2)(n ¡ x)N 2)F(n; k)=G(n; k +1)¡ G(n; k); 1But what is the little icon in the right margin? See page 9. x A Quick Start ::: where N is the forward shift operator and G is a certain function that we will ignore for the moment. In customary mathematical notation, zeil will have found that (n +2)(n ¡ x)F(n; k) ¡ (n +2)(n ¡ x)F (n +2;k)=G(n; k +1)¡ G(n; k): 3. The next step is to sum the recurrence that you just found over all the values of k that interest you. In this case you can sum over all integers k.Theright side telescopes to zero, and you end up with the recurrence that your unknown sum f(n) satis¯es, in the form f(n) ¡ f(n +2)=0: Since f(0) = 1 and f(1) = 0, you have found that f(n)=1,ifn is even, and f(n)=0,ifn is odd, and you're all ¯nished. If, on the other hand, you get a recurrence whose solution is not obvious to you because it is of order higher than the ¯rst and it does not have constant coe±cients, for instance, then go to step 5 below. 4. If you are using Mathematica, then get
Load more

Recommended publications
  • BERNOULLI NEWS, Vol 24 No 1 (2017)
    Vol. 24 (1), May 2017 Published twice per year by the Bernoulli Society ISSN 1360–6727 CONTENTS News from the Bernoulli Society p. 1 Awards and Prizes p. 2 New Executive Members A VIEW FROM THE PRESIDENT in the Bernoulli Society Dear Members of the Bernoulli Society, p. 3 As we all seem to agree, the role and image of statistics has changed dramatically. Still, it takes ones breath when realizing the huge challenges ahead. Articles and Letters Statistics has not always been considered as being very necessary. In 1848 the Dutch On Bayesian Measures of Ministry of Home Affairs established an ofŮice of statistics. And then, thirty years later Uncertainty in Large or InŮinite minister Kappeyne van de Coppelo abolishes the “superŮluous” ofŮice. The ofŮice was Dimensional Models p. 4 quite rightly put back in place in 1899, as “Centraal Bureau voor de Statistiek” (CBS). Statistics at the CBS has evolved from “simple” counting to an art requiring a broad range On the Probability of Co-primality of competences. Of course counting remains important. For example the CBS reports in of two Natural Numbers Chosen February 2017 that almost 1 out of 4 people entitled to vote in the Netherlands is over the at Random p. 7 age of 65. But clearly, knowing this generates questions. What is the inŮluence of this on the outcome of the elections? This calls for more data. Demographic data are combined with survey data and nowadays also with data from other sources, in part to release the “survey pressure” that Ůirms and individuals are facing.
    [Show full text]
  • Contemporary Mathematics 78
    CONTEMPORARY MATHEMATICS 78 Braids Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Artin's Braid Group held July 13-26. 1986 at the University of California, Santa Cruz, California Joan S. Birman Anatoly Libgober Editors http://dx.doi.org/10.1090/conm/078 Recent Titles in This Series 120 Robert S. Doran, Editor, Selfadjoint and nonselfadjoint operator algebras and operator theory, 1991 119 Robert A. Melter, Azriel Rosenfeld, and Prabir Bhattacharya, Editors, Vision geometry, 1991 118 Yan Shi-Jian, Wang Jiagang, and Yang Chung-chun, Editors, Probability theory and its applications in China, 1991 117 Morton Brown, Editor, Continuum theory and dynamical systems, 1991 116 Brian Harboume and Robert Speiser, Editors, Algebraic geometry: Sundance 1988, 1991 115 Nancy Flournoy an'il Robert K. Tsutakawa, Editors, Statistical multiple integration, 1991 114 Jeffrey C. Lagarias and Michael J. Todd, Editors, Mathematical developments arising from linear programming, 1990 113 Eric Grinberg and Eric Todd Quinto, Editors, Integral geometry and tomography, 1990 112 Philip J. Brown and Wayne A. Fuller, Editors, Statistical analysis of measurement error models and applications, 1990 Ill Earl S. Kramer and Spyros S. Magliveras, Editors, Finite geometries and combinatorial designs, I 990 II 0 Georgia Benkart and J. Marshall Osborn, Editors, Lie algebras and related topics, 1990 109 Benjamin Fine, Anthony Gaglione, and Francis C. Y. Tang, Editors, Combinatorial group theory, 1990 108 Melvyn S. Berger, Editor, Mathematics of nonlinear science, 1990 107 Mario Milman and Tomas Schonbek, Editors, Harmonic analysis and partial differential equations, 1990 I 06 Wilfried Sieg, Editor, Logic and computation, 1990 I 05 Jerome Kaminker, Editor, Geometric and topological invariants of elliptic operators, 1990 I 04 Michael Makkai and Robert Pare, Accessible categories: The foundations of categorical model theory, 1989 I 03 Steve Fisk, Coloring theories, 1989 I 02 Stephen McAdam, Primes associated to an ideal, 1989 101 S.-Y.
    [Show full text]
  • MAA Rocky Mountain Section Meeting
    e Mathematical Association of America Rocky Mountain Section Meeting April Ôâ and ÔÞ, òýÔý Colorado State University Fort Collins, CO Rocky Mountain Section Meeting Annual Book Sale April 16 & 17, 2010 All listed titles available for shipment, free shipping & handling. A full catalog and order form for shipped orders is included on the middle pages of this booklet for your convenience. Over 100 titles available (in limited quantities) for immediate purchase at our display. All catalog prices have been discounted 10% below membership prices, and 10% of book proceeds will be returned to the section – enlarge your book collection while benefiting the section today! To place an order, visit us in the Cherokee Park Ballroom. Discount prices available to all meeting participants; prices good only on orders placed at the meeting. All order forms must be returned to a display staff member for processing!! Payment by check, Visa or MasterCard accepted (sorry - no cash sales). Please make checks payable to: MAA Rocky Mountain Section. MATHEMATICAL ASSOCIATION OF AMERICA ò Schedule Friday, April Ôâ :ýý-Ôò:ýý Section NExT workshop (Virginia Dale) À:çý-ÔÔ:çý Workshop: Proposal writing for the NSF DUE (Lory òÔÞ) Stephanie Fitchett, NSF and University of Northern Colorado ÔÔ:¥ -Ôò:¥ Luncheon for Dept. Chairs and MAA Liaisons (Lory òçý) ÔÔ:ýý-¥:çý Registration (Lory òòÞ) Ô:ýý-Ô:Ôý Opening Remarks and Welcome (North Ballroom) Ô:Ôý-Ô: Burton W. Jones Teaching Award Lecture (North Ballroom) Richard Grassl, University of Northern Colorado Ô:ýý- :çý Publisher
    [Show full text]
  • Rafe Mazzeo Curriculum Vitæ Address
    Rafe Mazzeo Curriculum Vitæ Address: Department of Mathematics Room 383R Stanford University Stanford, California 94305-2125 Phone: (650) 723-1894 Mobile: (650) 714-5718 [email protected] http://math.stanford.edu/~mazzeo Research Interests: Geometric and Microlocal Analysis, Partial Differen- tial Equations. Education: B.S. Massachusetts Institute of Technology, 1982. Thesis under T. Shifrin: Shrinking tubes and variants of the Gauss-Bonnet formula Ph.D. Massachusetts Institute of Technology, 1986. Thesis under R.B. Melrose: Hodge cohomology of negatively curved manifolds Employment: 1986-1988 Stanford University, Szeg¨oAssistant Professor. 1988-1992 Stanford University, Assistant Professor (on leave, 1991-1993). 1991-1993 University of Washington, Assistant Professor. 1992-1997 Stanford University, Associate Professor. 1997- Stanford University, Professor. 2007-2010 Department Chair Awards and Fellowships: National Science Foundation Postdoctoral Fellow, 1987-1990 Alfred P. Sloan Research Fellow, 1991-1995 National Science Foundation Young Investigator, 1992-1997 Louis and Claude Rosenberg Jr. University Fellow in Undergraduate Education, 2013-2018. National Science Foundation DMS grants, 1990 { present. Other Major Administrative Activities: Faculty Director, Stanford Pre-Collegiate Studies/Stanford Online High School, 2010 { Director, Institute for Advanced Study/Park City Mathematics Insti- tute, September 2014 { August 2024. Other: Managing Editor, Communications in Partial Differential Equations, 1996 { Editorial Board, AMS series Graduate Studies in Mathematics, 2010{ 2016. Editorial Board, Notices of the American Mathematical Society, 2010 { 2013. Editorial Board, Annales de l'IHP { equations nonlineaire, 2010 {. Scientific Advisory Board, Banff International Research Station, 2008 { 2010. Scientific Advisory Board, American Institute of Mathematics, 2010 { Organizing Committee, MSRI Semester Spectral Invariants, Spring 2001. Nachdiplom Lectures, ETH Z¨urich, Summer Semester 2004.
    [Show full text]
  • Meetings of the MAA Ken Ross and Jim Tattersall
    Meetings of the MAA Ken Ross and Jim Tattersall MEETINGS 1915-1928 “A Call for a Meeting to Organize a New National Mathematical Association” was DisseminateD to subscribers of the American Mathematical Monthly and other interesteD parties. A subsequent petition to the BoarD of EDitors of the Monthly containeD the names of 446 proponents of forming the association. The first meeting of the Association consisteD of organizational Discussions helD on December 30 and December 31, 1915, on the Ohio State University campus. 104 future members attendeD. A three-hour meeting of the “committee of the whole” on December 30 consiDereD tentative Drafts of the MAA constitution which was aDopteD the morning of December 31, with Details left to a committee. The constitution was publisheD in the January 1916 issue of The American Mathematical Monthly, official journal of The Mathematical Association of America. Following the business meeting, L. C. Karpinski gave an hour aDDress on “The Story of Algebra.” The Charter membership included 52 institutions and 1045 inDiviDuals, incluDing six members from China, two from EnglanD, anD one each from InDia, Italy, South Africa, anD Turkey. Except for the very first summer meeting in September 1916, at the Massachusetts Institute of Technology (M.I.T.) in CambriDge, Massachusetts, all national summer anD winter meetings discussed in this article were helD jointly with the AMS anD many were joint with the AAAS (American Association for the Advancement of Science) as well. That year the school haD been relocateD from the Back Bay area of Boston to a mile-long strip along the CambriDge siDe of the Charles River.
    [Show full text]
  • Guide to Jewish Studies Resources at UT Austin
    the SCHUSTERMAN CENTER for JEWISH STUDIES presents A GUIDE TO THE COLLECTIONS JEWISH STUDIES RESOURCES at THE UNIVERSITY OF TEXAS AT AUSTIN table of CONTENTS I. A Message from the Director ............................................... 2 II. The Architecture and Planning Library ..................................... 4 III. The Fine Arts Library ........................,............................... 5 IV. The Perry-Castañeda Library .............................................. 6 V. The Tarlton Law Library ..................................................... 7 VI. The Nettie Lee Benson Latin American Collection . 8 VII. The Harry Ransom Center ............................................... 10 VIII. The Dolph Briscoe Center for American History . 20 a message from THE DIRECTOR ONE OF THE founding goals of the Schusterman Center for Jewish Studies at The University of Texas at Austin was to become a crossroads for the study of Jews and Jewish culture in all its aspects, with particular focus on Jewish life in the Americas. A crucial piece of this vision has been to make more visible to an international audience the rich research collections concerning Jews in the vari- ous archives and libraries on the Austin campus. We have prepared this guide to promote the use of these resources by both students and scholars based not only in Austin, but also elsewhere in the United States and around the world. Someone not familiar with the University of Texas may find astonishing the depth, breadth, and importance of these materials. Available for research are resources like the papers of Jewish writers, including Isaac Bashevis Singer, Arthur Miller, and Nor- man Mailer, local history collections of the Texas Jewish Historical Society, exten- ROBERT H. ABZUG, DIRECTOR sive holdings in Yiddish and Hebrew, and rare and unusual examples of Judaica.
    [Show full text]
  • Prof. Günter M. Ziegler, Phd a Freie Universität Berlin Institut Für
    Prof. Günter M. Ziegler, PhD Address Freie Universität Berlin Institut für Mathematik Arnimallee 2 D-14195 Berlin, Germany Tel. +49 30 838 75 668 E-Mail [email protected] WWW http://page.mi.fu-berlin.de/gmziegler/ Curriculum Vitae 1963 Born in München (Munich), Germany (Citizenship: German) 1981–1984 Studies in Mathematics and Physics, Ludwig-Maximilians-University, Munich 1984–1987 Graduate studies in Mathematics, Massachusetts Institute of Technology, Cambridge, MA 1987 PhD at M.I.T. Advisor: Prof. Anders Björner Thesis: “Algebraic Combinatorics of Hyperplane Arrangements” 1987–1991 Postdoc at Augsburg University, with Prof. Martin Grötschel 1991–1992 Special year on “Combinatorics”, Institut Mittag-Leffler, Djursholm, Sweden 1992–1994 Research scientist at Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) Deputy of Martin Grötschel for the “Combinatorial Optimization” department 1992 Habilitation, TU Berlin Thesis: “Combinatorics and Topology of Subspace Arrangements” 1992–1994 Privatdozent, TU Berlin 1993–2000 Faculty member, DFG Research Training Group “Algorithmic Discrete Mathematics” 1995–1998 Professor of Discrete Mathematics (C3/equivalent to associate professor), TU Berlin 1998–2011 Professor of Mathematics (C4/full professor), TU Berlin (since 2006: W3) 2001 Visiting Miller Research Professor, UC Berkeley 2000–2005 Faculty member, DFG International Research Training Group “Combinatorics, Geometry, and Computation” (Berlin–ETH Zurich) 2002– Member, DFG Research Center Matheon 2006–2017 Faculty member, DFG Research
    [Show full text]
  • Prizes and Awards
    SAN DIEGO • JAN 10–13, 2018 January 2018 SAN DIEGO • JAN 10–13, 2018 Prizes and Awards 4:25 p.m., Thursday, January 11, 2018 66 PAGES | SPINE: 1/8" PROGRAM OPENING REMARKS Deanna Haunsperger, Mathematical Association of America GEORGE DAVID BIRKHOFF PRIZE IN APPLIED MATHEMATICS American Mathematical Society Society for Industrial and Applied Mathematics BERTRAND RUSSELL PRIZE OF THE AMS American Mathematical Society ULF GRENANDER PRIZE IN STOCHASTIC THEORY AND MODELING American Mathematical Society CHEVALLEY PRIZE IN LIE THEORY American Mathematical Society ALBERT LEON WHITEMAN MEMORIAL PRIZE American Mathematical Society FRANK NELSON COLE PRIZE IN ALGEBRA American Mathematical Society LEVI L. CONANT PRIZE American Mathematical Society AWARD FOR DISTINGUISHED PUBLIC SERVICE American Mathematical Society LEROY P. STEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH American Mathematical Society LEROY P. STEELE PRIZE FOR MATHEMATICAL EXPOSITION American Mathematical Society LEROY P. STEELE PRIZE FOR LIFETIME ACHIEVEMENT American Mathematical Society SADOSKY RESEARCH PRIZE IN ANALYSIS Association for Women in Mathematics LOUISE HAY AWARD FOR CONTRIBUTION TO MATHEMATICS EDUCATION Association for Women in Mathematics M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE WOMEN IN MATHEMATICS Association for Women in Mathematics MICROSOFT RESEARCH PRIZE IN ALGEBRA AND NUMBER THEORY Association for Women in Mathematics COMMUNICATIONS AWARD Joint Policy Board for Mathematics FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY AN UNDERGRADUATE STUDENT American Mathematical Society Mathematical Association of America Society for Industrial and Applied Mathematics BECKENBACH BOOK PRIZE Mathematical Association of America CHAUVENET PRIZE Mathematical Association of America EULER BOOK PRIZE Mathematical Association of America THE DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY TEACHING OF MATHEMATICS Mathematical Association of America YUEH-GIN GUNG AND DR.CHARLES Y.
    [Show full text]
  • SOURCE New Archives of American Mathematics Collections
    HISTORIA MATHEMATICA 20 (t993), 92-94 SOURCE This department welcomes correspondence, brief announcements, and article-length descriptions of collections of publications, correspondence, and archival material relevant to the history of mathematics. Manuscripts describing major collections (covering such matters as acquisition, size, scope, state of cataloging, and current and future availability) should follow the same standards as other articles. They will be abstracted and indexed like other articles, and authors will be supplied with free reprints. New Archives of American Mathematics Collections FREDERIC F. BURCHSTED Archives of American Mathematics, Center for American History, SRH 2.109, University of Texas, Austin, Texas 78713 The Archives of American Mathematics of the University of Texas at Austin Center for American History announces the availability of the following collec- tions. This list supplements the list which appeared in Historia Mathematica 14 (1987), 366-374. Benedict (Harry Yandell), papers, 1895-1898. l0 in. In the papers of University of Texas at Austin professor and president Harry Yandell Benedict are his Harvard University graduate student notebooks (1895-1898) on courses of M. Brcher, W. E. Byerly, A. Hall, W. F. Osgood, and B. O. Peirce. Bing (R. H.), papers, 1948-1986.4 ft. R. H. Bing (1914-1986) was a geometric topologist at the University of Wisconsin (1947-1972) and the University of Texas at Austin (1973-1986). Records of Bing's research chiefly relate to the preparation of his The Geometric Topology of 3-Manifolds (1983). The correspondence is chiefly from the last few years of Bing's life. Included are research and lecture notes, drafts of publications, teaching material, newspaper clippings, preprints, and reprints.
    [Show full text]
  • Finite Fields: Theory, Applications, and Algorithms
    CoNTEMPORARY MATHEMATICS 168 Finite Fields: Theory, Applications, and Algorithms Second International Conference on Finite Fields: Theory, Applications, and Algorithms August 17-2L 1993 Las Vegas, Nevada Gary L. Mullen Peter Jau-Shyong Shiue Editors http://dx.doi.org/10.1090/conm/168 Recent Titles in This Series 168 Gary L. Mullen and Peter Jau-Shyong Shiue, Editors, Finite fields: Theory, applications, and algorithms, 1994 167 RobertS. Doran, Editor, C*-Algebras: 1943-1993, 1994 166 George E. Andrews, David M. Bressoud, and L. Alayne Parson, Editors, The Rademacher legacy to mathematics, 1994 165 Barry Mazur and Glenn Stevens, Editors, p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, 1994 164 Cameron Gordon, Yoav Moriah, and Bronislaw Wajnryb, Editors, Geometric topology, 1994 163 Zhong-Ci Shi and Chung-Chun Yang, Editors, Computational mathematics in China, 1994 162 Ciro Ciliberto, E. Laura Livorni, and Andrew J. Sommese, Editors, Classification of algebraic varieties, 1994 161 Paul A. Schweitzer, S. J., Steven Hurder, Nathan Moreira dos Santos, and Jose Luis Arraut, Editors, Differential topology, foliations, and group actions, 1994 160 Niky Kamran and Peter J. Olver, Editors, Lie algebras, cohomology, and new applications to quantum mechanics, 1994 159 William J. Heinzer, Craig L. Huneke, and Judith D. Sally, Editors, Commutative algebra: Syzygies, multiplicities, and birational algebra, 1994 158 Eric M. Friedlander and Mark E. Mahowald, Editors, Topology and representation theory, 1994 157 Alfio Quarteroni, Jacques Periaux, Yuri A. Kuznetsov, and Olof B. Widlund, Editors, Domain decomposition methods in science and engineering, 1994 156 Steven R. Givant, The structure of relation algebras generated by relativizations, 1994 155 William B.
    [Show full text]
  • Dedekind Sums
    VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS 16 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 16 Dedekind Sums Dedekind Sums Rademacher | Grosswald Noncommutative Rings provides a cross-section of ideas, techniques, and results that give the reader an idea of that part of algebra which concerns itself with noncommutative rings. In the space of 200 pages, Dedekind Sums Herstein covers the Jacobson radical, semisimple rings, commuta- tivity theorems, simple algebras, representations of nite groups, polynomial identities, Goldie's theorem, and the Golod–Shafarevitch theorem. Almost every practicing ring theorist has studied portions of Hans Rademacher this classic monograph. and Herstein's book is a guided tour through a gallery of masterpieces. The author's style is always elegant and his proofs always enlight- Emil Grosswald ening… I had a lot of pleasure when I rst read this book while I was an undergraduate student attending to a course given by C. Procesi at the University of Rome. Today, I appreciate even more the author's mastery and real gift for exposition. —Fabio Mainardi, MAA Reviews AMS / MAA PRESS 4-color Process 119 pages spine: 1/4" finish size: 5.5" X 8.5" 50 lb stock 10.1090/car/016 DEDEKIND SUMS By HANS RADEMACHER and EMIL GROSSWALD THE CARUS MATHEMATICAL MONOGRAPHS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications IVAN NIVEN, Chairman Subcommittee on Carus Monographs R. G. BARTLE, Chairman R. P. BOAS D. T. FINKBEINER II HE CARUS MATHEMATICAL MONOGRAPHS are an expression of the desire of Mrs. Mary Hegeler Carus, and of T her son, Dr.
    [Show full text]
  • R Ational Period Functions for the M Odular Group and Related Discrete
    Order Number 9S07755 Rational period functions for the modular group and related discrete groups Gethner, Ellen, Ph.D. The Ohio State University, 1992 UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 R ational P eriod F unctions for the M o dular G roup a n d R elated D iscrete G roups dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ellen Gethner, A.B., M.A. The Ohio State University 1992 Dissertation Committee: Approved by L. Alayne Parson Francis W. Carroll ndvifidvisor Robert Gold Department of Mathematics A cknowledgements I am deeply grateful to my advisor Alayne Parson for her generosity with her time and her patience, and for many enticing mathematical discussions. I am indebted to Frank Carroll and Bob Gold for serving on my dissertation committee, and par­ ticularly to Frank Carroll for his moral and mathematical support throughout my tenure at Ohio State University. I would like to thank Marvin Knopp who made a conjecture, the proof of which evolved into the material in Chapter 2, and who sug­ gested the problem which ultimately became the body of Chapter 3. Special thanks go to Stan Wagon who graciously supplied the Mathematica code which generated the graphics in Figure 4. This project could not have been completed without the patience, good cheer, and moral support of my family and close friends. I especially thank my family for their unwavering belief in me. I am grateful to Bobby Craighead, Griff Elder, Alan Foreman, Meg Hartenstein, Steffanie Sabbaj, Lynn Singleton, and Barry Spieler, each of whom helped in their own unique way.
    [Show full text]