Mathematical Circus & 'Martin Gardner
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Kissing Numbers: Surprises in Dimension Four Günter M
Kissing numbers: Surprises in Dimension Four Günter M. Ziegler, TU Berlin The “kissing number problem” is a basic geometric problem that got its name from billards: Two balls “kiss” if they touch. The kissing number problem asks how many other balls can touch one given ball at the same time. If you arrange the balls on a pool ta- ble, it is easy to see that the answer is exactly six: Six balls just perfectly surround a given ball. Graphic: Detlev Stalling, ZIB Berlin and at the same time Vladimir I. Levenstein˘ in Russia proved that the correct, exact maximal numbers for the kissing number prob- lem are 240 in dimension 8, and 196560 in dimension 24. This is amazing, because these are also the only two dimensions where one knows a precise answer. It depends on the fact that mathemati- cians know very remarkable configurations in dimensions 8 and 24, which they call the E8 lattice and the Leech lattice, respectively. If, however, you think about this as a three-dimensional problem, So the kissing number problem remained unsolved, in particular, the question “how many balls can touch a given ball at the same for the case of dimension four. The so-called 24-cell, a four- time” becomes much more interesting — and quite complicated. In dimensional “platonic solid” of remarkable beauty (next page), fact, The 17th century scientific genius Sir Isaac Newton and his yields a configuration of 24 balls that would touch a given one in colleague David Gregory had a controversy about this in 1694 — four-dimensional space. -
Iterations of the Words-To-Numbers Function
1 2 Journal of Integer Sequences, Vol. 21 (2018), 3 Article 18.9.2 47 6 23 11 Iterations of the Words-to-Numbers Function Matthew E. Coppenbarger School of Mathematical Sciences 85 Lomb Memorial Drive Rochester Institute of Technology Rochester, NY 14623-5603 USA [email protected] Abstract The Words-to-Numbers function takes a nonnegative integer and maps it to the number of characters required to spell the number in English. For each k = 0, 1,..., 8, we determine the minimal nonnegative integer n such that the iterates of n converge to the fixed point (through the Words-to-Numbers function) in exactly k iterations. Our travels take us from some simple answers, to an etymology of the names of large numbers, to Conway/Guy’s established system representing the name of any integer, and finally use generating functions to arrive at a very large number. 1 Introduction and initial definitions The idea for the problem originates in a 1965 book [3, p. 243] by recreational linguistics icon Dmitri Borgmann (1927–85) and in a 1966 article [16] by another icon, recreational mathematician Martin Gardner (1914–2010). Gardner’s article appears again in a book [17, pp. 71–72, 265–267] containing reprinted Gardner articles from Scientific American along with his comments on his reader’s responses. Both Borgmann and Gardner identify the only English number that requires exactly the same number of characters to spell the number as the number represents. Borgmann goes further by identifying numbers in 16 other languages with this property. Borgmann calling such numbers truthful and Gardner calling them honest. -
Donald Knuth Fletcher Jones Professor of Computer Science, Emeritus Curriculum Vitae Available Online
Donald Knuth Fletcher Jones Professor of Computer Science, Emeritus Curriculum Vitae available Online Bio BIO Donald Ervin Knuth is an American computer scientist, mathematician, and Professor Emeritus at Stanford University. He is the author of the multi-volume work The Art of Computer Programming and has been called the "father" of the analysis of algorithms. He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process he also popularized the asymptotic notation. In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar,[4] Knuth created the WEB and CWEB computer programming systems designed to encourage and facilitate literate programming, and designed the MIX/MMIX instruction set architectures. As a member of the academic and scientific community, Knuth is strongly opposed to the policy of granting software patents. He has expressed his disagreement directly to the patent offices of the United States and Europe. (via Wikipedia) ACADEMIC APPOINTMENTS • Professor Emeritus, Computer Science HONORS AND AWARDS • Grace Murray Hopper Award, ACM (1971) • Member, American Academy of Arts and Sciences (1973) • Turing Award, ACM (1974) • Lester R Ford Award, Mathematical Association of America (1975) • Member, National Academy of Sciences (1975) 5 OF 44 PROFESSIONAL EDUCATION • PhD, California Institute of Technology , Mathematics (1963) PATENTS • Donald Knuth, Stephen N Schiller. "United States Patent 5,305,118 Methods of controlling dot size in digital half toning with multi-cell threshold arrays", Adobe Systems, Apr 19, 1994 • Donald Knuth, LeRoy R Guck, Lawrence G Hanson. -
Modified Moments for Indefinite Weight Functions [2Mm] (A Tribute
Modified Moments for Indefinite Weight Functions (a Tribute to a Fruitful Collaboration with Gene H. Golub) Martin H. Gutknecht Seminar for Applied Mathematics ETH Zurich Remembering Gene Golub Around the World Leuven, February 29, 2008 Martin H. Gutknecht Modified Moments for Indefinite Weight Functions My education in numerical analysis at ETH Zurich My teachers of numerical analysis: Eduard Stiefel [1909–1978] (first, basic NA course, 1964) Peter Läuchli [b. 1928] (ALGOL, 1965) Hans-Rudolf Schwarz [b. 1930] (numerical linear algebra, 1966) Heinz Rutishauser [1917–1970] (follow-up numerical analysis course; “selected chapters of NM” [several courses]; computer hands-on training) Peter Henrici [1923–1987] (computational complex analysis [many courses]) The best of all worlds? Martin H. Gutknecht Modified Moments for Indefinite Weight Functions My education in numerical analysis (cont’d) What did I learn? Gauss elimination, simplex alg., interpolation, quadrature, conjugate gradients, ODEs, FDM for PDEs, ... qd algorithm [often], LR algorithm, continued fractions, ... many topics in computational complex analysis, e.g., numerical conformal mapping What did I miss to learn? (numerical linear algebra only) QR algorithm nonsymmetric eigenvalue problems SVD (theory, algorithms, applications) Lanczos algorithm (sym., nonsym.) Padé approximation, rational interpolation Martin H. Gutknecht Modified Moments for Indefinite Weight Functions My first encounters with Gene H. Golub Gene’s first two talks at ETH Zurich (probably) 4 June 1971: “Some modified eigenvalue problems” 28 Nov. 1974: “The block Lanczos algorithm” Gene was one of many famous visitors Peter Henrici attracted. Fall 1974: GHG on sabbatical at ETH Zurich. I had just finished editing the “Lectures of Numerical Mathematics” of Heinz Rutishauser (1917–1970). -
Gazette Des Mathématiciens
JUILLET 2018 – No 157 la Gazette des Mathématiciens • Autour du Prix Fermat • Mathématiques – Processus ponctuels déterminantaux • Raconte-moi... le champ libre gaussien • Tribune libre – Quelle(s) application(s) pour le plan Torossian-Villani la GazetteComité de rédaction des Mathématiciens Rédacteur en chef Sébastien Gouëzel Boris Adamczewski Université de Nantes Institut Camille Jordan, Lyon [email protected] [email protected] Sophie Grivaux Rédacteurs Université de Lille [email protected] Thomas Alazard École Normale Supérieure de Paris-Saclay [email protected] Fanny Kassel IHÉS Maxime Bourrigan [email protected] Lycée Saint-Geneviève, Versailles [email protected] Pauline Lafitte Christophe Eckès École Centrale, Paris Archives Henri Poincaré, Nancy [email protected] [email protected] Damien Gayet Romain Tessera Institut Fourier, Grenoble Université Paris-Sud [email protected] [email protected] Secrétariat de rédaction : smf – Claire Ropartz Institut Henri Poincaré 11 rue Pierre et Marie Curie 75231 Paris cedex 05 Tél. : 01 44 27 67 96 – Fax : 01 40 46 90 96 [email protected] – http://smf.emath.fr Directeur de la publication : Stéphane Seuret issn : 0224-8999 À propos de la couverture. Cette image réalisée par Alejandro Rivera (Institut Fourier) est le graphe de la partie positive du champ libre gaussien sur le tore plat 2=(2á2) projeté sur la somme des espaces propres du laplacien de valeur propre plus petite que 500. Les textures et couleurs ajoutées sur le graphe sont pure- ment décoratives. La fonction a été calculée avec C++ comme une somme de polynômes trigonométriques avec des poids aléatoires bien choisis, le graphe a été réalisé sur Python et les textures et couleurs ont été réalisées avec gimp. -
Aliens of Marvel Universe
Index DEM's Foreword: 2 GUNA 42 RIGELLIANS 26 AJM’s Foreword: 2 HERMS 42 R'MALK'I 26 TO THE STARS: 4 HIBERS 16 ROCLITES 26 Building a Starship: 5 HORUSIANS 17 R'ZAHNIANS 27 The Milky Way Galaxy: 8 HUJAH 17 SAGITTARIANS 27 The Races of the Milky Way: 9 INTERDITES 17 SARKS 27 The Andromeda Galaxy: 35 JUDANS 17 Saurids 47 Races of the Skrull Empire: 36 KALLUSIANS 39 sidri 47 Races Opposing the Skrulls: 39 KAMADO 18 SIRIANS 27 Neutral/Noncombatant Races: 41 KAWA 42 SIRIS 28 Races from Other Galaxies 45 KLKLX 18 SIRUSITES 28 Reference points on the net 50 KODABAKS 18 SKRULLS 36 AAKON 9 Korbinites 45 SLIGS 28 A'ASKAVARII 9 KOSMOSIANS 18 S'MGGANI 28 ACHERNONIANS 9 KRONANS 19 SNEEPERS 29 A-CHILTARIANS 9 KRYLORIANS 43 SOLONS 29 ALPHA CENTAURIANS 10 KT'KN 19 SSSTH 29 ARCTURANS 10 KYMELLIANS 19 stenth 29 ASTRANS 10 LANDLAKS 20 STONIANS 30 AUTOCRONS 11 LAXIDAZIANS 20 TAURIANS 30 axi-tun 45 LEM 20 technarchy 30 BA-BANI 11 LEVIANS 20 TEKTONS 38 BADOON 11 LUMINA 21 THUVRIANS 31 BETANS 11 MAKLUANS 21 TRIBBITES 31 CENTAURIANS 12 MANDOS 43 tribunals 48 CENTURII 12 MEGANS 21 TSILN 31 CIEGRIMITES 41 MEKKANS 21 tsyrani 48 CHR’YLITES 45 mephitisoids 46 UL'LULA'NS 32 CLAVIANS 12 m'ndavians 22 VEGANS 32 CONTRAXIANS 12 MOBIANS 43 vorms 49 COURGA 13 MORANI 36 VRELLNEXIANS 32 DAKKAMITES 13 MYNDAI 22 WILAMEANIS 40 DEONISTS 13 nanda 22 WOBBS 44 DIRE WRAITHS 39 NYMENIANS 44 XANDARIANS 40 DRUFFS 41 OVOIDS 23 XANTAREANS 33 ELAN 13 PEGASUSIANS 23 XANTHA 33 ENTEMEN 14 PHANTOMS 23 Xartans 49 ERGONS 14 PHERAGOTS 44 XERONIANS 33 FLB'DBI 14 plodex 46 XIXIX 33 FOMALHAUTI 14 POPPUPIANS 24 YIRBEK 38 FONABI 15 PROCYONITES 24 YRDS 49 FORTESQUIANS 15 QUEEGA 36 ZENN-LAVIANS 34 FROMA 15 QUISTS 24 Z'NOX 38 GEGKU 39 QUONS 25 ZN'RX (Snarks) 34 GLX 16 rajaks 47 ZUNDAMITES 34 GRAMOSIANS 16 REPTOIDS 25 Races Reference Table 51 GRUNDS 16 Rhunians 25 Blank Alien Race Sheet 54 1 The Universe of Marvel: Spacecraft and Aliens for the Marvel Super Heroes Game By David Edward Martin & Andrew James McFayden With help by TY_STATES , Aunt P and the crowd from www.classicmarvel.com . -
Local Covering Optimality of Lattices: Leech Lattice Versus Root Lattice E8
Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8 Achill Sch¨urmann, Frank Vallentin ∗ 10th November 2004 Abstract We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice E8. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E8. The new lattice yields a sphere covering which is more than 12% less dense than the formerly ∗ best known given by the lattice A8. Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a non-simplicial Delone subdivision. We hereby in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid our answer is even strongest possible in a sense. 1 Introduction The Leech lattice is the exceptional lattice in dimension 24. Soon after its discovery by Leech [Lee67] it was conjectured that it is extremal for several geometric problems in R24: the kissing number problem, the sphere packing problem and the sphere covering problem. In 1979, Odlyzko and Sloane and independently Levenshtein solved the kissing number problem in dimension 24 by showing that the Leech lattice gives an optimal solution. Two years later, Bannai and Sloane showed that it gives the unique solution up to isometries (see [CS88], Ch. 13, 14). Unlike the kissing number problem, the other two problems are still open. Recently, Cohn and Kumar [CK04] showed that the Leech lattice gives the unique densest lattice sphere packing in R24. -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n. -
NEW THIS WEEK from MARVEL... Spider-Woman #1 Outlawed #1
NEW THIS WEEK FROM MARVEL... Spider-Woman #1 Outlawed #1 Star Wars #4 Guardians of the Galaxy #3 Fantastic Four #20 Captain Marvel #16 Captain America #20 Excalibur #9 X-Force #9 Deadpool #4 Morbius #5 Amazing Mary Jane #6 Marvels X #3 (of 6) 2020 Iron Age #1 Atlantis Attacks #3 (of 5) Conan the Barbarian #14 Ghost-Spider #8 Valkyrie Jane Foster #9 Runaways #31 2020 Machine Man #2 (of 2) Aero #9 Thor #3 (2nd print) Star Wars Rise of Kylo Ren #2 (3rd print) Star Wars Rise of Kylo Ren #3 (2nd print) "Dollar" Empyre Galactus "Dollar" Empyre Swordman Taskmaster the Right Price GN Immortal Hulk Vol. 6 GN King Thor GN Conan the Barbarian Vol. 2 GN X-Men Milestones Messiah Complex GN Into the Spider-Verse Casual Miles Morales Funko-Pop NEW THIS WEEK FROM DC... Robin 80th Anniversary 100-Page Super Spectacular #1 Batman #91 Dceased Unkillables #2 (of 3) Justice League #43 Nightwing #70 Year of the Villain Hell Arisen #4 (of 4) Brave and the Bold #28 Facsimile Edition Aquaman #58 Plunge #2 (of 6) Teen Titans #40 1 Titans Giant Low Low Woods #4 (of 6) Superman's Pal Jimmy Olsen #9 (of 12) He Man and the Masters of the Multiverse #5 (of 6) Lucifer #18 Dollar Comics Justice League #1 (1987) Dollar Comics JLA #1 Superman Vol. 2 GN NEW THIS WEEK FROM IMAGE... 18 Undiscovered Country #5 Tartarus #2 Family Tree #5 Middlewest #16 Ascender #10 Die Die Die #9 Spawn #306 Bitter Root #7 Hardcore Reloaded #4 (of 5) Oblivion Song Vol. -
Volume 1 a Collection of Essays Presented at the First Frances White Ewbank Colloquium on C.S
Inklings Forever Volume 1 A Collection of Essays Presented at the First Frances White Ewbank Colloquium on C.S. Lewis & Article 1 Friends 1997 Full Issue 1997 (Volume 1) Follow this and additional works at: https://pillars.taylor.edu/inklings_forever Part of the English Language and Literature Commons, History Commons, Philosophy Commons, and the Religion Commons Recommended Citation (1997) "Full Issue 1997 (Volume 1)," Inklings Forever: Vol. 1 , Article 1. Available at: https://pillars.taylor.edu/inklings_forever/vol1/iss1/1 This Full Issue is brought to you for free and open access by the Center for the Study of C.S. Lewis & Friends at Pillars at Taylor University. It has been accepted for inclusion in Inklings Forever by an authorized editor of Pillars at Taylor University. For more information, please contact [email protected]. INKLINGS FOREVER A Collection of Essays Presented at tlte First FRANCES WHITE EWBANK COLLOQUIUM on C.S. LEWIS AND FRIENDS II ~ November 13-15, 1997 Taylor University Upland, Indiana ~'...... - · · .~ ·,.-: :( ·!' '- ~- '·' "'!h .. ....... .u; ~l ' ::-t • J. ..~ ,.. _r '· ,. 1' !. ' INKLINGS FOREVER A Collection of Essays Presented at the Fh"St FRANCES WHITE EWBANK COLLOQliTUM on C.S. LEWIS AND FRIENDS Novem.ber 13-15, 1997 Published by Taylor University's Lewis and J1nends Committee July1998 This collection is dedicated to Francis White Ewbank Lewis scholar, professor, and friend to students for over fifty years ACKNOWLEDGMENTS David Neuhauser, Professor Emeritus at Taylor and Chair of the Lewis and Friends Committee, had the vision, initiative, and fortitude to take the colloquium from dream to reality. Other committee members who helped in all phases of the colloquium include Faye Chechowich, David Dickey, Bonnie Houser, Dwight Jessup, Pam Jordan, Art White, and Daryl Yost. -
The Entropy Conundrum: a Solution Proposal
OPEN ACCESS www.sciforum.net/conference/ecea-1 Conference Proceedings Paper – Entropy The Entropy Conundrum: A Solution Proposal Rodolfo A. Fiorini 1,* 1 Politecnico di Milano, Department of Electronics, Information and Bioengineering, Milano, Italy; E- Mail: [email protected] * E-Mail: [email protected]; Tel.: +039-02-2399-3350; Fax: +039-02-2399-3360. Received: 11 September 2014 / Accepted: 11 October 2014 / Published: 3 November 2014 Abstract: In 2004, physicist Mark Newman, along with biologist Michael Lachmann and computer scientist Cristopher Moore, showed that if electromagnetic radiation is used as a transmission medium, the most information-efficient format for a given 1-D signal is indistinguishable from blackbody radiation. Since many natural processes maximize the Gibbs- Boltzmann entropy, they should give rise to spectra indistinguishable from optimally efficient transmission. In 2008, computer scientist C.S. Calude and physicist K. Svozil proved that "Quantum Randomness" is not Turing computable. In 2013, academic scientist R.A. Fiorini confirmed Newman, Lachmann and Moore's result, creating analogous example for 2-D signal (image), as an application of CICT in pattern recognition and image analysis. Paradoxically if you don’t know the code used for the message you can’t tell the difference between an information-rich message and a random jumble of letters. This is an entropy conundrum to solve. Even the most sophisticated instrumentation system is completely unable to reliably discriminate so called "random noise" from any combinatorially optimized encoded message, which CICT called "deterministic noise". Entropy fundamental concept crosses so many scientific and research areas, but, unfortunately, even across so many different disciplines, scientists have not yet worked out a definitive solution to the fundamental problem of the logical relationship between human experience and knowledge extraction.