DOCSLIB.ORG

Numeration 2016

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Numeration 2016 Numeration 2016 Prague, Czech Republic, May 23–27 2016 Department of Mathematics FNSPE Czech Technical University in Prague Numeration 2016 Prague, Czech Republic, May 23–27 2016 Department of Mathematics FNSPE Czech Technical University in Prague Print: Česká technika — nakladatelství ČVUT Editor: Petr Ambrož [email protected] Katedra matematiky Fakulta jaderná a fyzikálně inženýrská České vysoké učení technické v Praze Trojanova 13 120 00 Praha 2 List of Participants Rafael Alcaraz Barrera [email protected] IME, University of São Paulo Karam Aloui [email protected] Institut Elie Cartan de Nancy / Faculté des Sciences de Sfax Petr Ambrož petr.ambroz@fjfi.cvut.cz FNSPE, Czech Technical University in Prague Hamdi Ammar [email protected] Sfax University Myriam Amri [email protected] Sfax University Hamdi Aouinti [email protected] Université de Tunis Simon Baker [email protected] University of Reading Christoph Bandt [email protected] University of Greifswald Attila Bérczes [email protected] University of Debrecen Anne Bertrand-Mathis [email protected] University of Poitiers Dávid Bóka [email protected] Eötvös Loránd University Horst Brunotte [email protected] Marta Brzicová [email protected] FNSPE, Czech Technical University in Prague Péter Burcsi [email protected] Eötvös Loránd University Francesco Dolce [email protected] Université Paris-Est Art¯urasDubickas [email protected] Vilnius University Lubomira Dvořáková [email protected] FNSPE, Czech Technical University in Prague Jordan Emme [email protected] Aix-Marseille University Sébastien Ferenczi [email protected] IMPA – CNRS, Rio de Janeiro, Brasil Aviezri Fraenkel [email protected] Weizmann Institute of Science Christiane Frougny [email protected] LIAFA / IRIF, Paris Kevin Hare [email protected] University of Waterloo Pavel Heller [email protected] Université Paris-Est Yuke Huang [email protected] School of Mathematics and Systems Science, BUAA Peter Hudoba [email protected] Eötvös Loránd University, Budapest Amara Chandoul [email protected] Sfax University Koji Chinen [email protected] Kindai University, Osaka, Japan Maria Rita Iaco [email protected] Université Paris Diderot Jonas Jankauskas [email protected] Montanuniversität Leoben Kan Jiang [email protected] Utrecht University Shoichi Kamada [email protected] Kumamoto University Rania Kammoun [email protected] Université de Sfax, Tunisie Kamil Keprt [email protected] University of Ostrava Karel Klouda karel.klouda@fit.cvut.cz FIT, CTU in Prague Ondřej Kolouch [email protected] University of Ostrava Attila Kovács [email protected] Eötvös Loránd University Daniel Krenn [email protected] AAU Klagenfurt, Austria Tamás Krutki [email protected] Eötvös Loránd University Petr Kůrka [email protected] Center for Theoretical Study, Prague DoYong Kwon [email protected] Chonnam National University Revekka Kyriakoglou [email protected] Université Paris Est Marne la Vallée (LIGM) Niels Langeveld [email protected] University of Leiden Manfred Madritsch [email protected] Université de Lorraine Faiza Mahjoub [email protected] Sfax University Merkhi Malika [email protected] Sfax University Bill Mance [email protected] Polish Academy of Sciences Zuzana Masáková zuzana.masakova@fjfi.cvut.cz FNSPE, Czech Technical University in Prague David W. Matula [email protected] Southern Methodist University Mohamed Mkaouar [email protected] Sfax University Hbaib Mohamed [email protected] Sfax University Clemens Müllner [email protected] Technical University of Vienna Gábor Nagy [email protected] Department of Computer Algebra, ELTE IK Radhakrishnan Nair [email protected] The University of Liverpool Lukáš Novotný [email protected] University of Ostrava Marco Pedicini [email protected] Roma Tre University Edita Pelantová edita.pelantova@fjfi.cvut.cz FNSPE, Czech Technical University in Prague Attila Pethő [email protected] University of Debrecen Joel Rivat [email protected] Université d’Aix-Marseille Driss Sana [email protected] Sfax University Zhang Shuqin [email protected] University of Leoben Klaus Scheicher [email protected] BOKU, Vienna Johannes Schleischitz [email protected] BOKU Vienna Nikita Sidorov [email protected] The University of Manchester Bernd Sing [email protected] The University of the West Indies Lukas Spiegelhofer [email protected] Universite de Lorraine, France Štěpán Starosta stepan.starosta@fit.cvut.cz FIT, CTU in Prague Wolfgang Steiner [email protected] CNRS, University Paris 7 Paul Surer [email protected] BOKU Vienna Milena Svobodová [email protected] FNSPE, Czech Technical University in Prague Jan Šustek [email protected] University of Ostrava Szabolcs Tengely [email protected] University of Debrecen Tomas Vavra [email protected] FNSPE, Czech Technical University in Prague Jean-Louis Verger-Gaugry [email protected] CNRS, University Savoie Mont Blanc Walid Wannes [email protected] Sfax University Mario Weitzer [email protected] TU Graz Invited talks Algebraic Aspects of Canonical Number Systems H. Brunotte .................................. 1 Pisot Numbers and Their Conjugates A. Dubickas .................................. 6 Freezing Arithmetic Algorithms into Number Representation Theorems D.W. Matula ................................. 10 Self-Affine Sets: Topology, Uniqueness, Simultaneous Expansions N. Sidorov ................................... 11 Contributed talks Normality of Different Orders for the Cantor Series Expansions D. Airey, B. Mance .............................. 17 Topological Transitivity and Entropy of Unique β-Expansions R. Alcaraz Barrera, S. Baker, D. Kong ................... 18 Strongly q-Additive Functions and Largest Prime Factor of an Integer M. Amri .................................... 19 Unique Expansions and Intersections of Cantor Sets S. Baker .................................... 20 Eventually Periodic Representations in Positional Number Systems S. Baker, Z. Masáková, E. Pelantová, T. Vávra . 21 Kneading Sequences, Bernoulli Convolutions, and β-Expansion C. Bandt ................................... 25 Rauzy’s Tilings in the Light of Uniform Distribution; +β and β Case A. Bertrand-Mathis ..............................− 26 Reconstruction of Matrix-Based Numeration Systems from Some Expansions D. Bóka, P. Burcsi .............................. 28 On-line Mutliplication and Division in Number Systems with Complex Bases M. Brzicová, C. Frougny, E. Pelantová, M. Svobodová . 29 Permutations and Negative Beta-Shifts É. Charlier, W. Steiner ........................... 30 Return Words and Palindromes in Specular Sets F. Dolce .................................... 34 On the Density of Sets Defined by Sum-of-Digit Function in Base 2 J. Emme, A. Prikhod’ko ........................... 41 Numeration Systems and Their Uses A.S. Fraenkel ................................. 44 Beta-Representations of 0 and Pisot Numbers C. Frougny, E. Pelantová .......................... 50 Base d expansions with digits 0 to q 1 K.G. Hare ...................................− 54 The Numbers of Distinct Squares and Cubes in the Tribonacci Sequence Y. Huang, Z. Wen .............................. 59 Some Improvements on Number Expansion Computations P. Hudoba, A. Kovács ............................ 65 Approximation Problems and Knapsack Cryptosystems in p-Adic Numberlands H. Inoue, S. Kamada, K. Naito, K. Shiromoto . 67 Two properties related to β-expansions M.R. Iacò, W. Steiner, R. Tichy ...................... 73 Pisot-Salem numbers, Interlacing and 0; 1 -Words J. Jankauskas .................................f g 76 Multiple Expansions of Real Numbers Over Digit Set 0; 1; q K. Jiang ....................................f g 77 Critical Bases for Ternary Alphabets V. Komornik, M. Pedicini .......................... 78 Redundant Expansions, Continued Fractions, and (Non-)Minimal Expansions D. Krenn ................................... 79 On Simultaneous Number Systems with 3 Bases T. Krutki, G. Nagy .............................. 83 Sturmian Word, Cantor Set, and Unique Expansions over Ternary Alphabets D.Y. Kwon .................................. 85 The Hausdorff Dimension of Non-Normal Numbers M. Madritsch, I. Petrykiewicz ........................ 86 On Distribution of Subsequences of the Natural Numbers on the p-Adic Integers R. Nair .................................... 88 Algorithmic Questions on Canonical Number Systems and Shift Radix Systems A. Pethő .................................... 90 Distribution Modulo 1 and the Cardinality Gap Phenomenon J. Schleischitz ................................. 91 Exploring the Intersection of the Pieces of Rauzy Fractals B. Sing .................................... 92 Divisibility of Binomial Coefficients by Powers of Primes L. Spiegelhofer ................................ 93 Continued Fractions and Spectra of Certain Operators Š. Starosta ................................... 94 Substitutions, Shifts of Finite Type, and Numeration P. Surer .................................... 99 On Height One Trinomials and Limit Dynamical Zeta Functions J.-L. Verger-Gaugry ............................. 101 Algebraic Aspects of Canonical Number Systems Horst Brunotte
Load more

Recommended publications
  • Residue Arithmetic in Digital Computers
    r* 6 l't ¡r "RESIDUT ARITHMETIC IN DIGITAL COIIPUTERS" RAMESH CHANDRA DEBNATH/ M.SC. BEING A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF ELECTRICAL ENGINEERING THE UNIVERSITY OF ADELAIDE IVIARCl-i I979 SUIq¡4ARY The number systems used have great impact on the speed of a computer, At present at least 4 number systems (binary, negabinary, ternary and residue) are in use either for general purposes or for special purposes. The calîry plîopagation delay which j-s inherent to all weighted systems (binary, negabi-nary and ternary) is a limiting factor on the speed of computations. tlnlike the weighted system, the residue system is carry free which is the m¿rln attractj-ve feature of this system and therefore its potentiality in a general purpose computer, in the light of present day tech- nology, has been investigated. As the background of the investigation the binary and negabinary arithmetic have been thoroughly studied. A ne',r method for fractional negabinary division, and a correction on negabinary multiplì-cation have been proposed. It has been observed that the binary arithmetic operations are fas- ter thair thcse ir. negabinary syslem. The residue system has got. problems too. The sign detectj-on, operations j-nvolving sign detection (comparison, overflow detection, division) and the resj-due-to--decimal- con- version are difficult. Moreover, the residue systemT being an i.nteger: systetn, makes floati-ng poinc arithmetic operatlcns very difficult. The def:ails of the existing algorithms,/ methods to solve those problems have been thoroughly inves- tigated and drawbacks of those algoritlrms/methods have been di scussed.
    [Show full text]
  • On the Behavior of Mahler's Measure Under Iteration 1
    ON THE BEHAVIOR OF MAHLER'S MEASURE UNDER ITERATION PAUL FILI, LUKAS POTTMEYER, AND MINGMING ZHANG Abstract. For an algebraic number α we denote by M(α) the Mahler measure of α. As M(α) is again an algebraic number (indeed, an algebraic integer), M(·) is a self-map on Q, and therefore defines a dynamical system. The orbit size of α, denoted #OM (α), is the cardinality of the forward orbit of α under M. We prove that for every degree at least 3 and every non-unit norm, there exist algebraic numbers of every orbit size. We then prove that for algebraic units of degree 4, the orbit size must be 1, 2, or infinity. We also show that there exist algebraic units of larger degree with arbitrarily large but finite orbit size. 1. Introduction The Mahler measure of an algebraic number α with minimal polynomial f(x) = n anx + ··· + a0 2 Z[x] is defined as: n n Y Y M(α) = janj maxf1; jαijg = ±an αi: i=1 i=1 jαij>1 Qn where f(x) = an i=1(x − αi) 2 C[x]. It is clear that M(α) ≥ 1 is a real algebraic integer, and it follows from Kronecker's theorem that M(α) = 1 if and only if α is a root of unity. Moreover, we will freely use the facts that M(α) = M(β) whenever α and β have the same minimal polynomial, and that M(α) = M(α−1). D.H. Lehmer [10] famously asked in 1933 if the Mahler measure for an algebraic number which is not a root of unity can be arbitrarily close to 1.
    [Show full text]
  • The Smallest Perron Numbers 1
    MATHEMATICS OF COMPUTATION Volume 79, Number 272, October 2010, Pages 2387–2394 S 0025-5718(10)02345-8 Article electronically published on April 26, 2010 THE SMALLEST PERRON NUMBERS QIANG WU Abstract. A Perron number is a real algebraic integer α of degree d ≥ 2, whose conjugates are αi, such that α>max2≤i≤d |αi|. In this paper we com- pute the smallest Perron numbers of degree d ≤ 24 and verify that they all satisfy the Lind-Boyd conjecture. Moreover, the smallest Perron numbers of degree 17 and 23 give the smallest house for these degrees. The computa- tions use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in C 1. Introduction Let α be an algebraic integer of degree d, whose conjugates are α1 = α, α2,...,αd and d d−1 P = X + b1X + ···+ bd−1X + bd, its minimal polynomial. A Perron number, which was defined by Lind [LN1], is a real algebraic integer α of degree d ≥ 2 such that α > max2≤i≤d |αi|.Any Pisot number or Salem number is a Perron number. From the Perron-Frobenius theorem, if A is a nonnegative integral matrix which is aperiodic, i.e. some power of A has strictly positive entries, then its spectral radius α is a Perron number. Lind has proved the converse, that is to say, if α is a Perron number, then there is a nonnegative aperiodic integral matrix whose spectral radius is α.Lind[LN2] has investigated the arithmetic of the Perron numbers.
    [Show full text]
  • **4************ from the Original Document
    DoCUMENT/ RESUME ED 220 67)9 CE 033 642 TITLE Military Curriculum Materials for Vocational and (14Technical Educaticin. Fundamentals of Digital Logic, 7-1. INSTITUTION Marine Corps, Washington, D.C.; Ohio State Univ., Columbus. National Center for Research in Vocational Education, SPONS AGENCY Office of Education (DHEW), Washington, D.C4. PUB DATE (76) NOTE 146p. EDR's PRICE MF01/P606 PlusPostase. DESCRIPTORS Arithmetic; *Digital'IComputers; *Electric Circuits; *Electronic Technicians; !Equipment Maintenance; Individualized Instruction; Instructional Materials; *Mathematical Lpgic; Mathematics; Military Personnel; Military TrainVng; Postsecondary Educationp Secondary Education; *Technical Education IDENTIFIERS Binary Arithmetic; Boolean Algebra; *Computer Logic; Military Curriculum Project ABSTRACT 'Tarqeted for grades0'through adult, these military-developed curriculum mater als consist ofa student lesson book with text readings and review pierVes designed to prepare electronic.per,sonnel for further trainin in digital'techniques. Covered,in the five lessons are binary arithmeti8 (number systems, decimal systemi, the mathematical form of notation, number system conversion, matheinatical operat4ons, specially coded systems); Boolean.a,lgebra (basic 'functions, signal levels, application of L Boolean algebra, Boolean equations, drawing logic diagrams, theorems and truth tables, simplifying Boolean equations); logic gates (diode logic circuits, transistor logic ,ci.rcuits, NOT circuits, EXCLUSIVE OR circu0s, operational analysis); logic
    [Show full text]
  • Appendix: the Cost of Computing: Big-O and Recursion
    Appendix: The Cost of Computing: Big-O and Recursion Big-O otation In the previous section we have seen a chart comparing execution times for various sorting algorithms. The chart, however, did not show any absolute numbers to tell us the actual time it took, say, to sort an array of 10,000 double numbers using the insertion sort algorithm. That is because the actual time depends, among other things, on the processor of the machine the program is running on, not only on the algorithm used. For example, the worst possible sorting algorithm executing on a Pentium II 450 MHz chip may finish a lot quicker in some situations than the best possible sorting algorithm running on an old Intel 386 chip. Even when two algorithms execute on the same machine, other processes that are running on the system can considerably skew the results one way or another. Therefore, execution time is not really an adequate measurement of the "goodness" of an algorithm. We need a more mathematical way to determine how good an algorithm is, and which of two algorithms is better. In this section, we will introduce the "big-O" notation, which does provide just that: a mathematical description that allows you to rate an algorithm. The Math behind Big-O There is a little background mathematics required to work with this concept, but it is not much, and you should quickly understand the major concepts involved. The most important concept you may need to review is that of the limit of a function. Definition: Limit of a Function We say that a function f converges to a limit L as x approaches a number c if f(x) gets closer and closer to L if x gets closer and closer to c, and we use the notation lim f (x) = L x→c Similarly, a function f converges to a limit L as x approaches positive (negative) infinity if f(x) gets closer and closer to L as x is positive (negative) and gets bigger (smaller) and bigger (smaller).
    [Show full text]
  • On Expansions in Non-Integer Base
    SAPIENZA UNIVERSITA` DI ROMA DOTTORATO DI RICERCA IN “MODELLI E METODI MATEMATICI PER LA TECNOLOGIA E LA SOCIETA` ” XXII CICLO AND UNIVERSITE´ PARIS 7-PARIS DIDEROT DOCTORAT D’INFORMATIQUE On expansions in non-integer base Anna Chiara Lai Dipartimento di Metodi e Modelli Matematici per le Scienze applicate Via Antonio Scarpa, 16 - 00161 Roma. and Laboratoire d’Informatique Algorithmique: Fondements et Applications CNRS UMR 7089, Universit´eParis Diderot - Paris 7, Case 7014 75205 Paris Cedex 13 A. A. 2008-2009 Contents Introduction 3 Chapter 1. Background results on expansions in non-integer base 6 1. Our toolbox 6 2. Positional number systems 10 3. Positional numeration systems with positive integer base 11 4. Expansions in non-integer bases 12 Chapter 2. Expansions in complex base 17 1. Introduction 17 2. Geometrical background 18 3. Characterization of the convex hull of representable numbers 24 4. Representability in complex base 35 5. Overview of original contributions, conclusions and further developments 37 Chapter 3. Expansions in negative base 40 1. Introduction 40 2. Preliminaries on expansions in non-integer negative base 41 3. Symbolic dynamical systems and the alternate order 44 4. A characterization of sofic ( q)-shifts and ( q)-shifts of finite type 46 − − 5. Entropy of the ( q)-shift 47 − 6. The Pisot case 49 7. A conversion algorithm from positive to negative base 52 8. Overview of original contributions, conclusions and further developments 56 Chapter4. GeneralizedGoldenMeanforternaryalphabets 58 1. Introduction 58 2. Expansions in non-integer base with alphabet with deleted digits 58 3. Critical bases 60 4. Normal ternary alphabets 62 5.
    [Show full text]
  • 3958 Weird Numbers
    3958 Weird Numbers Binary numbers form the principal basis of computer science. Most of you have heard of other systems, such as ternary, octal, or hexadecimal. You probably know how to use these systems and how to convert numbers between them. But did you know that the system base (radix) could also be negative? One assistant professor at the Czech Technical University has recently met negabinary numbers and other systems with a negative base. Will you help him to convert numbers to and from these systems? A number N written in the system with a positive base R will always appear as a string of digits between 0 and R − 1, inclusive. A digit at the position P (positions are counted from right to left and starting with zero) represents a value of RP . This means the value of the digit is multiplied by RP and values of all positions are summed together. For example, if we use the octal system (radix R = 8), a number written as 17024 has the following value: 1 ∗ 84 + 7 ∗ 83 + 0 ∗ 82 + 2 ∗ 81 + 4 ∗ 80 = 1 ∗ 4096 + 7 ∗ 512 + 2 ∗ 8 + 4 ∗ 1 = 7700 With a negative radix −R, the principle remains the same: each digit will have a value of (−R)P . For example, a negaoctal (radix R = −8) number 17024 counts as: 1 ∗ (−8)4 + 7 ∗ (−8)3 + 0 ∗ (−8)2 + 2 ∗ (−8)1 + 4 ∗ (−8)0 = 1 ∗ 4096 − 7 ∗ 512 − 2 ∗ 8 + 4 ∗ 1 = 500 One big advantage of systems with a negative base is that we do not need a minus sign to express negative numbers.
    [Show full text]
  • 1 Introduction 2 Place Value Representation
    Exotic Arithmetic, 2019 Fusing Dots, Antidots, and Black Holes 1 Introduction This paper discusses several methods of representing numbers, and several ways to understand these methods of representation.1 We begin with what is called decimal representation, the ordinary method we use to represent integers and fractions. Because the method of representation is an important starting point in learning the arithmetic of integers and decimals, we shall explore alternative methods of representation, that is, representation using bases other than our usual base 10 ten. This is roughly akin to the idea that one does not really understand one's own language until we learn a second language. Instead of trying to develop representation in an arbitrary base b, we select a specific base for the sake of clarity. This is base 5 representation. Also, we'll spend some time learning Martian numeration, base 6. Later we will discuss other representation including those for which the base b is not a positive integer. We also explore the system of enumeration when b is a rational number but not an integer, and then when b is a negative integer. Finally, we'll also see that it is even possible for b to be irrational. 2 Place Value Representation The place value interpretation of 4273 is 4000 + 200 + 70 + 3, which is a sum of multiples of powers of 10. The relevant powers of 10 are 103 = 1000; 102 = 100; 101 = 10; and 100 = 1. Each one has a coefficient or multiplier, 4; 2; 7; and 3, respectively. Thus 4 · 103; 2 · 102; 7 · 101; and 3 · 100 are multiples of powers of 10 and therefore 4273 is a sum of multiples of powers of 10.
    [Show full text]
  • New Families of Pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant
    AN ABSTRACT OF THE THESIS OF Hieu Trung Do for the degree of Doctor of Philosophy in Mathematics presented on May 27, 2016. Title: New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant. Abstract approved: Thomas A. Schmidt Translation surfaces can be viewed as polygons with parallel and equal sides identified. An affine homeomorphism φ from a translation surface to itself is called pseudo-Anosov when its derivative is a constant matrix in SL2(R) whose trace is larger than 2 in absolute value. In this setting, the eigendirections of this matrix defines the stable and unstable flow on the translation surface. Taking a transversal to the stable flows, the first return map of the flow induces an interval exchange transformation T . The Sah-Arnoux-Fathi invariant of φ is the sum of the wedge product between the lengths of the subintervals of T and their translations. This wedge product does not depend on the choice of transversal. We apply Veech's construction of pseudo-Anosov homeomorphisms to produce infinite families of pseudo-Anosov maps in the stratum H(2; 2) with vanishing Sah-Arnoux-Fathi invariant, as well as sporadic examples in other strata. ©Copyright by Hieu Trung Do May 27, 2016 All Rights Reserved New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant by Hieu Trung Do A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented May 27, 2016 Commencement June 2016 Doctor of Philosophy thesis of Hieu Trung Do presented on May 27, 2016 APPROVED: Major Professor, representing Mathematics Chair of the Department of Mathematics Dean of the Graduate School I understand that my thesis will become part of the permanent collection of Oregon State University libraries.
    [Show full text]
  • Combinatorics, Automata and Number Theory
    Combinatorics, Automata and Number Theory CANT Edited by Val´erie Berth´e LIRMM - Univ. Montpelier II - CNRS UMR 5506 161 rue Ada, 34392 Montpellier Cedex 5, France Michel Rigo Universit´edeLi`ege, Institut de Math´ematiques Grande Traverse 12 (B 37), B-4000 Li`ege, Belgium 2 Number representation and finite automata Christiane Frougny Univ. Paris 8 and LIAFA, Univ. Paris 7 - CNRS UMR 7089 Case 7014, F-75205 Paris Cedex 13, France Jacques Sakarovitch LTCI, CNRS/ENST - UMR 5141 46, rue Barrault, F-75634 Paris Cedex 13, France. Contents 2.1 Introduction 5 2.2 Representation in integer base 8 2.2.1 Representation of integers 8 2.2.2 The evaluator and the converters 11 2.2.3 Representation of reals 18 2.2.4 Base changing 23 2.3 Representation in real base 23 2.3.1 Symbolic dynamical systems 24 2.3.2 Real base 27 2.3.3 U-systems 35 2.3.4 Base changing 43 2.4 Canonical numeration systems 48 2.4.1 Canonical numeration systems in algebraic number fields 48 2.4.2 Normalisation in canonical numeration systems 50 2.4.3 Bases for canonical numeration systems 52 2.4.4 Shift radix systems 53 2.5 Representation in rational base 55 2.5.1 Representation of integers 55 2.5.2 Representation of the reals 61 2.6 A primer on finite automata and transducers 66 2.6.1 Automata 66 2.6.2 Transducers 68 2.6.3 Synchronous transducers and relations 69 2.6.4 The left-right duality 71 3 4 Ch.
    [Show full text]
  • Ito-Sadahiro Numbers Vs. Parry Numbers
    Acta Polytechnica Vol. 51 No. 4/2011 Ito-Sadahiro numbers vs. Parry numbers Z. Mas´akov´a, E. Pelantov´a Abstract We consider a positional numeration system with a negative base, as introduced by Ito and Sadahiro. In particular, we focus on the algebraic properties of negative bases −β for which the corresponding dynamical system is sofic, which β happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of − is eventually periodic. We call β +1 such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in thesamecontextfortheR´enyi positive base numeration system. Keywords: numeration systems, negative base, Pisot number, Parry number. ∈AN 1 Introduction According to Parry, the string x1x2x3 ... rep- resents the β-expansion of a number x ∈ [0, 1) if and The expansion of a real number in the positional only if number system with base β>1, as defined by R´enyi [12] is closely related to the transformation ∗ xixi+1xi+2 ...≺ d (1) (2) T :[0, 1) → [0, 1), given by the prescription T (x):= β βx −#βx$.Everyx ∈ [0, 1) is a sum of the infinite for every i =1, 2, 3,... series D { | ∞ Condition (2) ensures that the set β = dβ(x) xi − ∈ } D x = , where x = #βT i 1(x)$ (1) x [0, 1) is shift invariant, and so the closure of β βi i N i=1 in A , denoted by Sβ, is a subshift of the full shift N for i =1, 2, 3,... A . The notion of β-expansion can naturally be ex- Directly from the definition of the transformation T tended to all non-negative real numbers: The expres- we can derive that the ‘digits’ x take values in the set i sion of a positive real number y in the form {0, 1, 2,...,%β&−1} for i =1, 2, 3,...
    [Show full text]
  • On Expansions in Non-Integer Base
    SAPIENZA UNIVERSITA` DI ROMA DOTTORATO DI RICERCA IN “MODELLI E METODI MATEMATICI PER LA TECNOLOGIA E LA SOCIETA` ” XXII CICLO AND UNIVERSITE´ PARIS 7 - PARIS DIDEROT DOCTORAT D’INFORMATIQUE Abstract of the thesis On expansions in non-integer base Anna Chiara Lai Dipartimento di Metodi e Modelli Matematici per le Scienze applicate Via Antonio Scarpa, 16 - 00161 Roma. and Laboratoire d’Informatique Algorithmique: Fondements et Applications CNRS UMR 7089, Universit´eParis Diderot - Paris 7, Case 7014 75205 Paris Cedex 13 1 CONTENTS Introduction 1 1. Expansions in complex base 4 1.1. Introduction 4 1.2. Characterization of the convex hull of representable numbers 4 1.3. Representability in complex base 6 1.4. Conclusions and further developments 7 2. Expansions in negative base 7 2.1. Introduction 7 2.2. Symbolic dynamical systems and the alternate order 7 2.3. A characterization of sofic (¡q)-shifts and (¡q)-shifts of finite type 8 2.4. Entropy of the (¡q)-shift 8 2.5. The Pisot case 8 2.6. A conversion algorithm from positive to negative base 8 2.7. Conclusions and further developments 8 3. Generalized Golden Mean for ternary alphabets 8 3.1. Introduction 8 3.2. Critical bases 9 3.3. Normal ternary alphabets 9 3.4. Critical bases for ternary alphabets 9 3.5. Critical base and m-admissible sequences 10 3.6. Conclusions and further developments 10 4. Univoque sequences for ternary alphabets 10 Conclusions and further developments 11 References 11 INTRODUCTION This dissertation is devoted to the study of developments in series in the form ¥ x (1) i ∑ i i=1 q with the coefficients xi belonging to a finite set of positive real values named alphabet and the ratio q, named base, being either a real or a complex number.
    [Show full text]