DOCSLIB.ORG

Reciprocal Divisor Method 2007

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Reciprocal Divisor Method 2007 1 The Reciprocal Divisor Method for Abbreviation of Multiplication Tables 0 9 1 8 2 7 3 6 4 5 2 This booklet is dedicated to my wife, Laura Ann. First Edition, 3 October 2007 This booklet and all its art are copyright 2007 by author Michael Thomas De Vlieger, all rights reserved. No presentation or reproduction of any portion of this booklet is to be made without express permission from the author. This first edition of “Reciprocal Divisor Method” is a draft of the work and is presented exclusively at the 6 October 2007 meeting of the Dozenal Society of America on Long Island, New York. Michael Thomas De Vlieger Vinci LLC 5750 Delor Street 5106 Hampton Avenue Suite 205 Saint Louis, Missouri 63109-2830 Saint Louis, Missouri 63109-3315 314-351-7456 [email protected] www.vincico.com 3 0 7 1 6 2 5 3 4 The Reciprocal Divisor Method for Abbreviation of Multiplication Tables w x 0 1 2 0 u v 3 4 t 5 9 1 s 6 r 7 q 8 8 2 p 9 o A 7 3 n B m C 6 4 l D 5 k E j F 0 i G Ç 1 h H ç 2 g I f J e K 9 3 d L c M 8 4 b N a O 7 5 Z P 6 Y X Q W V U T S R F 0 1 E 2 D 3 C 4 B 5 A 6 9 8 7 Michael De Vlieger • 3 October 2007 • Saint Louis, Missouri 4 Contents Contents .............................................................................................................................. 4 Index of Figures and Tables............................................................................................ 6 Preface................................................................................................................................. 7 Part 1 • Properties of Integers ............................................................................................. 8 Integral Bases..................................................................................................................8 Prime Composition ..................................................................................................... 8 Divisors..................................................................................................................... 10 Compounding Divisibility and Rank ........................................................................ 11 Magnitude and Human Scale.................................................................................... 14 Reciprocal Divisors................................................................................................... 14 Totatives.................................................................................................................... 14 Part 2 • The Multiplication Fact Table.............................................................................. 19 Introduction................................................................................................................... 19 Rhythms in the Multiplication Table ............................................................................ 21 Multiples or First-Rank Digits.................................................................................. 21 Units or Zero-Rank Digits ........................................................................................ 22 Classification of the Factors f of Any Integer Base r.................................................... 23 Factors which are Divisors of the Base (fd) .................................................................. 25 Cyclical Unit Digits in Products of fd ....................................................................... 25 Complementary Relationships within Reciprocal Divisor Pairs .............................. 25 The “Unity-Identity” Divisors as Factors ..................................................................... 26 Factors which are Unity (f1)...................................................................................... 26 Factors which are “Identity” (fr )............................................................................... 26 Factor Classes for all fd ............................................................................................. 27 Summary – Divisor Factors ...................................................................................... 27 Higher Rank Divisors ............................................................................................... 28 Factors which are both Nondivisors and Nontotative (fn) ............................................ 29 Summary – Nontotative Nondivisor Factors ............................................................ 31 Totative Factors (ft)....................................................................................................... 32 Factor Classes for the Totative Factors ft.................................................................. 32 Summary – Totative Factors..................................................................................... 32 Factor Classes ............................................................................................................... 33 Factor Class A: The Unity-Identity Divisors............................................................ 33 Factor Class B: The Effective Divisors .................................................................... 33 Factor Class C: The Effective Non Divisor Non Totatives ...................................... 34 Factor Class D: Totatives and Ineffective Nontotative Nondivisors ........................ 34 Summary....................................................................................................................... 36 Part 3 • Abbreviation of Multiplication Fact Tables......................................................... 38 Introduction................................................................................................................... 38 The Full Multiplication Table....................................................................................... 38 Truncation of the Table to the First Period................................................................... 39 Reduction to Eliminate Restatements ........................................................................... 39 The Minimal Abbreviated Table............................................................................... 39 The Crossing Abbreviated Table .............................................................................. 40 5 Abbreviated Tables ................................................................................................... 41 Summary................................................................................................................... 44 Part 4 • Operation Classes................................................................................................. 45 Know Thy Base’s Divisor Pairs.................................................................................... 45 Operation Class A: Products within the Abbreviated Table......................................... 45 Computation Process ................................................................................................ 46 Dozenal Examples .................................................................................................... 46 Operation Class B: Effective Divisor Factors............................................................... 47 Operation Class B • Dozenal Examples.................................................................... 48 Operation Class B • Sexagesimal Examples............................................................. 48 Operation Class C: Effective Non Divisor Non Totative Factors................................. 50 Operation Class C • Dozenal Examples.................................................................... 51 Operation Class C • Sexagesimal Examples............................................................. 52 Method for Determining Effective Factors fe ........................................................... 53 Operation Class D: Totatives and fn Involving Totatives ............................................. 54 Operation Class D • Dozenal Examples ................................................................... 56 Operation Class D • Sexagesimal Examples............................................................. 57 Multiplication Processes ............................................................................................... 58 Division Processes ........................................................................................................ 59 Operation Class A: Dividends within the Abbreviated Table ...................................... 59 Computation Process ................................................................................................ 59 Operation Class B: Effective Divisor Problem-Divisors.............................................. 60 Operation Class B • Dozenal Examples.................................................................... 60 Operation Class B • Sexagesimal Examples............................................................. 62 Operation Class C: Non Totative Non Divisor Problem-Divisors................................ 63 Operation Class C • Dozenal Examples.................................................................... 64 Operation Class C • Sexagesimal Examples............................................................. 65 Operation Class D: Totative Problem-Divisors ............................................................ 66 Operation Class D • Examples.................................................................................. 66 Range of Effectiveness for the RDM...........................................................................
Load more

Recommended publications
  • Dynamical Directions in Numeration Tome 56, No 7 (2006), P
    R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Guy BARAT, Valérie BERTHÉ, Pierre LIARDET & Jörg THUSWALDNER Dynamical directions in numeration Tome 56, no 7 (2006), p. 1987-2092. <http://aif.cedram.org/item?id=AIF_2006__56_7_1987_0> © Association des Annales de l’institut Fourier, 2006, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 56, 7 (2006) 1987-2092 DYNAMICAL DIRECTIONS IN NUMERATION by Guy BARAT, Valérie BERTHÉ, Pierre LIARDET & Jörg THUSWALDNER (*) Abstract. — This survey aims at giving a consistent presentation of numer- ation from a dynamical viewpoint: we focus on numeration systems, their asso- ciated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to β-numeration and its gener- alisations, abstract numeration systems and shift radix systems, as well as G-scales and odometers.
    [Show full text]
  • Nieuw Archief Voor Wiskunde
    Nieuw Archief voor Wiskunde Boekbespreking Kevin Broughan Equivalents of the Riemann Hypothesis Volume 1: Arithmetic Equivalents Cambridge University Press, 2017 xx + 325 p., prijs £ 99.99 ISBN 9781107197046 Kevin Broughan Equivalents of the Riemann Hypothesis Volume 2: Analytic Equivalents Cambridge University Press, 2017 xix + 491 p., prijs £ 120.00 ISBN 9781107197121 Reviewed by Pieter Moree These two volumes give a survey of conjectures equivalent to the ber theorem says that r()x asymptotically behaves as xx/log . That Riemann Hypothesis (RH). The first volume deals largely with state- is a much weaker statement and is equivalent with there being no ments of an arithmetic nature, while the second part considers zeta zeros on the line v = 1. That there are no zeros with v > 1 is more analytic equivalents. a consequence of the prime product identity for g()s . The Riemann zeta function, is defined by It would go too far here to discuss all chapters and I will limit 3 myself to some chapters that are either close to my mathematical 1 (1) g()s = / s , expertise or those discussing some of the most famous RH equiv- n = 1 n alences. Most of the criteria have their own chapter devoted to with si=+v t a complex number having real part v > 1 . It is easily them, Chapter 10 has various criteria that are discussed more brief- seen to converge for such s. By analytic continuation the Riemann ly. A nice example is Redheffer’s criterion. It states that RH holds zeta function can be uniquely defined for all s ! 1.
    [Show full text]
  • The Modal Logic of Potential Infinity, with an Application to Free Choice
    The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity.
    [Show full text]
  • Basic Computer Arithmetic
    BASIC COMPUTER ARITHMETIC TSOGTGEREL GANTUMUR Abstract. First, we consider how integers and fractional numbers are represented and manipulated internally on a computer. The focus is on the principles behind the algorithms, rather than on implementation details. Then we develop a basic theoretical framework for analyzing algorithms involving inexact arithmetic. Contents 1. Introduction 1 2. Integers 2 3. Simple division algorithms 6 4. Long division 11 5. The SRT division algorithm 16 6. Floating point numbers 18 7. Floating point arithmetic 21 8. Propagation of error 24 9. Summation and product 29 1. Introduction There is no way to encode all real numbers by using finite length words, even if we use an alphabet with countably many characters, because the set of finite sequences of integers is countable. Fortunately, the real numbers support many countable dense subsets, and hence the encoding problem of real numbers may be replaced by the question of choosing a suitable countable dense subset. Let us look at some practical examples of real number encodings. ● Decimal notation. Examples: 36000, 2:35(75), −0:000072. ● Scientific notation. Examples: 3:6⋅104, −72⋅10−6. The general form is m⋅10e. In order to have a unique (or near unique) representation of each number, one can impose a normalization, such as requiring 1 ≤ ∣m∣ < 10. 2 −1 ● System with base/radix β. Example: m2m1m0:m−1 = m2β +m1β +m0 +m−1β . The dot separating the integer and fractional parts is called the radix point. ● Binary (β = 2), octal (β = 8), and hexadecimal (β = 16) numbers. ● Babylonian hexagesimal (β = 60) numbers.
    [Show full text]
  • Sequences and Their Limits
    Sequences and their limits c Frank Zorzitto, Faculty of Mathematics University of Waterloo The limit idea For the purposes of calculus, a sequence is simply a list of numbers x1; x2; x3; : : : ; xn;::: that goes on indefinitely. The numbers in the sequence are usually called terms, so that x1 is the first term, x2 is the second term, and the entry xn in the general nth position is the nth term, naturally. The subscript n = 1; 2; 3;::: that marks the position of the terms will sometimes be called the index. We shall deal only with real sequences, namely those whose terms are real numbers. Here are some examples of sequences. • the sequence of positive integers: 1; 2; 3; : : : ; n; : : : • the sequence of primes in their natural order: 2; 3; 5; 7; 11; ::: • the decimal sequence that estimates 1=3: :3;:33;:333;:3333;:33333;::: • a binary sequence: 0; 1; 0; 1; 0; 1;::: • the zero sequence: 0; 0; 0; 0;::: • a geometric sequence: 1; r; r2; r3; : : : ; rn;::: 1 −1 1 (−1)n • a sequence that alternates in sign: 2 ; 3 ; 4 ;:::; n ;::: • a constant sequence: −5; −5; −5; −5; −5;::: 1 2 3 4 n • an increasing sequence: 2 ; 3 ; 4 ; 5 :::; n+1 ;::: 1 1 1 1 • a decreasing sequence: 1; 2 ; 3 ; 4 ;:::; n ;::: 3 2 4 3 5 4 n+1 n • a sequence used to estimate e: ( 2 ) ; ( 3 ) ; ( 4 ) :::; ( n ) ::: 1 • a seemingly random sequence: sin 1; sin 2; sin 3;:::; sin n; : : : • the sequence of decimals that approximates π: 3; 3:1; 3:14; 3:141; 3:1415; 3:14159; 3:141592; 3:1415926; 3:14159265;::: • a sequence that lists all fractions between 0 and 1, written in their lowest form, in groups of increasing denominator with increasing numerator in each group: 1 1 2 1 3 1 2 3 4 1 5 1 2 3 4 5 6 1 3 5 7 1 2 4 5 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;::: 2 3 3 4 4 5 5 5 5 6 6 7 7 7 7 7 7 8 8 8 8 9 9 9 9 It is plain to see that the possibilities for sequences are endless.
    [Show full text]
  • The Mayan Long Count Calendar Thomas Chanier
    The Mayan Long Count Calendar Thomas Chanier To cite this version: Thomas Chanier. The Mayan Long Count Calendar. 2015. hal-00750006v11 HAL Id: hal-00750006 https://hal.archives-ouvertes.fr/hal-00750006v11 Preprint submitted on 8 Dec 2015 (v11), last revised 16 Dec 2015 (v12) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Mayan Long Count Calendar Thomas Chanier∗1 1 Universit´ede Brest, 6 avenue Victor le Gorgeu, F-29285 Brest Cedex, France The Mayan Codices, bark-paper books from the Late Postclassic period (1300 to 1521 CE) contain many astronomical tables correlated to ritual cycles, evidence of the achievement of Mayan naked- eye astronomy and mathematics in connection to religion. In this study, a calendar supernumber is calculated by computing the least common multiple of 8 canonical astronomical periods. The three major calendar cycles, the Calendar Round, the Kawil and the Long Count Calendar are shown to derive from this supernumber. The 360-day Tun, the 365-day civil year Haab’ and the 3276-day Kawil-direction-color cycle are determined from the prime factorization of the 8 canonical astronomical input parameters.
    [Show full text]
  • Basic Analysis I: Introduction to Real Analysis, Volume I
    Basic Analysis I Introduction to Real Analysis, Volume I by Jiríˇ Lebl June 8, 2021 (version 5.4) 2 Typeset in LATEX. Copyright ©2009–2021 Jiríˇ Lebl This work is dual licensed under the Creative Commons Attribution-Noncommercial-Share Alike 4.0 International License and the Creative Commons Attribution-Share Alike 4.0 International License. To view a copy of these licenses, visit https://creativecommons.org/licenses/ by-nc-sa/4.0/ or https://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons PO Box 1866, Mountain View, CA 94042, USA. You can use, print, duplicate, share this book as much as you want. You can base your own notes on it and reuse parts if you keep the license the same. You can assume the license is either the CC-BY-NC-SA or CC-BY-SA, whichever is compatible with what you wish to do, your derivative works must use at least one of the licenses. Derivative works must be prominently marked as such. During the writing of this book, the author was in part supported by NSF grants DMS-0900885 and DMS-1362337. The date is the main identifier of version. The major version / edition number is raised only if there have been substantial changes. Each volume has its own version number. Edition number started at 4, that is, version 4.0, as it was not kept track of before. See https://www.jirka.org/ra/ for more information (including contact information, possible updates and errata). The LATEX source for the book is available for possible modification and customization at github: https://github.com/jirilebl/ra Contents Introduction 5 0.1 About this book ....................................5 0.2 About analysis ....................................7 0.3 Basic set theory ....................................8 1 Real Numbers 21 1.1 Basic properties ...................................
    [Show full text]
  • What Will Count As Mathematics in 2100?
    What will count as mathematics in 2100? Keith Devlin What is mathematics today? What is mathematics? That’s one of the most basic questions in the philosophy of mathematics. The answer has changed several times throughout history. Up to 500 B.C. or thereabouts, mathematics was—if it was anything to be given a name—the systematic use of numbers. This was the period of Egyptian, Babylonian, and early Chinese and Japanese mathematics. In those civilizations, mathematics consisted primarily of arithmetic. It was largely utilitarian, and very much of a cookbook variety. (“Do such and such to a number and you will get the answer.”) Modern mathematics, as an area of study, traces its lineage to the ancient Greeks of the period from around 500 B.C. to 300 A.D. From the perspective of what is classified as mathematics today, the ancient Greeks focused on properties of number and shape (geometry). [The word “mathematics” itself comes from the Greek for “that which is learnable.” As always when interpreting one culture or age with another, it has to be acknowledged that things often appear quite different to those within a particular culture or age than when viewed from the other.] It was with the Greeks that mathematics came into being as an identifiable discipline, and not just a collection of techniques for measuring, counting, and accounting. Greek interest in mathematics was not just utilitarian; they regarded mathematics as an intellectual pursuit having both aesthetic and religious elements. Around 500 B.C., Thales of Miletus (now part of Turkey) introduced the idea that the precisely stated assertions of mathematics could be logically proved by a formal argument.
    [Show full text]
  • Ed 040 737 Institution Available from Edrs Price
    DOCUMENT RESUME ED 040 737 LI 002 060 TITLE Automatic Data Processing Glossary. INSTITUTION Bureau of the Budget, Washington, D.C. NOTE 65p. AVAILABLE FROM Reprinted and distributed by Datamation Magazine, 35 Mason St., Greenwich, Conn. 06830 ($1.00) EDRS PRICE EDRS Price MF-$0.50 HC-$3.35 DESCRIPTORS *Electronic Data Processing, *Glossaries, *Word Lists ABSTRACT The technology of the automatic information processing field has progressed dramatically in the past few years and has created a problem in common term usage. As a solution, "Datamation" Magazine offers this glossary which was compiled by the U.S. Bureau of the Budget as an official reference. The terms appear in a single alphabetic sequence, ignoring commas or hyphens. Definitions are given only under "key word" entries. Modifiers consisting of more than one word are listed in the normally used sequence (record, fixed length). In cases where two or more terms have the same meaning, only the preferred term is defined, all synonylious terms are given at the end of the definition.Other relationships between terms are shown by descriptive referencing expressions. Hyphens are used sparingly to avoid ambiguity. The derivation of an acronym is shown by underscoring the appropriate letters in the words from which the acronym is formed. Although this glossary is several years old, it is still considered the best one available. (NH) Ns U.S DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGINATING IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECES- SARILY REPRESENT OFFICIAL OFFICE OF EDU- CATION POSITION OR POLICY automatic data processing GLOSSA 1 I.R DATAMATION Magazine reprints this Glossary of Terms as a service to the data processing field.
    [Show full text]
  • Surprises from Mathematics Education Research: Student (Mis)Use of Mathematical Definitions Barbara S
    Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Definitions Barbara S. Edwards and Michael B. Ward 1. INTRODUCTION. The authors of this paper met at a summer institute sponsored by the Oregon Collaborative for Excellence in the Preparation of Teachers (OCEPT). Edwards is a researcher in undergraduate mathematics education. Ward, a pure math- ematician teaching at an undergraduate institution, had had little exposure to math- ematics education research prior to the OCEPT program. At the institute, Edwards described to Ward the results of her Ph.D. dissertation [5] on student understanding and use of definitions in undergraduate real analysis. In that study, tasks involving the definitions of “limit” and “continuity,” for example, were problematic for some of the students. Ward’s intuitive reaction was that those words were “loaded” with conno- tations from their nonmathematical use and from their less than completely rigorous use in elementary calculus. He said, “I’ll bet students have less difficulty or, at least, different difficulties with definitions in abstract algebra. The words there, like ‘group’ and ‘coset,’ are not so loaded.” Eventually, with OCEPT support, the authors studied student understanding and use of definitions in an introductory abstract algebra course populated by undergradu- ate mathematics majors and taught by Ward. The “surprises” in the title are outcomes that surprised Ward, among others. He was surprised to see his algebra students having difficulties very similar to those of Edwards’s analysis students. (So he lost his bet.) In particular, he was surprised to see difficulties arising from the students’ understanding of the very nature of mathematical definitions, not just from the content of the defini- tions.
    [Show full text]
  • Analogues of the Robin-Lagarias Criteria for the Riemann Hypothesis
    Analogues of the Robin-Lagarias Criteria for the Riemann Hypothesis Lawrence C. Washington1 and Ambrose Yang2 1Department of Mathematics, University of Maryland 2Montgomery Blair High School August 12, 2020 Abstract Robin's criterion states that the Riemann hypothesis is equivalent to σ(n) < eγ n log log n for all integers n ≥ 5041, where σ(n) is the sum of divisors of n and γ is the Euler-Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that σ(n) < eγ 4 3 2 2 n log log n for all odd numbers n ≥ 3 · 5 · 7 · 11 ··· 67. Lagarias's criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to σ(n) < Hn +exp Hn log Hn for all integers n ≥ 1, where Hn is the nth harmonic number. We establish an analogue to Lagarias's 0 criterion for the Riemann hypothesis by creating a new harmonic series Hn = 2Hn − H2n and 3n 0 0 demonstrating that the Riemann hypothesis is equivalent to σ(n) ≤ log n + exp Hn log Hn for all odd n ≥ 3. We prove stronger analogues to Robin's inequality for odd squarefree numbers. Furthermore, we find a general formula that studies the effect of the prime factorization of n and its behavior in Robin's inequality. 1 Introduction The Riemann hypothesis, an unproven conjecture formulated by Bernhard Riemann in 1859, states that all non-real zeroes of the Riemann zeta function 1 X 1 ζ(s) = ns n=1 1 arXiv:2008.04787v1 [math.NT] 11 Aug 2020 have real part 2 .
    [Show full text]
  • The Reverse Mathematics of Non-Decreasing Subsequences Ludovic Patey
    The reverse mathematics of non-decreasing subsequences Ludovic Patey To cite this version: Ludovic Patey. The reverse mathematics of non-decreasing subsequences. Archive for Mathematical Logic, Springer Verlag, 2017, 56 (5-6), pp.491 - 506. 10.1007/s00153-017-0536-9. hal-01888765 HAL Id: hal-01888765 https://hal.archives-ouvertes.fr/hal-01888765 Submitted on 5 Oct 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THE REVERSE MATHEMATICS OF NON-DECREASING SUBSEQUENCES LUDOVIC PATEY Abstract. Every function over the natural numbers has an infinite subdo- main on which the function is non-decreasing. Motivated by a question of Dzhafarov and Schweber, we study the reverse mathematics of variants of this statement. It turns out that this statement restricted to computably bounded functions is computationally weak and does not imply the existence of the halting set. On the other hand, we prove that it is not a consequence of Ramsey's theorem for pairs. This statement can therefore be seen as an arguably natural principle between the arithmetic comprehension axiom and stable Ramsey's theorem for pairs. 1. Introduction A non-decreasing subsequence for a function f : N ! N is a set X ⊆ N such that f(x) ≤ f(y) for every pair x < y 2 X.
    [Show full text]