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![Arxiv:1906.06684V1 [Math.NA]](https://docslib.b-cdn.net/cover/7444/arxiv-1906-06684v1-math-na-37444.webp)
Arxiv:1906.06684V1 [Math.NA]
Randomized Computation of Continuous Data: Is Brownian Motion Computable?∗ Willem L. Fouch´e1, Hyunwoo Lee2, Donghyun Lim2, Sewon Park2, Matthias Schr¨oder3, Martin Ziegler2 1 University of South Africa 2 KAIST 3 University of Birmingham Abstract. We consider randomized computation of continuous data in the sense of Computable Analysis. Our first contribution formally confirms that it is no loss of generality to take as sample space the Cantor space of infinite fair coin flips. This extends [Schr¨oder&Simpson’05] and [Hoyrup&Rojas’09] considering sequences of suitably and adaptively biased coins. Our second contribution is concerned with 1D Brownian Motion (aka Wiener Process), a prob- ability distribution on the space of continuous functions f : [0, 1] → R with f(0) = 0 whose computability has been conjectured [Davie&Fouch´e’2013; arXiv:1409.4667,§6]. We establish that this (higher-type) random variable is computable iff some/every computable family of moduli of continuity (as ordinary random variables) has a computable probability distribution with respect to the Wiener Measure. 1 Introduction Randomization is a powerful technique in classical (i.e. discrete) Computer Science: Many difficult problems have turned out to admit simple solutions by algorithms that ‘roll dice’ and are efficient/correct/optimal with high probability [DKM+94,BMadHS99,CS00,BV04]. Indeed, fair coin flips have been shown computationally universal [Wal77]. Over continuous data, well-known closely connected to topology [Grz57] [Wei00, 2.2+ 3], notions of proba- § § bilistic computation are more subtle [BGH15,Col15]. 1.1 Overview Section 2 resumes from [SS06] the question of how to represent Borel probability measures. -
1 Mersenne Primes and Perfect Numbers
1 Mersenne Primes and Perfect Numbers Basic idea: try to construct primes of the form an − 1; a, n ≥ 1. e.g., 21 − 1 = 3 but 24 − 1=3· 5 23 − 1=7 25 − 1=31 26 − 1=63=32 · 7 27 − 1 = 127 211 − 1 = 2047 = (23)(89) 213 − 1 = 8191 Lemma: xn − 1=(x − 1)(xn−1 + xn−2 + ···+ x +1) Corollary:(x − 1)|(xn − 1) So for an − 1tobeprime,weneeda =2. Moreover, if n = md, we can apply the lemma with x = ad.Then (ad − 1)|(an − 1) So we get the following Lemma If an − 1 is a prime, then a =2andn is prime. Definition:AMersenne prime is a prime of the form q =2p − 1,pprime. Question: are they infinitely many Mersenne primes? Best known: The 37th Mersenne prime q is associated to p = 3021377, and this was done in 1998. One expects that p = 6972593 will give the next Mersenne prime; this is close to being proved, but not all the details have been checked. Definition: A positive integer n is perfect iff it equals the sum of all its (positive) divisors <n. Definition: σ(n)= d|n d (divisor function) So u is perfect if n = σ(u) − n, i.e. if σ(u)=2n. Well known example: n =6=1+2+3 Properties of σ: 1. σ(1) = 1 1 2. n is a prime iff σ(n)=n +1 p σ pj p ··· pj pj+1−1 3. If is a prime, ( )=1+ + + = p−1 4. (Exercise) If (n1,n2)=1thenσ(n1)σ(n2)=σ(n1n2) “multiplicativity”. -
Prime Divisors in the Rationality Condition for Odd Perfect Numbers
Aid#59330/Preprints/2019-09-10/www.mathjobs.org RFSC 04-01 Revised The Prime Divisors in the Rationality Condition for Odd Perfect Numbers Simon Davis Research Foundation of Southern California 8861 Villa La Jolla Drive #13595 La Jolla, CA 92037 Abstract. It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k + 4m+1 ℓ 2αi 1) i=1 qi to establish that there do not exist any odd integers with equality (4k+1)4m+2−1 between σ(N) and 2N. The existence of distinct prime divisors in the repunits 4k , 2α +1 Q q i −1 i , i = 1,...,ℓ, in σ(N) follows from a theorem on the primitive divisors of the Lucas qi−1 sequences and the square root of the product of 2(4k + 1), and the sequence of repunits will not be rational unless the primes are matched. Minimization of the number of prime divisors in σ(N) yields an infinite set of repunits of increasing magnitude or prime equations with no integer solutions. MSC: 11D61, 11K65 Keywords: prime divisors, rationality condition 1. Introduction While even perfect numbers were known to be given by 2p−1(2p − 1), for 2p − 1 prime, the universality of this result led to the the problem of characterizing any other possible types of perfect numbers. It was suggested initially by Descartes that it was not likely that odd integers could be perfect numbers [13]. After the work of de Bessy [3], Euler proved σ(N) that the condition = 2, where σ(N) = d|N d is the sum-of-divisors function, N d integer 4m+1 2α1 2αℓ restricted odd integers to have the form (4kP+ 1) q1 ...qℓ , with 4k + 1, q1,...,qℓ prime [18], and further, that there might exist no set of prime bases such that the perfect number condition was satisfied. -
Ces`Aro's Integral Formula for the Bell Numbers (Corrected)
Ces`aro’s Integral Formula for the Bell Numbers (Corrected) DAVID CALLAN Department of Statistics University of Wisconsin-Madison Medical Science Center 1300 University Ave Madison, WI 53706-1532 [email protected] October 3, 2005 In 1885, Ces`aro [1] gave the remarkable formula π 2 cos θ N = ee cos(sin θ)) sin( ecos θ sin(sin θ) ) sin pθ dθ p πe Z0 where (Np)p≥1 = (1, 2, 5, 15, 52, 203,...) are the modern-day Bell numbers. This formula was reproduced verbatim in the Editorial Comment on a 1941 Monthly problem [2] (the notation Np for Bell number was still in use then). I have not seen it in recent works and, while it’s not very profound, I think it deserves to be better known. Unfortunately, it contains a typographical error: a factor of p! is omitted. The correct formula, with n in place of p and using Bn for Bell number, is π 2 n! cos θ B = ee cos(sin θ)) sin( ecos θ sin(sin θ) ) sin nθ dθ n ≥ 1. n πe Z0 eiθ The integrand is the imaginary part of ee sin nθ, and so an equivalent formula is π 2 n! eiθ B = Im ee sin nθ dθ . (1) n πe Z0 The formula (1) is quite simple to prove modulo a few standard facts about set par- n titions. Recall that the Stirling partition number k is the number of partitions of n n [n] = {1, 2,...,n} into k nonempty blocks and the Bell number Bn = k=1 k counts n k n n k k all partitions of [ ]. -
ON SOME GAUSSIAN PELL and PELL-LUCAS NUMBERS Abstract
Ordu Üniv. Bil. Tek. Derg., Cilt:6, Sayı:1, 2016,8-18/Ordu Univ. J. Sci. Tech., Vol:6, No:1,2016,8-18 ON SOME GAUSSIAN PELL AND PELL-LUCAS NUMBERS Serpil Halıcı*1 Sinan Öz2 1Pamukkale Uni., Science and Arts Faculty,Dept. of Math., KınıklıCampus, Denizli, Turkey 2Pamukkale Uni., Science and Arts Faculty,Dept. of Math., Denizli, Turkey Abstract In this study, we consider firstly the generalized Gaussian Fibonacci and Lucas sequences. Then we define the Gaussian Pell and Gaussian Pell-Lucas sequences. We give the generating functions and Binet formulas of Gaussian Pell and Gaussian Pell- Lucas sequences. Moreover, we obtain some important identities involving the Gaussian Pell and Pell-Lucas numbers. Keywords. Recurrence Relation, Fibonacci numbers, Gaussian Pell and Pell-Lucas numbers. Özet Bu çalışmada, önce genelleştirilmiş Gaussian Fibonacci ve Lucas dizilerini dikkate aldık. Sonra, Gaussian Pell ve Gaussian Pell-Lucas dizilerini tanımladık. Gaussian Pell ve Gaussian Pell-Lucas dizilerinin Binet formüllerini ve üreteç fonksiyonlarını verdik. Üstelik, Gaussian Pell ve Gaussian Pell-Lucas sayılarını içeren bazı önemli özdeşlikler elde ettik. AMS Classification. 11B37, 11B39.1 * [email protected], 8 S. Halıcı, S. Öz 1. INTRODUCTION From (Horadam 1961; Horadam 1963) it is well known Generalized Fibonacci sequence {푈푛}, 푈푛+1 = 푝푈푛 + 푞푈푛−1 , 푈0 = 0 푎푛푑 푈1 = 1, and generalized Lucas sequence {푉푛} are defined by 푉푛+1 = 푝푉푛 + 푞푉푛−1 , 푉0 = 2 푎푛푑 푉1 = 푝, where 푝 and 푞 are nonzero real numbers and 푛 ≥ 1. For 푝 = 푞 = 1, we have classical Fibonacci and Lucas sequences. For 푝 = 2, 푞 = 1, we have Pell and Pell- Lucas sequences. -
Mersenne and Fermat Numbers 2, 3, 5, 7, 13, 17, 19, 31
MERSENNE AND FERMAT NUMBERS RAPHAEL M. ROBINSON 1. Mersenne numbers. The Mersenne numbers are of the form 2n — 1. As a result of the computation described below, it can now be stated that the first seventeen primes of this form correspond to the following values of ra: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281. The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127—1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in Archibald [l]; see also Kraitchik [4]. The next five Mersenne primes were found in 1952; they are at present the five largest known primes of any form. They were announced in Lehmer [7] and discussed by Uhler [13]. It is clear that 2" —1 can be factored algebraically if ra is composite; hence 2n —1 cannot be prime unless w is prime. Fermat's theorem yields a factor of 2n —1 only when ra + 1 is prime, and hence does not determine any additional cases in which 2"-1 is known to be com- posite. On the other hand, it follows from Euler's criterion that if ra = 0, 3 (mod 4) and 2ra + l is prime, then 2ra + l is a factor of 2n— 1. -
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. -
MAT344 Lecture 6
MAT344 Lecture 6 2019/May/22 1 Announcements 2 This week This week, we are talking about 1. Recursion 2. Induction 3 Recap Last time we talked about 1. Recursion 4 Fibonacci numbers The famous Fibonacci sequence starts like this: 1; 1; 2; 3; 5; 8; 13;::: The rule defining the sequence is F1 = 1;F2 = 1, and for n ≥ 3, Fn = Fn−1 + Fn−2: This is a recursive formula. As you might expect, if certain kinds of numbers have a name, they answer many counting problems. Exercise 4.1 (Example 3.2 in [KT17]). Show that a 2 × n checkerboard can be tiled with 2 × 1 dominoes in Fn+1 many ways. Solution: Denote the number of tilings of a 2 × n rectangle by Tn. We check that T1 = 1 and T2 = 2. We want to prove that they satisfy the recurrence relation Tn = Tn−1 + Tn−2: Consider the domino occupying the rightmost spot in the top row of the tiling. It is either a vertical domino, in which case the rest of the tiling can be interpreted as a tiling of a 2 × (n − 1) rectangle, or it is a horizontal domino, in which case there must be another horizontal domino under it, and the rest of the tiling can be interpreted as a tiling of a 2 × (n − 2) rectangle. Therefore Tn = Tn−1 + Tn−2: Since the number of tilings satisfies the same recurrence relation as the Fibonacci numbers, and T1 = F2 = 1 and T2 = F3 = 2, we may conclude that Tn = Fn+1. -
Mathematical Circus & 'Martin Gardner
MARTIN GARDNE MATHEMATICAL ;MATH EMATICAL ASSOCIATION J OF AMERICA MATHEMATICAL CIRCUS & 'MARTIN GARDNER THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, DC 1992 MATHEMATICAL More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments from Scientific American with a Preface by Donald Knuth, A Postscript, from the Author, and a new Bibliography by Mr. Gardner, Thoughts from Readers, and 105 Drawings and Published in the United States of America by The Mathematical Association of America Copyright O 1968,1969,1970,1971,1979,1981,1992by Martin Gardner. All riglhts reserved under International and Pan-American Copyright Conventions. An MAA Spectrum book This book was updated and revised from the 1981 edition published by Vantage Books, New York. Most of this book originally appeared in slightly different form in Scientific American. Library of Congress Catalog Card Number 92-060996 ISBN 0-88385-506-2 Manufactured in the United States of America For Donald E. Knuth, extraordinary mathematician, computer scientist, writer, musician, humorist, recreational math buff, and much more SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications ANDREW STERRETT, JR.,Chairman Spectrum Editorial Board ROGER HORN, Chairman SABRA ANDERSON BART BRADEN UNDERWOOD DUDLEY HUGH M. EDGAR JEANNE LADUKE LESTER H. LANGE MARY PARKER MPP.a (@ SPECTRUM Also by Martin Gardner from The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, D. C. 20036 (202) 387- 5200 Riddles of the Sphinx and Other Mathematical Puzzle Tales Mathematical Carnival Mathematical Magic Show Contents Preface xi .. Introduction Xlll 1. Optical Illusions 3 Answers on page 14 2. Matches 16 Answers on page 27 3. -
And Its Properties on the Sequence Related to Lucas Numbers
Mathematica Aeterna, Vol. 2, 2012, no. 1, 63 - 75 On the sequence related to Lucas numbers and its properties Cennet Bolat Department of Mathematics, Art and Science Faculty, Mustafa Kemal University, 31034, Hatay,Turkey Ahmet I˙pek Department of Mathematics, Art and Science Faculty, Mustafa Kemal University, 31034, Hatay,Turkey Hasan Köse Department of Mathematics, Science Faculty, Selcuk University 42031, Konya,Turkey Abstract The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recur- rence relation. In this article, we study a new generalization {Lk,n}, with initial conditions Lk,0 = 2 and Lk,1 = 1, which is generated by the recurrence relation Lk,n = kLk,n−1 + Lk,n−2 for n ≥ 2, where k is integer number. Some well-known sequence are special case of this generalization. The Lucas sequence is a special case of {Lk,n} with k = 1. Modified Pell-Lucas sequence is {Lk,n} with k = 2. We produce an extended Binet’s formula for {Lk,n} and, thereby, identities such as Cassini’s, Catalan’s, d’Ocagne’s, etc. using matrix algebra. Moreover, we present sum formulas concerning this new generalization. Mathematics Subject Classi…cation: 11B39, 15A23. Keywords: Classic Fibonacci numbers; Classic Lucas numbers; k-Fibonacci numbers; k-Lucas numbers, Matrix algebra. 64 C. Bolat, A. Ipek and H. Kose 1 Introduction In recent years, many interesting properties of classic Fibonacci numbers, clas- sic Lucas numbers and their generalizations have been shown by researchers and applied to almost every …eld of science and art. -
Adopting Number Sequences for Shielding Information
Manish Bansal et al, International Journal of Computer Science and Mobile Computing, Vol.3 Issue.11, November- 2014, pg. 660-667 Available Online at www.ijcsmc.com International Journal of Computer Science and Mobile Computing A Monthly Journal of Computer Science and Information Technology ISSN 2320–088X IJCSMC, Vol. 3, Issue. 11, November 2014, pg.660 – 667 RESEARCH ARTICLE Adopting Number Sequences for Shielding Information Mrs. Sandhya Maitra, Mr. Manish Bansal, Ms. Preety Gupta Associate Professor - Department of Computer Applications and Institute of Information Technology and Management, India Student - Department of Computer Applications and Institute of Information Technology and Management, India Student - Department of Computer Applications and Institute of Information Technology and Management, India [email protected] , [email protected] , [email protected] Abstract:-The advancement of technology and global communication networks puts up the question of safety of conveyed data and saved data over these media. Cryptography is the most efficient and feasible mode to transfer security services and also Cryptography is becoming effective tool in numerous applications for information security. This paper studies the shielding of information with the help of cryptographic function and number sequences. The efficiency of the given method is examined, which assures upgraded cryptographic services in physical signal and it is also agile and clear for realization. 1. Introduction Cryptography is the branch of information security. The basic goal of cryptography is to make communication secure in the presence of third party (intruder). Encryption is the mechanism of transforming information into unreadable (cipher) form. This is accomplished for data integrity, forward secrecy, authentication and mainly for data confidentiality. -
A Guide on Probability Distributions
powered project A guide on probability distributions R-forge distributions Core Team University Year 2008-2009 LATEXpowered Mac OS' TeXShop edited Contents Introduction 4 I Discrete distributions 6 1 Classic discrete distribution 7 2 Not so-common discrete distribution 27 II Continuous distributions 34 3 Finite support distribution 35 4 The Gaussian family 47 5 Exponential distribution and its extensions 56 6 Chi-squared's ditribution and related extensions 75 7 Student and related distributions 84 8 Pareto family 88 9 Logistic ditribution and related extensions 108 10 Extrem Value Theory distributions 111 3 4 CONTENTS III Multivariate and generalized distributions 116 11 Generalization of common distributions 117 12 Multivariate distributions 132 13 Misc 134 Conclusion 135 Bibliography 135 A Mathematical tools 138 Introduction This guide is intended to provide a quite exhaustive (at least as I can) view on probability distri- butions. It is constructed in chapters of distribution family with a section for each distribution. Each section focuses on the tryptic: definition - estimation - application. Ultimate bibles for probability distributions are Wimmer & Altmann (1999) which lists 750 univariate discrete distributions and Johnson et al. (1994) which details continuous distributions. In the appendix, we recall the basics of probability distributions as well as \common" mathe- matical functions, cf. section A.2. And for all distribution, we use the following notations • X a random variable following a given distribution, • x a realization of this random variable, • f the density function (if it exists), • F the (cumulative) distribution function, • P (X = k) the mass probability function in k, • M the moment generating function (if it exists), • G the probability generating function (if it exists), • φ the characteristic function (if it exists), Finally all graphics are done the open source statistical software R and its numerous packages available on the Comprehensive R Archive Network (CRAN∗).