WHAT LIES BETWEEN + AND (and beyond)? H.P.Williams

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1 Working Par LSEOR ISSN (Onlin) WHAT LIES BETWEEN + AND (and byond)? HPWilliams London School of Economics hwilliams@lsacuk

2 First ublishd in Grat Britain in 2010 by th Orational Rsarch Grou, Dartmnt of Managmnt London School of Economics and Political Scinc Coyright Th London School of Economics and Political Scinc, 2010 Th contributors hav assrtd thir moral rights All rights rsrvd No art of this ublication may b rroducd, stord in a rtrival systm, or transmittd in any form or by any mans, without th rior rmission in writing of th ublishr, nor b circulatd in any form of binding or covr othr than that in which it is ublishd

3 WHAT LIES BETWEEN + AND (and byond)? HPWilliams London School of Economics hwilliams@lsacuk Abstract W attmt to crat a continuum of functions btwn addition and multilication (and byond) Such a function could hav ractical alications Addition, multilication, xonntiation, ttration tc ar all articular cass of a gnralisation of Ackrmann s function for succssiv intgral valus of on of th argumnts Intrmdiat functions can b viwd as rsults of a fractional valu of this argumnt Ways of sking such functions with givn rortis ar invstigatd It is notd that som othr intgrally dfind functions of mathmatics can b xtndd to fractional argumnts Two ossibl aroachs, which ar considrd hr, ar (i) to considr Gauss s Arithmtic-Gomtric man and (ii) to considr solutions of th functional quation ff(x) = x Kywords: Ackrmann s function, ttration, Gauss s man, functional quations

4 1 INTRODUCTION Multilication can b rgardd as succssiv addition, xonntiation as succssiv multilication tc Th nxt oration (ttration) rquirs a stiulation about th ordr in which succssiv xonntiations ar carrid out (sinc xonntiation is not commutativ) W thrfor considr th hirarchy of functions a+b axb = a + a + a (b tims) a b = a x a x x a (b tims) b a = a a a (b tims) Th fourth oration is usually known as ttration and somtims writtn as indicatd Th convntion is adotd of assuming brackting from th toth nam was coind by Goodstin [4] It has rcivd som attntion (s th wb sit of Gislr [3]) In articular dfining th function for fractional b rsnts a roblm, as discussd blow Th hirarchy of ths functions has also bn considrd by Rubtsov and Romrio [5] W could introduc th succssor function a + 1 as th zroth function if w wishd In ordr to motivat th roblm w will considr a numrical xaml tc = 5 2 x 3 = = = 16 Markd on th grah in figur 1 it can b sn that it is ossibl to draw a smooth curv though ths valus whr r = 1, 2, 3, 4 for addition, multilication, xonntiation, ttration rsctivly In ordr to sk th valu for an oration halfway btwn + and w sk th valu at th oint indicatd by th arrow

5 2r r Figur 1 Succssiv Arithmtic Orations Alid to 2 and 3

6 2 ACKERMANN S FUNCTION A gnralisation of Ackrmann s function [1] can b dfind asily by th following rcursion f(a+1, b+1, c) = f(a, f(a+1,b,c), c) with initial conditions f(0, b, c) = b+1 f(1, 0, c) = c f(2, 0,c) = 0 f(a+1, 0, c) = 1 for a> 1 It is asy to vrify that f(0, b, c) = b+1 f(1, b, c) = b+c f(2, b, c) = bxc f(3, b, c) = c b b f(4, b, c) = c th succssor function addition multilication xonntiation ttration W sk f( 3 / 2, b, c) Ackrmann s function is usually xrssd as a function of 2 argumnts by fixing c at (say) 2 It is a doubly rcursiv function which grows fastr than any rimitiv rcursiv function: g in ordr to valuat f(a+1,, ) w nd to valuat f(a+1,,) for smallr argumnts and f(a,, ) for much largr argumnts Its xlosiv growth is dmonstratd by f(0, 3, 2) = 4 f(1, 3, 2) = 5 f(2, 3, 2) = 6 f(3, 3, 2) = 8 f(4, 3, 2) = 16 f(5, 3, 2) = 65536

7 It is intrsting to not that f(a, b, c) is not wll dfind for fractional b ithr Eg what is (½)2? W can, howvr, dfin ( 1 / ) 2 It is 2 sinc 2 = 2 3 GAUSS S ARITHMETIC-GEOMETRIC MEAN Lt A(a, b) = (a+b)/2 th arithmtic man G(a,b) = (a x b) th gomtric man M(a, b), Gauss s Arithmtic-Gomtric man (s g Cox [2]) is halfway btwn A(a, b) and G(a, b) and is dfind, itrativly, by a 1 = G(a, b), b 1 = A(a, b) a n+1 = G(a n, b n ), b n+1 = A(a n, b n ) M(a, b) = Lt n -> a n = Lt n -> b n For xaml g(2,128) = 16, A(2,128) = 65, M(2,128) = 3626 Sinc a + b = A(a, b) x 2 = A(a, b) 2 2 a x b = G(a, b) 2 = G(a, b) 3 2 Lt a 3 / 2 b = M(a, b) 5 / 2 2 = M(a, b) 3 / 2 M(a, b)) M(a, b) has an analytic solution in trms of llitic intgrals But thr is a difficulty Considr th following valus on a lin a M(a,M(a,b)) M(a,b) M(M(a,b),b) b Whil M(a,M(a,b)) is th man of a and M(a,b) and M(M(a,b),b) is th man of M(a,b) and b, M(a,b) M(M(a,M(a,b)), M(M(a,b),b))

8 4 THE FUNCTIONAL EQUATION ff(x) = x A bridg btwn addition and multilication is rovidd by th xonntial function (or its invrs, th logarithmic function) sinc (a+b) = a x b Lt ff(x) = x W can dfin f(a 3 / 2 b) as f(a) x f(b) Thrfor w sk solutions of ff(x) = x f(x) is a function btwn x (i th idntity function) and x This functional quation has bn xamind by a numbr of authors For xaml Hammrsly [5] and subsqunt corrsondnc in th IMA Bulltin Howvr w sk a solution with a numbr of rasonabl conditions In articular w initially dmand that f(x) (dfind on th nonngativ rals) satisfis (i) (ii) (iii) (iv) x < f(x) < x f(x) is monotonic strictly incrasing f(x) is continuous and infinitly diffrntiabl Th drivativs ar monotonic strictly incrasing Lt us dfin f(0) = Thn f() = 0 = 1 f(1) = f( ) = f(1) This givs th following tabl of valus x 0 1 f(x) 1 As ncssary (discrt) conditions for th abov (continuous) conditions w dmand that f(x), its gradints, gradints of gradints tc ar monotonic incrasing and tak intrmdiat valus btwn th corrsonding valus of x and f(x) This imlis 1 < (1 ) / < ( 1) /(1 ) giving 0469 < < 05 It would aar that f(x) is not uniqu

9 W lot ossibl valus of f(x) if w st at (say) 049 giving x f(x) ff(x) = x f(x) x x Figur 2 Th functions x, f(x) and ff(x) = x 5 REFERENCES 1 Ackrmann, W, 1928 Zum Hilbrtschn Aufbau dr rlln Zahln, Mathmatisch Annaln Cox, DA, 1985 Gauss and th gomtric-arithmtic man, Notics Amr Math Soc 32(2) Gislr, D, 2009 Ttration wb sit, htt://wwwttrationorg 4 Goodstin, RL, 1947 Tranfinit ordinals in rcursiv numbr thory, Journal of Symbolic Logic 12 5 Hammrsly, JM, 1983 Functional roots and indicial smigrous, Bulltin of th IMA Rubtsov, CA, and Romrio, GF, 2004 Ackrmann s function and nw arithmtical orations, htt://wwwrotarysaluzzoit/filpd=/irorazi%20(1)df