Working Par LSEOR 10-119 ISSN 2041-4668 (Onlin) WHAT LIES BETWEEN + AND (and byond)? HPWilliams London School of Economics hwilliams@lsacuk
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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
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 2 + 3 = 5 2 x 3 = 6 2 3 = 8 3 2 = 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
2r3 16 15 13 14 12 11 10 9 8 7 6 5 1 2 3 4 r Figur 1 Succssiv Arithmtic Orations Alid to 2 and 3
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
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))
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
W lot ossibl valus of f(x) if w st at (say) 049 giving x 0 049 1 163 272 510 1518 f(x) 049 1 163 272 510 1518 16402 16 ff(x) = x f(x) x 14 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 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 99 118-133 2 Cox, DA, 1985 Gauss and th gomtric-arithmtic man, Notics Amr Math Soc 32(2) 147-151 3 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 19 194-196 6 Rubtsov, CA, and Romrio, GF, 2004 Ackrmann s function and nw arithmtical orations, htt://wwwrotarysaluzzoit/filpd=/irorazi%20(1)df