G. H. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc. (2) 10 (1912) 54--90.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that and # are scale changes which preserve the set of transmonomials. Note that f 6= f and f #6= f . These compositions are used to consider transbasis starting with level one (b 1 = x) which is particularly useful for di erential calculus (see below) 1.4. A conjecture of Hardy. In [2] a conjecture states that the functional inverse of log x log log x is not equivalent to any exp log function over R for x 1. The Theorem 1.2 of [3] illustrate the interest of transseries by a proof of this conjecture. 2. Di erential algebraic polynomials Let P = d P d be a di erential ....

Hardy (G.H.). { Properties of logarithmico-exponential functions. Proceedings of the London mathematical society - 10:p. 54-90, 1911.

.... n p . Let be a continuous function from U into R and let (x) e Gamma1 1;1 f1 (x) Gamma1 p;np fp (x) Then (x) sup x2U n (x) Proof. We first notice that we will be able to apply theorem 3 on our input data: by a well known theorem, which goes back to Hardy (see [Har 11] the germs at infinity of f 1 ; Delta Delta Delta ; f p lie in a common Hardy field. Consequently, f 1 OEOE s Delta Delta Delta OEOE s f p , and f 1 ; Delta Delta Delta ; f p are strictly increasing in a suitable neighbourhood of infinity. The mapping is defined in a neighbourhood V ....

G.H. Hardy. Properties of logarithmico-exponential functions. Proceedings of the London mathematical society 10,2 (p 54-90).

....nies pour x susamment grand) forment un corps totalement ordonn e. L int er et principal des L fonctions est que beaucoup de fonctions que l on rencontre dans la pratique (par exemple comme solutions d equations di erentielles) se d eveloppent dans une echelle de L fonctions. Hardy se demanda [2, 3] quel genre de fonctions ne pourrait pas se d evelopper dans une telle echelle. Il construit notamment une solution de l equation fonctionnelle E(x 1) e E(x) qui cro t plus vite que toute exponentielle it er ee. Il conjectura egalement l inexistence d une L fonction asymptotique a ....

G.H. Hardy. Properties of logarithmico-exponential functions. Proceedings of the London Mathematical Society, 10(2):54-90, 1911.

....function built up from x and the rational numbers Q by the field operations, exponentiation and logarithm. In this paper, we shall only consider exp log functions which are defined in a neigbourhood of infinity. Hardy has shown that, ultimately, such functions are either negative, zero or positive [11, 12]. In other words, the germs of exp log functions at infinity form a totally ordered field. But how to decide whether a given exp log function is asymptotically superior to another one in a neighbourhood of infinity More generally, is it possible to compute an asymptotic expansion of a given ....

G.H. Hardy. Properties of logarithmico-exponential functions. Proceedings of the London Mathematical Society, 10(2):54--90, 1911.

....functions. In particular, Cai and Selman [CS96] showed the following result. Let t; T : IN IN be logarithmico exponential functions 2 such that t is bounded above by a polynomial and T is fully time constructible. If t(n) log t(n) o(T (n) then ADTime(t(n) 6= ADTime(T (n) 2 Hardy [Har11] rst de ned and studied logarithmico exponential functions to provide a scale of in nities . He showed, among other things, that a logarithmico exponential function cannot increase more slowly than every iterated logarithm function, nor faster than every iterated exponential function, and ....

G. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc., 10:54-90, 1911.

.... best lower bounds to be obtained are not easily expressible in terms of conventional mathematical notation. Usually, lower bounds in complexity theory can be described with expressions involving the operations f ; exp; logg only. Growth rates so expressible are called L functions by Hardy [8, 9]. Unfortunately, the answers to the questions posed in the introduction involves functions that are not approximated well by any L function. For instance, the best lower bound that can be shown using current techniques for the classes MA exp , ZPEXP NP , or exp 2 seems to be ....

G.H. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc. 2, 54-90 (1912).

....any t(x) and s(x) in L, either t(x) o(s(x) or s(x) o(t(x) or there exists a nonzero constant c, such that lim x## t(x) s(x) c. Let f (#) denote the function that iterates # applications of f . That is, f (1) x) f(x) and f (# 1) x) f(f (#) x) for # # 1. Hardy proved [9] that for every function t # L, if lim x## t(x) #, then there is some constant # so that log (#) x) o(t(x) as well as t(x) o(exp (#) x) Informally, a logarithmico exponential function that goes to infinity cannot increase more slowly than every iterated logarithm function nor ....

G. Hardy, Properties of logarithmico-exponential functions, Proc. London Math. Soc., 10 (1911), pp. 54--90.

....serves as a base for generalisations to more general classes of functions. Key words: Asymptotic expansion, exp log function, transseries, algorithm. 1 Introduction The problem of expanding exp log functions has been considered by several authors and goes back to Hardy s work on L functions [Har 11] However, the computational version of the problem has been settled only recently. Shackell [Sh 90] was the first to give an explicit algorithm and Gonnet and Gruntz [GoGr 92] were the first to give one, which has been implemented in practice. A variant of this latter algorithm was rediscovered ....

G.H. Hardy. Properties of logarithmico-exponential functions. Proceedings of the London mathematical society 10,2 (p 54-90).

....algebraic construction of nonstandard models of T an;exp . These models have surprising applications. In x4 we will show that the compositional inverse to x 7 (log x) log log x) is not asymptotic to a composition of semialgebraic functions, log and exp. This answers a question of Hardy posed in [9]. In Partially supported by NSF grants DMS 9202833 and INT 9224546. Partially supported by EPSRC Senior Research Fellowship. Partially supported by NSF grants DMS 9306159 and INT 9224546 and an AMS Centennial Fellowship. 1991 Mathematics Subject Classification. Primary 03H05; Secondary ....

....in this sense, and also has these properties. Let i(x) be the compositional inverse of the function x log x. Then e i(x) is the inverse of the function (log x) log log x) We will prove that e i(x) is not asymptotic to a logarithmic exponential function. This was conjectured by Hardy in [9] and proved for e e i(x) instead of e i(x) by Shackell in [21] 4.1) We first show that i(x) is asymptotic to x log x . Let x = y log y. Then lim x 1 i(x) log x x = lim y 1 y(log y log log y) y log y = lim y 1 1 log log y log y = 1: We next argue that i(x) itself ....

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G.H. Hardy, Properties of logarithmico-exponential functions, Proc. London Math. Soc. 10 (1912), 54-90.

....LE is very useful for examining the asymptotic behavior of definable functions as the formal series representing the function reflects its asymptotic expansion. Consider the function f(x) log x) log log x) and let g be an compositional inverse to f defined on (r; 1) for some r 2 R. In [H] Hardy conjectured that g is not asymptotic to a composition of exp, log and semialgebraic functions. Shackell ( Sh] came close to verifying this conjecture by proving that the inverse to (log log x) log log log x) is not asymptotic to such a composition. 3 Using the series expansion for g in ....

G.H. Hardy, Properties of logarithmico-exponential functions, Proc. London Math. Soc. 10 (1912), 54-90.

....For any t(x) and s(x) in L, either t(x) o(s(x) or s(x) o(t(x) or there exists a non zero constant c, such that lim x 1 t(x) s(x) c. Let f ( denote the function that iterates applications of f . That is, f (1) x) f(x) and f ( 1) x) f(f ( x) for 1. Hardy proved [Har11] that for every function t 2 L, 4 if lim x 1 t(x) 1, then there is some constant so that log ( x) o(t(x) as well as t(x) o(exp ( x) informally, a logarithmico exponential function that goes to infinity cannot increase more slowly than every iterated logarithm function, nor ....

G. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc., (2),10:54--90, 1911.

....tight as in 1. Average Case Computational Complexity Theory 27 the worst case. Cai and Selman [CS95] obtained such a result when the time functions involved are restricted to be logarithmico exponential. The class L of logarithmico exponential (log exp, in short) functions, first studied by Hardy [Har11], is the smallest class of functions f : IR IR containing every constant function f(x) c and the identity function f(x) x such that if f(x) and g(x) are in Gamma, then so are f(x) Gamma g(x) exp(f(x) i.e. e f(x) and ln f(x) if f(x) is eventually positive) It follows that ....

....that every function in L is either eventually positive or eventually negative or identically zero. It follows that every function in L is eventually monotonic by the fact that L is closed under differentiation. Define f (l 1) x) f ffi f (l) x) l 1) and f (1) x) 1. It was shown in [Har11] that if t 2 L tends to infinity, then there is a constant l such that log (l) x) o(f(x) as well as f(x) o(exp (l) x) A function T is a log exp time function if for some f 2 L and for all n 2 IN, T (n) bf(n)c. It is easy to see that almost all commonly used time functions are ....

G. Hardy. Properties of logarithmico-exponential functions. Proceedings of London Mathematical Society, 10:54--90, 1911.

....of functions concerned. This brings particular difficulties regarding constants. As is the normal practice in this area we shall use an oracle for the determination of signs of these. We discuss this matter more fully in Section 1.4. Inverse functions have long been problematic in asymptotics [5, 8]. However in [18] the authors gave an algorithm for inverting nested forms which solves the problem of expressing the asymptotic behaviour of inverse functions. In the present paper we treat the more general problem of implicit 2 BRUNO SALVY AND JOHN SHACKELL functions. More precisely, let ....

....1 and g 1, we know that f=g 1, but if g 1 we can of course make no deduction regarding the limit of f=g. In order to obtain a calculus, we need some measure of the rapidity with which a function tends to its limit. This has long been recognized; the problem was extensively studied by Hardy [7, 8], and some of the ideas used there go back to the work of du Bois Reymond. Nested forms and expansions are based on Hardy s orders of infinity, but have a more formal recursive structure which is suitable for algorithmic work. We use the classical notations e k (x) for the exponential iterated k ....

Hardy, G. H. Properties of logarithmico-exponential functions. Proceedings of the London Mathematical Society 10, 2 (1911), 54--90.

....any t(x) and s(x) in L, either t(x) o(s(x) or s(x) o(t(x) or there exists a non zero constant c, such that lim x 1 t(x) s(x) c. Let f ( denote the function that iterates applications of f . That is, f (1) x) f(x) and f ( 1) x) f(f ( x) for 1. Hardy proved [Har11] that for every function t 2 L, if lim x 1 t(x) 1, then there is some constant so that log ( x) o(t(x) as well as t(x) o(exp ( x) informally, a logarithmicoexponential function that goes to infinity cannot increase more slowly than every iterated logarithm function, nor ....

G. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc., (2),10:54--90, 1911.

....In particular, Cai and Selman [CS96] showed the following result. Let t; T : IN IN be logarithmico exponential functions 2 such that t is bounded above by a polynomial and T is fully time constructible. If t(n) log t(n) o(T (n) then ADTime(t(n) ae F NaN 6= ADTime(T (n) 2 Hardy [Har11] first defined and studied logarithmico exponential functions to provide a scale of infinities . He showed, among other things, that a logarithmico exponential function cannot increase more slowly than every iterated logarithm function, nor faster than every iterated exponential function, and ....

G. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc., 10:54--90, 1911.

....any t(x) and s(x) in L, either t(x) o(s(x) or s(x) o(t(x) or there exists a non zero constant c, such that lim x t(x) s(x) c. Let f ( denote the function that iterates applications of f . That is, f (1) x) f (x) and f ( 1) x) f ( f ( x) for 1. Hardy proved [Har11] that for every function t 2 L, if lim x t(x) then there is some constant so that log ( x) o(t(x) as well as t(x) o(exp ( x) informally, a logarithmico exponential function that goes to infinity cannot increase more slowly than every iterated logarithm function, nor ....

G. Hardy. Properties of logarithmico-exponential functions. Proceedings of the London Mathematical Society, (2),10:54--90, 1911.

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G. H. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc. (2) 10 (1912) 54--90.

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G.H. Hardy, Properties of logarithmico-exponential functions, Proc. London Math. Soc. 10 (1912), 54-90.

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G.H. Hardy, Properties of logarithmico-exponential functions, Proc. London Math. Soc. 10 (1912), 54-90.

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Hardy, G. H. Properties of logarithmicoexponential functions. Proceedings of the London Mathematical Society 10, 2 (1911), 54--90.

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