asymp1 http://www.openmath.org/cd/asymp1.ocd 1999-10-19 2 0 experimental alg1 arith1 fns1 limit1 linalg1 logic1 nums1 quant1 relation1 setname1 setname3 set1 This CD provides a representation of various asymptotic set constructors (O, \Omega, etc.) The constructors represent sets of functions : R -> R. O The O symbol represents a unary function which constructs a set of certain functions of type reals to reals. The condition f(n)=O(g(n)) is intended to express an upper bound condition on f. O(g) = { f:reals -> reals | exists c in positive reals and M in the naturals such that forall n geq M. |f(n)| leq c*g(n)} o The o symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The condition f(n) = o(g(n)) is intended to express a lower bouund condition on f. Formally we say that f(n) = o(g(n)) if and only if the limit as n tends to infinity of f(n)/g(n) exists and is equal to 0. o(g) = {f : reals -> reals | the limit as x tends to infinity of f(x)/g(x) is 0} theta The theta symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The theta symbol represents a set of functions which all have the same 'rate of growth'. Formally we say that f(x) = theta(g(x)) if and only if there are constants c_1 not= 0 and c_2 not= 0 and x_0 such that for all x > x_0 it is true that c_1*g(x) < f(x) < c_2*g(x). f(x) = theta(g(x)) if and only if there are constants c_1 not= 0 and c_2 not= 0 and x_0 such that for all x > x_0 it is true that c_1*g(x) < f(x) < c_2*g(x) asymptotic The asymptotic symbol represents a binary relation between two functions of type reals to reals. The asymptotic relation between two functions returns true if the two functions have the same rate of growth and more precisely there ratio approaches 1 as the variable approaches infinity. Formally we say that f(x) is asymptotic to g(x) if and only if the limit as x tends to infinity of f(x)/g(x) = 1. f(x) is asymptotic g(x) if and only if the limit as x tends to infinity of f(x)/g(x) = 1 omega The omega symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The omega symbol represents a set of functions such that for any function in the set omega(g(x)), f(x); it is not true that f(x) is in o(g(x)). Formally we say that f(x) = omega(g(x)) if and only if there is an epsilon > 0 and an infinite sequence x_1, x_2, x_3, ... such that for all j then abs(f(x_j)) > epsilon * g(x_j). f(x) is omega(g(x)) if and only if it is not true that f(x) is o(g(x)) f(x) = omega(g(x)) if and only if there is an epsilon > 0 and an infinite sequence x_1, x_2, x_3, ... such that for all j then abs(f(x_j)) > epsilon * g(x_j). Omega The Omega symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The Omega symbol represents a set of functions such that for any function in the set Omega(g(x)), f(x); it is not true that f(x) is in O(g(x)). f(x) is Omega(g(x)) if and only if it is not true that f(x) is O(g(x))