OpenMath

OpenMath is an extensible standard for representing the semantics of mathematical objects.

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OpenMath Content Dictionary: asymp1

Canonical URL:
http://www.openmath.org/cd/asymp1.ocd
CD File:
asymp1.ocd
CD as XML Encoded OpenMath:
asymp1.omcd
Defines:
O, Omega, asymptotic, o, omega, theta
Date:
1999-10-19
Version:
2 (Revision 1)
Review Date:
2017-12-31
Status:
experimental

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          then the derived work must include a different CDName element,
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          Dictionary Group whose name is, for example, `math' containing
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  Author: Bill Naylor

This CD provides a representation of various asymptotic set constructors (O, \Omega, etc.) The constructors represent sets of functions : R -> R.

O

Description:

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.

Commented Mathematical property (CMP):
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)}
Formal Mathematical property (FMP):
O ⁡ ( g ) = { f ∈ function_set ⁡ ( R , R ) | ∃   c , M . c ∈ R ∧ c > 0 ∧ M ∈ N ∧ ∀   n . n ∈ N ∧ n ≥ M ⇒ | f ⁡ ( n ) | ≤ c ⁢ g ⁡ ( n ) }
Signatures:
sts

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o

Description:

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.

Commented Mathematical property (CMP):
o(g) = {f : reals -> reals | the limit as x tends to infinity of f(x)/g(x) is 0}
Formal Mathematical property (FMP):
o ⁡ ( g ) = { f ∈ function_set ⁡ ( R , R ) | limit x → ∞ - f ⁡ ( x ) g ⁡ ( x ) = 0 }
Signatures:
sts

[Next: theta] [Previous: O] [Top]

theta

Description:

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).

Commented Mathematical property (CMP):
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)
Formal Mathematical property (FMP):
theta ⁡ ( g ) = { f ∈ function_set ⁡ ( R , R ) | ∃   c_ 1 , c_ 2 , x_ 0 . c_ 1 ≠ 0 ∧ c_ 2 ≠ 0 ∧ ∀   x . x > x_ 0 ∧ c_ 1 ⁢ g ⁡ ( x ) < f ⁡ ( x ) ∧ f ⁡ ( x ) < c_ 2 ⁢ g ⁡ ( x ) }
Signatures:
sts

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asymptotic

Description:

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.

Commented Mathematical property (CMP):
f(x) is asymptotic g(x) if and only if the limit as x tends to infinity of f(x)/g(x) = 1
Formal Mathematical property (FMP):
asymptotic ⁡ ( f , g ) ≡ ( limit x → ∞ - f ⁡ ( x ) g ⁡ ( x ) = 1 )
Signatures:
sts

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omega

Description:

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).

Commented Mathematical property (CMP):
f(x) is omega(g(x)) if and only if it is not true that f(x) is o(g(x))
Formal Mathematical property (FMP):
omega ⁡ ( g ) = { f ∈ function_set ⁡ ( R , R ) | ¬ ( f ⁡ ( x ) ∈ o ⁡ ( g ) ) }
Commented Mathematical property (CMP):
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).
Formal Mathematical property (FMP):
f ∈ omega ⁡ ( g ) ≡ ∃   epsilon , seq . epsilon > 0 ∧ ∀   j . | f ⁡ ( seq j ) | > epsilon ⁢ g ⁡ ( seq j )
Signatures:
sts

[Next: Omega] [Previous: asymptotic] [Top]

Omega

Description:

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)).

Commented Mathematical property (CMP):
f(x) is Omega(g(x)) if and only if it is not true that f(x) is O(g(x))
Formal Mathematical property (FMP):
Omega ⁡ ( g ) = { f ∈ function_set ⁡ ( R , R ) | ¬ ( f ⁡ ( x ) ∈ O ⁡ ( g ) ) }
Signatures:
sts

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