- We say that
is in the ORDER OF MAGNITUDE
of
and we write
if
there exist two (strictly) positive constants
and
such that for
big enough we have
(2)

- We say that
is an ASYMPTOTIC UPPER BOUND
of
and we write
if there exists
a (strictly) positive constants
such that for
big enough we have
(3)

- We say that
is an ASYMPTOTIC LOWER BOUND
of
and we write
if there exists
a (strictly) positive constants
such that for
big enough we have
(4)

- With
and
we have
Indeed we have
(5)

for with and . - Assume that there exists a positive integer
such that
and
for every
.
Then we have
(6)

Indeed we have(7)

- Assume
and
are positive real constants.
Then we have
(8)

Indeed for we have(9)

Hence we can choose and .

- holds iff and hold together.
- Each of the predicates , and define a reflexive and transitive binary relation among the -to- functions. Moreover is symmetric.
- We have the following TRANSPOSITION FORMULA
(10)

(11) |

(12) |

- if then ,
- if then ,
- if then .

(13) |

*
*

2008-01-07