functions with other functions as input
transkawa
or itself, as input and returns only a set of values?
e.g
y ( f(x)) = x where x is a set of values which could include y itself.
this is for some research work.
I would be happy to be informed on this set of functions i have liberally
categorised in the above manner that returns a set while accepting functions
as input. sorry, i am not a mathematician, just an analyst.
xnt
Arturo Magidin
> can anyone describe to me a class of functions that accept other functions,
> or itself, as input and returns only a set of values?
They are those functions whose domain is a set of functions. There
really is no other general description.
>
> e.g
> y ( f(x)) = x where x is a set of values which could include y itself.
>
> this is for some research work.
Some examples:
If V is the vector space of continuous functions defined on the
interval [0,1], then there is a function (a linear transformation)
that accepts elements of V as inputs (functions as inputs), and
returns their integral from 0 to 1; that is, the function F is
F(f) = integra_0^1 f(t)dt.
Similarly, if V is the vector space of differentiable functions
defined on the real line, the function D defined by D(f) = f' is a
function that accepts functions as inputs and returns functions as
outputs.
> I would be happy to be informed on this set of functions i have liberally
> categorised in the above manner that returns a set while accepting functions
> as input.
There is no such "set"; the class of all such functions is a proper
class, not a set. And given *any* nonempty collection of functions A,
and *any* nonempty set B, you can always define functions with domain
A and codomain B, so you can always have functions whose inputs are
elements of A (functions) and outputs are elements of B (whatever you
want them to be). The concept of a "function" is too general for the
kind of categorization you seem to be hoping for.
--
Arturo Magidin
Frederick Williams
>
> can anyone describe to me a class of functions that accept other functions,
> or itself, as input and returns only a set of values?
You may be interest in the work of Scott and Domain Theory.
>
> e.g
> y ( f(x)) = x where x is a set of values which could include y itself.
>
> this is for some research work.
> I would be happy to be informed on this set of functions i have liberally
> categorised in the above manner that returns a set while accepting functions
> as input. sorry, i am not a mathematician, just an analyst.
> xnt
--
Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.
Frederick Williams
>
> can anyone describe to me a class of functions that accept other functions,
> or itself, as input and returns only a set of values?
--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting