# Chaotic functions

Acolyte
Posers and Puzzles 05 Mar '05 08:02
1. Acolyte
Now With Added BA
05 Mar '05 08:02
Describe a function from the reals to the reals such that the image of any interval intersects with every other interval.

Can such a function have an inverse?
2. 05 Mar '05 15:36
Originally posted by Acolyte
the image of any interval intersects with every other interval.
what does that mean?
3. Acolyte
Now With Added BA
07 Mar '05 08:36
Originally posted by fearlessleader
what does that mean?
It means: if I pick any four numbers a<b, c<d, there'll be some number x between a and b such that f(x) lies between c and d.
4. 07 Mar '05 21:05
Originally posted by Acolyte
It means: if I pick any four numbers a<b, c<d, there'll be some number x between a and b such that f(x) lies between c and d.
theoreticaly all quantum wave-forms are of this nature (to my understanding the range of the image is inversly proportional to the size of the interval examined.)

how about:

limit(a-->0) (x/a)-[a(x+1)]

infintie slopes and greatest integer functions are both mathmaticaly clumsy, but i belive this fits the specifications (in theory).
5. Acolyte
Now With Added BA
09 Mar '05 20:42
Originally posted by fearlessleader
theoreticaly all quantum wave-forms are of this nature (to my understanding the range of the image is inversly proportional to the size of the interval examined.)

how about:

limit(a-->0) (x/a)-[a(x+1)]

infintie slopes and greatest integer functions are both mathmaticaly clumsy, but i belive this fits the specifications (in theory).
"(to my understanding the range of the image is inversly proportional to the size of the interval examined.)"

Huh? If A is a subset of B, then f(A) is a subset of f(B), so I don't see what you're trying to say.

"limit(a-->0) (x/a)-[a(x+1)]"

What do the square brackets mean? If they're a rounding function, this limit is not defined for x other than 0.
6. 09 Mar '05 21:28
Originally posted by Acolyte
It means: if I pick any four numbers a<b, c<d, there'll be some number x between a and b such that f(x) lies between c and d.
I assume you just want a function of x, rather than one of a and b and x?
7. 09 Mar '05 21:42
wow is this a type of math or something b/c im completly out of it and can anyone teach me what this is?
8. Scheel
<blank>
09 Mar '05 23:05
Originally posted by Acolyte
Describe a function from the reals to the reals such that the image of any interval intersects with every other interval.

Can such a function have an inverse?
Can't realy think of a function that would satisfy this describtion - but I see no reason why it should not be possible to construct one.

It will be difficult to ensure that it has an inverse.
If its not injective the inverse will not be well defined.
So eventhough we require that the image of any interval intersects with every other interval, we can not go for a function that is surjective in to R from any subinterval. We will have to look for a function where the image of any interval is dense in R.

If we could think of a function that reorders the real numbers in a way that seems completely random then that might be a candidate, but reordering that that is not wellordered seems difficult.
9. 11 Mar '05 12:26
Huh? If A is a subset of B, then f(A) is a subset of f(B), so I don't see what you're trying to say.

What do the square brackets mean? If they're a rounding function, this limit is not defined for x other than 0.
in quantum mechanics the higher the resolution with which you examin a thing, the greater it varies from the overall appearence. thus, if examined on a mathmaticaly perfect level, all quamtum mechanical equations would be infinitly erratic. i may be misunderstanding it.

correct me if i'm wrong: what we are looking for is a function which includes all reals on every interval, no matter how small.

the [ ] are a greatest intiger funtion. prehaps the stated equation would be undifined, but i think it would fit the bill. any funtion that fits these standereds would have to have an infinite number of points for which it was undifined in every interval, at least if it was to have any sembilence of continuity.
10. 11 Mar '05 13:32
Maybe a sine wave of infinite frequency and infinite amplitude is required
11. AThousandYoung
All My Soldiers...
11 Mar '05 14:32
Originally posted by benkoboy
wow is this a type of math or something b/c im completly out of it and can anyone teach me what this is?
I've studied multivariable calculus and a little linear algebra and differential equations and I am clueless. This must be at least upper division undergraduate math stuff.
12. 11 Mar '05 20:25
Originally posted by AThousandYoung
I've studied multivariable calculus and a little linear algebra and differential equations and I am clueless. This must be at least upper division undergraduate math stuff.
i dont think it's that complex, just weird.
iamatiger has the best idea so far, it's similar to mine, except it dosn't require greatest integer functions, which are discontinueus.
13. 11 Mar '05 21:063 edits
f:R->R such that f(x)=g(x+Q)
where
g:R->R = the bijection between R and the uncountably infinite set of coset representatives of the Abelian group R/Q
(which must exist by the Axiom of Choice).

All bijective functions have inverses.

14. Acolyte
Now With Added BA
11 Mar '05 22:34
Originally posted by THUDandBLUNDER
f:R->R such that f(x)=g(x+Q)
where
g:R->R = the bijection between R and the uncountably infinite set of coset representatives of the Abelian group R/Q
(which must exist by the Axiom of Choice).

All bijective functions have inverses.

Sounds like a good idea, but some explanation would help. For example, how do you know that |R| = |R/Q| ? Saying they're both uncountable isn't the same as saying they're the same size.

I must admit I was thinking also of what is possible in a choice-free world. It's possible to give an explicit (read: choice-free) example of a function which takes every interval to a dense set. But can you construct such a function that is also bijective? If not, why not?
15. 12 Mar '05 04:181 edit
Originally posted by Acolyte
Sounds like a good idea, but some explanation would help. For example, how do you know that |R| = |R/Q| ? Saying they're both uncountable isn't the same as saying they're the same size.

I must admit I was thinking also of what is po ...[text shortened]... ruct[/i] such a function that is also bijective? If not, why not?
Firstly, I should say that the function I suggested was not my idea.

Originally posted by Acolyte
Sounds like a good idea, but some explanation would help. For example, how do you know that |R| = |R/Q| ? Saying they're both uncountable isn't the same as saying they're the same size.

{R} is uncountable.
{Q} is countable.
Hence {R/Q} has the same cardinality as {R}.

The original question said 'describe', not 'construct'.

.