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Nov 30, 2004, 5:30:11 PM11/30/04

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f:R*R->R .Let us start with a simple case:

g(x,g(x,y))= f(x,y) f given (1)

or f(x,y)^[1/2]=g(x,y)

Example: f(x,y)=y/(1+x*y) , g(x,y)=2*y/(2+x*y) verifies (1).

g(x,g(x,y))= f(x,y) f given (1)

or f(x,y)^[1/2]=g(x,y)

Example: f(x,y)=y/(1+x*y) , g(x,y)=2*y/(2+x*y) verifies (1).

By the same way will have:f(x,y)^[1/p]=p*y/(p+x*y)=g(x,y) p integer

verifying g(x,g(x....)))n times =f(x,y)=y/(1+x*y).

We may generalize:

if f(x,y)=phi^-1(phi(y)+n(x)) or m^[n(x)](y) ,

phi(m(y))=phi(y)+1 ;

then f(x,y)^[r]=phi^-1(phi(y)+r.n(x)) or m^[r.n(x)](y) ,

here r is a positive real number.

Please your comments and ideas,alain.

Dec 14, 2004, 7:48:56 PM12/14/04

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Alain, my tetration.org web site is dedicated to continuous iteration.

See http://www.tetration.org/Dynamics/index.html for resources on

continuous iteration and

http://www.tetration.org/Combinatorics/index.html for an outline of my

combinatorial approach to the subject.

For ideas consider Stephen Wolfram's question:

> How can one extend recursive function definitions to continuous

> numbers? What is the continuous analog of the Ackermann function? The

> symbolic forms of the Ackermann function with a fixed first argument

> seem to have obvious interpretations for arbitrary real or complex

> values of the second argument. But is there a general way to extend

> these kinds of recursive definitions to continuous cases? Given a way

> to do this, how does it apply to recursive definitions like those on

> page 130? What happens to all the irregularities when one is between

> integer values? Or is it only possible to find simple continuous

> generalizations to functions that show fundamentally simple behavior?

> Can this be used as a characterization of when the behavior is simple?

For my non-peer-reviewed response to Wolfram's question on the NKS Forum

see http://forum.wolframscience.com/showthread.php?s=&threadid=488

Here's why I'm interested in the subject. A theory of continuously

iterated smooth matrix functions would probably encompass all of

dynamics in physics. A theory of continuously iterated smooth complex

functions would be powerful enough to extend the Ackermann function to

complex numbers.

Daniel Geisler

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