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[UFO] Attracting Fixpoints or attracting line?
#11
(04/09/2010, 06:43 AM)bo198214 Wrote:
(04/08/2010, 10:03 PM)tommy1729 Wrote: your plot gives the fixed points of exp( t_k + k2pi i ) = t_k

but should that not be periodic with k ?

since exp( x + k i ) is periodic with real k ?

im a bit confused now ...

No, the fixed points of exp, i.e. the points with exp(z)=z, can be obtained by considering , i.e. the fixed points of the branches of the logarithm. All the non-real fixed points of exp are repelling, i.e. |exp(z)|>1, thats why these fixed points are attracting for the logarithm and can be obtained by iterating the corresponding branch.

Gottfried's question was now what happens if we choose non-integer ; and found that the corresponding "fixed points" lie smoothly on a line between the proper fixed points of .

well , Bo , apart from the " no " i agree on what you said.

in fact i already understood that.

however , isnt t_k the same for e.g. k = 2 and k = 4 in the fixed points of exp( t_k + k2pi i ) = t_k ?
Reply
#12
(04/09/2010, 12:11 PM)tommy1729 Wrote: in fact i already understood that.

No, you still didnt.

Quote:however , isnt t_k the same for e.g. k = 2 and k = 4 in the fixed points of exp( t_k + k2pi i ) = t_k ?

If you look at my explanations above, you see that the is not inside the exponential. The fixed point equation is exp(t_k)=t_k. If you apply the logarithm to both sides you have to deal with the branches as the exponential is not injective on the complex plane.
And these branches introduce the corresponding .
And yes the fixed points are different for and .
Somwhere on the forum is an animated picture of the fixed points of .
Reply
#13
Hmm, perhaps it is interesting to add some context to that whole idea.

I was playing with two other aspects of iteration of the exponential.


First aspect is the spiraling of the orbit of a complex initial value in the complex plane when repeatedly exponentiated. This suggests, that a polar form of representation of the complex numbers could be more enlightening for the generalization to fractional iterations. And especially, a polar form with a fixpoint as origin.
Indeed this has some charme, even more if one introduces the log-polar-form for a complex number:

where and the arc of the angle.

Then multiplication and division of two values x and y is simply reflected by addition/subtraction of the log-polar components, k'th iterated multiplication (powers) just multiplication of the components by k and exponentiation is and the logarithm results from the inverse of this operation.

Fractional iterates of complex initial numbers look then like a very smooth and even nearly linear interpolation of that lengthes and arcs - but only on first glance: the spiral of the flow is distorted by something which looks a bit like a perihel-effect. It diminuishes (and the flow smoothes) when the fixpoint is approached and extends/chaotizes(?) when the fixpoint is left by iteration.

But leave this here as just another curious observation. More in the focus was the log-polar-representation itself: here the second component (the arc of the angle) exceeds very naturally the interval 0..2*Pi once we introduce iterated addition and multiplication on the log-polar-coefficients and operations back and forth can be meaningfully concatenated without loss of information.

This leads then to the ...
... second aspect, the idea of the introduction of a "winding-number".

Once the idea (and the verbal term) is there, you find related material online; for instance a couple of very good articles of R.M.Corless et al (see below) where they just deal with that winding-number and its possible significance for an extended representation of complex numbers. In the above log-polar-representation we see it nicely that the complex arithmetic via exponentials and logarithms is just somehow modular to modulus 2*Pi (in the arc-parameter) and it is suggestive to introduce a representation for complex numbers and complex algebra which leaves the modular arithmetic behind. It is really nice to see this when playing around a bit. (The observation of "fix-points" vs. "fix-line" as described in this thread was just a detail in the context of exercising in that framework).

However, as also Corless et al discuss in their article: by buying simplicity for multiplication and exponentiation using the log-polar-form we pay with complication for addition. Actually, Corless et al, as I understood them, gave up because of intractability of addition when the winding-number is introduced.

That was also my idea and also I give up here.

We might extend the arc-parameter phi, possibly rescale it to the 2Pi-unit and then allow not only the interval (modulo 1) but the full real line: then a/the "winding-number" is represented by the integer part. But this does not help much: as long as we have no meaningful interpretation of the addition of two log-polarforms, where the winding-number is kept significant such that we can, for instance, invert the operations.
I felt, it would really be a nice toy-project to experiment with this a bit, however... my time (and energy) is somehow more limited than in the previous years.

Gottfried

-------------


Corless, R. M., and Jeffrey, D. J.
The unwinding number.
Sigsam Bulletin 30, 2 (June 1996), 28-35.

[online-version:
http://www.apmaths.uwo.ca/~djeffrey/Offp...ditors.pdf
Editor's Corner: The Unwinding Number
]


Corless, R. M., and Jeffrey, D. J.
The Wright omega function
Ontario Research Centre for Computer Algebra and the Department of Applied Mathematics
University of Western Ontario, London, Canada
Reference: J. Calmet et al (Eds.): AISC-Calculemus 2002, LNAI 2385, pp. 76-89, 2002, Springer-Verlag


Reasoning about the elementary functions of complex analysis
Corless, R.M., Jeffrey, D.J. and Davenport, J.H.
[can be found online]
Abstract:
There are many problems with the simplification of elementary functions, particularly over the complex plane. Systems tend to make “howlers” or not to simplify enough. In this paper we outline the “unwinding number” approach to such problems, and show how it can be used to systematise such simplification, even though we have not yet reduced it to a complete algorithm. The unsolved problems are probably more amenable to the techniques of artificial intelligence and theorem proving than the original problem of complex-variable analysis.
Keywords: Elementary functions; Branch cuts; Complex identities.
Gottfried Helms, Kassel
Reply
#14
(04/09/2010, 04:13 PM)bo198214 Wrote:
(04/09/2010, 12:11 PM)tommy1729 Wrote: in fact i already understood that.

No, you still didnt.

Quote:however , isnt t_k the same for e.g. k = 2 and k = 4 in the fixed points of exp( t_k + k2pi i ) = t_k ?

If you look at my explanations above, you see that the is not inside the exponential. The fixed point equation is exp(t_k)=t_k. If you apply the logarithm to both sides you have to deal with the branches as the exponential is not injective on the complex plane.
And these branches introduce the corresponding .
And yes the fixed points are different for and .
Somwhere on the forum is an animated picture of the fixed points of .

thanks. i think i understand now.

maybe gottfried should make more plots , say from k in [-3,3]

i wonder where the animated picture (bo mentioned) is btw.

regards

tommy1729
Reply
#15
(04/09/2010, 08:30 PM)Gottfried Wrote: Hmm, perhaps it is interesting to add some context to that whole idea.

I was playing with two other aspects of iteration of the exponential.


First aspect is the spiraling of the orbit of a complex initial value in the complex plane when repeatedly exponentiated. This suggests, that a polar form of representation of the complex numbers could be more enlightening for the generalization to fractional iterations. And especially, a polar form with a fixpoint as origin.
Indeed this has some charme, even more if one introduces the log-polar-form for a complex number:

where and the arc of the angle.

Then multiplication and division of two values x and y is simply reflected by addition/subtraction of the log-polar components, k'th iterated multiplication (powers) just multiplication of the components by k and exponentiation is and the logarithm results from the inverse of this operation.

Fractional iterates of complex initial numbers look then like a very smooth and even nearly linear interpolation of that lengthes and arcs - but only on first glance: the spiral of the flow is distorted by something which looks a bit like a perihel-effect. It diminuishes (and the flow smoothes) when the fixpoint is approached and extends/chaotizes(?) when the fixpoint is left by iteration.

But leave this here as just another curious observation. More in the focus was the log-polar-representation itself: here the second component (the arc of the angle) exceeds very naturally the interval 0..2*Pi once we introduce iterated addition and multiplication on the log-polar-coefficients and operations back and forth can be meaningfully concatenated without loss of information.

This leads then to the ...
... second aspect, the idea of the introduction of a "winding-number".

Once the idea (and the verbal term) is there, you find related material online; for instance a couple of very good articles of R.M.Corless et al (see below) where they just deal with that winding-number and its possible significance for an extended representation of complex numbers. In the above log-polar-representation we see it nicely that the complex arithmetic via exponentials and logarithms is just somehow modular to modulus 2*Pi (in the arc-parameter) and it is suggestive to introduce a representation for complex numbers and complex algebra which leaves the modular arithmetic behind. It is really nice to see this when playing around a bit. (The observation of "fix-points" vs. "fix-line" as described in this thread was just a detail in the context of exercising in that framework).

However, as also Corless et al discuss in their article: by buying simplicity for multiplication and exponentiation using the log-polar-form we pay with complication for addition. Actually, Corless et al, as I understood them, gave up because of intractability of addition when the winding-number is introduced.

That was also my idea and also I give up here.

We might extend the arc-parameter phi, possibly rescale it to the 2Pi-unit and then allow not only the interval (modulo 1) but the full real line: then a/the "winding-number" is represented by the integer part. But this does not help much: as long as we have no meaningful interpretation of the addition of two log-polarforms, where the winding-number is kept significant such that we can, for instance, invert the operations.
I felt, it would really be a nice toy-project to experiment with this a bit, however... my time (and energy) is somehow more limited than in the previous years.

Gottfried

-------------


Corless, R. M., and Jeffrey, D. J.
The unwinding number.
Sigsam Bulletin 30, 2 (June 1996), 28-35.

[online-version:
http://www.apmaths.uwo.ca/~djeffrey/Offp...ditors.pdf
Editor's Corner: The Unwinding Number
]


Corless, R. M., and Jeffrey, D. J.
The Wright omega function
Ontario Research Centre for Computer Algebra and the Department of Applied Mathematics
University of Western Ontario, London, Canada
Reference: J. Calmet et al (Eds.): AISC-Calculemus 2002, LNAI 2385, pp. 76-89, 2002, Springer-Verlag


Reasoning about the elementary functions of complex analysis
Corless, R.M., Jeffrey, D.J. and Davenport, J.H.
[can be found online]
Abstract:
There are many problems with the simplification of elementary functions, particularly over the complex plane. Systems tend to make “howlers” or not to simplify enough. In this paper we outline the “unwinding number” approach to such problems, and show how it can be used to systematise such simplification, even though we have not yet reduced it to a complete algorithm. The unsolved problems are probably more amenable to the techniques of artificial intelligence and theorem proving than the original problem of complex-variable analysis.
Keywords: Elementary functions; Branch cuts; Complex identities.

that looks very very familiar.

yes i considered it too , and also gave up because of the addition problem.

my friend Davenport and i used to consider so-called group rings , which is somewhat nicely introduced by amateur timothy golden's so-called polysigned numbers.

you probably read about that at sci.math.

as for 2-D numbers however , only complex and P3 are intresting , and those are even 'equivalent'.

one of the problems is topology , is you have holes or twists , addition becomes problematic.

and i still cant see a link to tetration Smile

regards

tommy1729
Reply
#16
(04/10/2010, 12:07 PM)tommy1729 Wrote: i wonder where the animated picture (bo mentioned) is btw.

It is here.
Reply
#17
how to easily explain the gray part of gottfrieds graph ?
Reply
#18
(04/09/2010, 08:30 PM)Gottfried Wrote: Hmm, perhaps it is interesting to add some context to that whole idea.

I was playing with two other aspects of iteration of the exponential.


First aspect is the spiraling of the orbit of a complex initial value in the complex plane when repeatedly exponentiated. This suggests, that a polar form of representation of the complex numbers could be more enlightening for the generalization to fractional iterations. And especially, a polar form with a fixpoint as origin.
Indeed this has some charme, even more if one introduces the log-polar-form for a complex number:

where and the arc of the angle.

Then multiplication and division of two values x and y is simply reflected by addition/subtraction of the log-polar components, k'th iterated multiplication (powers) just multiplication of the components by k and exponentiation is and the logarithm results from the inverse of this operation.

Fractional iterates of complex initial numbers look then like a very smooth and even nearly linear interpolation of that lengthes and arcs - but only on first glance: the spiral of the flow is distorted by something which looks a bit like a perihel-effect. It diminuishes (and the flow smoothes) when the fixpoint is approached and extends/chaotizes(?) when the fixpoint is left by iteration.

But leave this here as just another curious observation. More in the focus was the log-polar-representation itself: here the second component (the arc of the angle) exceeds very naturally the interval 0..2*Pi once we introduce iterated addition and multiplication on the log-polar-coefficients and operations back and forth can be meaningfully concatenated without loss of information.

This leads then to the ...
... second aspect, the idea of the introduction of a "winding-number".

Once the idea (and the verbal term) is there, you find related material online; for instance a couple of very good articles of R.M.Corless et al (see below) where they just deal with that winding-number and its possible significance for an extended representation of complex numbers. In the above log-polar-representation we see it nicely that the complex arithmetic via exponentials and logarithms is just somehow modular to modulus 2*Pi (in the arc-parameter) and it is suggestive to introduce a representation for complex numbers and complex algebra which leaves the modular arithmetic behind. It is really nice to see this when playing around a bit. (The observation of "fix-points" vs. "fix-line" as described in this thread was just a detail in the context of exercising in that framework).

However, as also Corless et al discuss in their article: by buying simplicity for multiplication and exponentiation using the log-polar-form we pay with complication for addition. Actually, Corless et al, as I understood them, gave up because of intractability of addition when the winding-number is introduced.

That was also my idea and also I give up here.

We might extend the arc-parameter phi, possibly rescale it to the 2Pi-unit and then allow not only the interval (modulo 1) but the full real line: then a/the "winding-number" is represented by the integer part. But this does not help much: as long as we have no meaningful interpretation of the addition of two log-polarforms, where the winding-number is kept significant such that we can, for instance, invert the operations.
I felt, it would really be a nice toy-project to experiment with this a bit, however... my time (and energy) is somehow more limited than in the previous years.

Gottfried

-------------


Corless, R. M., and Jeffrey, D. J.
The unwinding number.
Sigsam Bulletin 30, 2 (June 1996), 28-35.

[online-version:
http://www.apmaths.uwo.ca/~djeffrey/Offp...ditors.pdf
Editor's Corner: The Unwinding Number
]


Corless, R. M., and Jeffrey, D. J.
The Wright omega function
Ontario Research Centre for Computer Algebra and the Department of Applied Mathematics
University of Western Ontario, London, Canada
Reference: J. Calmet et al (Eds.): AISC-Calculemus 2002, LNAI 2385, pp. 76-89, 2002, Springer-Verlag


Reasoning about the elementary functions of complex analysis
Corless, R.M., Jeffrey, D.J. and Davenport, J.H.
[can be found online]
Abstract:
There are many problems with the simplification of elementary functions, particularly over the complex plane. Systems tend to make “howlers” or not to simplify enough. In this paper we outline the “unwinding number” approach to such problems, and show how it can be used to systematise such simplification, even though we have not yet reduced it to a complete algorithm. The unsolved problems are probably more amenable to the techniques of artificial intelligence and theorem proving than the original problem of complex-variable analysis.
Keywords: Elementary functions; Branch cuts; Complex identities.

first i want to try to explain why the sum ( addition ) is a problem.

( simple , no wiles-topology surreals Smile , and that 'simple' is the reason i try it ; to reach a larger audience and better commen understanding )

explaining a problem usually goes in terms of paradoxes.

the product is already understood.

now the sum.

since sums on a plane , or something that looks like a plane - the spiral from above - is done by straight vector sums , we just do the sum similar to complex sums.

lets enumerate the branches with the elements of Z.

and the elements as 'complex numbers on a branch ' => [C,Z]

note we dont use branches for products , but the products do affect wich branch we are getting to.

and lets say 1 = [1,0].

then a sum is now defined as [C1,Z1] + [C2,Z2] = [C1+C2,Z1+Z2]

and many properties similar to complex numbers exist - i think - .

havent tried proving properties such as distributive etc but the reason i didnt is the paradoxes below.

lets assume [0,Z1] = [0,Z2] = 0 , just like the complex 0 has no angle and there is just one "0".

consider the equation : X^2 = [1,-14]

does this still have 2 zero's ??

X^2 - [1,-14] = (X - sqrt[1,-14])(X + sqrt[1,-14])

i believe it fails , and only has one solution ! (think about it )

... with branch -7 clearly ...

now this might be clarified by doing :

suppose there is no single '0' ( only [0,Z] ) then :

X^2 - [1,-14] = 0

but there is no ' single 0 ' !

so x - x =/= 0 and x =/= x ?? [ paradox 1 ]

but remember we assumed a single '0' AND NOTE AT LEAST ONE ( single or multiple '0' ) NEEDS TO BE TRUE AND CONSISTANT ( otherwise -> paradox )

but this single zero fails too :

[1,-7] - [1,0] = [0,-7]

if we do [0,-7] and add [1,0] again we retrieve [1,-7]

BUT if there is only one '0' we have 0 = [0,0] = [0,-7]

and [1,-7] - [1,0] + [1,0] = [0,0] + [1,0] = [1,0]

DIFFERENT RESULT ! [ paradox 2 ]

also note that - [1,0] + [1,0] = 0 apparently does not hold since y + 0 = y.

but if there is only one '0' and x = [1,0] then x - x =/= 0 !! [ paradox 3 ]

*****

so any consistant idea of a sum and a zero at the same time , will not work here with these " spiral numbers " .


-----

but i wouldnt be tommy1729 if i wouldnt try to use the idea anyway !!

i have researched this long ago , so maybe i will regret posting " nonsense " when i remember what i wrote many years ago ... and what i wrote that lead me to drop the idea ...

but whatever :

maybe we can use these " spiral numbers " and consider the " spiral sum " as something different then sum.

we could try to , and hope not to fail , use the spiral numbers as an extension of R or C or similar. or visa versa.

lets call the spiral numbers S.

then maybe a number that has the topology/extension :

C * S

ALTHOUGH of course one usually extends something that makes more sense then these spiral numbers ...

give it a try ??


regards

tommy1729
Reply


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