Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
uniqueness
#11
Ansus Wrote:To be simple we have two sets of methods: those applicable for lower bases (and they are equal) and those applicable for higher bases (I suppose they also equal with each other). Probably we would not need a proof for equivalence of these two sets of methods because their areas of applicability do not overlap.

They do overlap, Matrix-Power and Intuitive Abel are applicable to all bases .
And even Cauchy-Integral we saw that there are similar methods for for both base ranges.

Quote:P.S. Where I could learn about matrix powers method?

See the pdf Gottfried posted here.
Reply
#12
Ansus Wrote:Does not it use Carleman matrices whic also used for partial iteration formula?

What is partial iteration?

Indeed the matrix power method coincides with regular iteration if applied to a fixed point.
Reply
#13
Ansus Wrote:You derived your Newton formula using Carleman matrices, and they also use Carleman matrices of arbitrary iteration of exp function.

thats exactly what i said: at fixed points both methods coincide.
Reply
#14
bo198214 Wrote:
tommy1729 Wrote:f(x) = (x + q(x)) ^2 = x^2 + 2q(x)x + q(x)^2

f(f(x)) = ( x^2 + 2q(x)x + q(x)^2 + q(x^2 + 2q(x)x + q(x)^2) ) ^2

q must appear inside q.

so ??

thats no proof or disproof of anything ?


regards

tommy1729
Reply
#15
tommy1729 Wrote:here are the equations that make half-iterate of exp(x) unique :

(under condition f(x) maps reals to reals and f(x) > x )

exp(x)

= f(f(x))

= D f(f(x)) = f ' (f(x)) * f ' (x)

= D^2 f(f(x)) = f '' (f(x)) * f ' (x)^2 + f ' (f(x)) * f '' (x)

= D^3 f(f(x)) = D^4 f(f(x))


regards

tommy1729

is there a solution where f ' (x) is not strictly rising but f(x) is Coo ??
Reply
#16
tommy1729 Wrote:
tommy1729 Wrote:here are the equations that make half-iterate of exp(x) unique :

(under condition f(x) maps reals to reals and f(x) > x )

exp(x)

= f(f(x))

= D f(f(x)) = f ' (f(x)) * f ' (x)

= D^2 f(f(x)) = f '' (f(x)) * f ' (x)^2 + f ' (f(x)) * f '' (x)

= D^3 f(f(x)) = D^4 f(f(x))


regards

tommy1729

is there a solution where f ' ' (x) is not strictly rising but f(x) is Coo ??

*corrected*
Reply
#17
tommy1729 Wrote:
tommy1729 Wrote:
tommy1729 Wrote:here are the equations that make half-iterate of exp(x) unique :

(under condition f(x) maps reals to reals and f(x) > x )

exp(x)

= f(f(x))

= D f(f(x)) = f ' (f(x)) * f ' (x)

= D^2 f(f(x)) = f '' (f(x)) * f ' (x)^2 + f ' (f(x)) * f '' (x)

= D^3 f(f(x)) = D^4 f(f(x))


regards

tommy1729

is there a solution where f ' ' (x) is not strictly rising but f(x) is Coo ??

*corrected*

ok.

i see now.

i was a bit confused ...

the new uniqueness conditions , i think ,

(under condition f(x) maps reals to reals and f(x) > x )

f ' (0) > 0

f '' (0) > 0

f ''' (0) > 0

...



and f(x) is Coo


regards

tommy1729
Reply
#18
Before uniqueness, one should show however existence, because a uniqueness criterion makes no sense if there are no functions satisfying this criterion ...
Reply
#19
bo198214 Wrote:Before uniqueness, one should show however existence, because a uniqueness criterion makes no sense if there are no functions satisfying this criterion ...

isnt that already done ?

by kneser ?

by robbins ?

doesnt robbins solution satisfy my uniqueness criterions ?

regards

tommy1729
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 9,909 02/17/2017, 05:21 AM
Last Post: JmsNxn
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 13,427 01/09/2017, 02:41 AM
Last Post: Gottfried
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,379 03/19/2016, 10:44 AM
Last Post: fivexthethird
  [2014] Uniqueness of periodic superfunction tommy1729 0 1,758 11/09/2014, 10:20 PM
Last Post: tommy1729
  Real-analytic tetration uniqueness criterion? mike3 25 19,369 06/15/2014, 10:17 PM
Last Post: tommy1729
  exp^[1/2](x) uniqueness from 2sinh ? tommy1729 1 2,035 06/03/2014, 09:58 PM
Last Post: tommy1729
  Uniqueness Criterion for Tetration jaydfox 9 10,477 05/01/2014, 10:21 PM
Last Post: tommy1729
  Uniqueness of Ansus' extended sum superfunction bo198214 4 6,242 10/25/2013, 11:27 PM
Last Post: tommy1729
  A question concerning uniqueness JmsNxn 3 5,179 10/06/2011, 04:32 AM
Last Post: sheldonison
  tetration bending uniqueness ? tommy1729 16 16,453 06/09/2011, 12:26 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)