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Apr
24
comment What is the Mathematical model FOR Scytale
@MattSamuel Frank's son and a co-writer filled in another half dozen books, three set long before Dune and, I think, two set after. This is from a big box of notes they found after Frank's death. Actually, looking online, it appears they wrote yet more books after I stopped paying attention. Worth reading once, certainly
Apr
23
comment What is the Mathematical model FOR Scytale
A character in the Dune books by Frank Herbert. en.wikipedia.org/wiki/Scytale_%28Dune%29
Mar
30
comment Factor matrix ${\bf A}$ into the product ${\bf B}{\bf C}$ where ${\bf C}$ has no negative entries and ${\bf B}$ has few non-zero entries
How about if you wait on the MSE question for a few days? Also, your link to it is faulty, there are some extraneous characters before the http. I tried to fix it with no luck math.stackexchange.com/questions/1719656/…
Mar
30
revised Factor matrix ${\bf A}$ into the product ${\bf B}{\bf C}$ where ${\bf C}$ has no negative entries and ${\bf B}$ has few non-zero entries
deleted 10 characters in body
Mar
12
revised a family of Pellian equations
added 318 characters in body
Mar
12
revised a family of Pellian equations
added 136 characters in body
Mar
12
answered a family of Pellian equations
Mar
12
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
added 193 characters in body
Mar
12
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
The picture is wonderful. Thank you.
Mar
12
comment a family of Pellian equations
found slides from a nice talk by Keith Matthews numbertheory.org/pdfs/dujella_slides.pdf
Mar
12
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
Gottfried, if you are interested, one thing that would be nice would be pictures of, say, the half iterate of sine, then the one third iterate, then maybe the two thirds iterate. I think the one third can be done directly from the Abel function and inverse function, maybe the two thirds can be done in the same way. I admit, the resulting pictures are not really dramatic.
Mar
11
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
Thank you, Gottfried, this is lovely. I did the computations in Pari, but I don't know how to program in it, so i had to find one new term at a time, in the asymptotic series for the Abel function. Good, maximum height 1.140179
Mar
9
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
@GottfriedHelms Zenit looked promising for a bit but then crumbled. Wolfsburg and Bayern Munich are still in it, I think.
Mar
9
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
@GottfriedHelms in that case, there could be some benefit to the arduous task of finding a few more terms in the expansion of $\alpha(z).$ I do give one more term above, that I did not use in the C++ program. You could try both versions in gp-pari and see if any gets substantially better when including the $91543 z^6 / something$ term
Mar
9
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
@GottfriedHelms I did find an old email address for you and sent the C++ program, less than a page and no included files (other than standard system files). Looking at it, the real work went into error estimates, finding initial endpoints for bisection, and so on. If you are using a language with arbitrary precision reals you might find ways to avoid all that extra work.
Mar
9
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
@GottfriedHelms I had to do that myself. I got computer versions of $\alpha()$ that worked. For the inverse function I numerically solved for $t$ in $\alpha(t) = A.$ That was a simple bisection root finder, but I remember adjusting interval limits appropriate for a target $A$ was not so simple. If I can find the code I will email you.
Mar
4
comment Combinatorial aspects of continued fractions
Have a look at the recent cambridge.org/us/academic/subjects/mathematics/number-theory/…
Feb
29
awarded  Nice Answer
Feb
29
comment Counting lattice points inside a three-dimensional ellipsoid
Fair enough....
Feb
28
comment Counting lattice points inside a three-dimensional ellipsoid
On second thought, no idea how bad this example is from your viewpoint. However, everything can be calculated here. The difference in representation counts for $T^2$ is $2 \delta T,$ where $\delta = 0$ if $T \equiv 0 \pmod 3,$ then $\delta = 1$ if $T \equiv 1,2 \pmod 6,$ and $\delta = -1$ if $T \equiv 4,5 \pmod 6.$