Comments for Rhapsody in Numbers

Comment on The Collatz Fractal by Xander

Xander — Tue, 24 Jan 2012 17:36:29 +0000

The general goal was to get a smooth function that behaved in a particular manner over the integers. Since [latex]i\sin(\pi z)[/latex] vanishes over the integers, we can get the behaviour that we want, i.e. [latex]C(x)|_{\mathbb{Z}} = f(x)[/latex] if we drop it. It was sloppy of me to not go over those details. Sorry.

Alternatively, I suppose that we could do a better job of choosing the characteristic functions in the first place. [latex]cos(\pi z/2) = \pm 1[/latex] when [latex]z[/latex] is even, and 0 otherwise; and [latex]\sin(\pi z/2) = \pm 1[/latex] when [latex]z[/latex] is odd, and 0 otherwise. Squaring these gives us the behaviour we want for the characteristic functions, and application of the power reducing formulae will give us the formula I originally gave.

The morals of the story: first, the smooth extensions of functions to larger domains need not be unique; and second, I shouldn’t try to work from three sets of notes at the same time.

Thanks for keeping me honest,
xander

Comment on The Collatz Fractal by Jason Liszka

Jason Liszka — Tue, 24 Jan 2012 05:27:44 +0000

Also, I could only get
C(z) = 1/4 * (2 + 7z – (2 + 5z) cos (pi *z))
from
C(z) = (-1^z + 1)/2 * (z / 2) – (-1^z – 1)/2 * (3*z + 1)
if I let -1^z = cos (pi * z). But -1^z = cos (pi * z) – i sin (pi * z).

cos(pi*z) behaves the same on positive integers as -1^z, so I guess the choice is arbitrary. I was just a little confused by the derivation. Using -1^z I get a different fractal that diverges almost everywhere.

Great post! Fun thinking about this.

Comment on The Collatz Fractal by Xander

Xander — Mon, 23 Jan 2012 15:12:28 +0000

You are correct. My nemesis, the dreaded WRONG SIGN has reared his ugly head once again. I have corrected the post.

Thank you,
xander

Comment on The Collatz Fractal by Jason Liszka

Jason Liszka — Mon, 23 Jan 2012 01:43:34 +0000

Interesting! I tried to replicate this and noticed you have a sign error in your first definition of C(z). You have
C(z) = (-1^z + 1)/2 * (z / 2) + (-1^z – 1)/2 * (3*z + 1)
but I think it should be
C(z) = (-1^z + 1)/2 * (z / 2) – (-1^z – 1)/2 * (3*z + 1).
With the first definition, C(5) = -16 and C^n(i) diverges very rapidly. With the second definition, C(5) = 16 and C^n(i) diverges, but not quite so dramatically.

Comment on Spider’s Web by Carrie

Carrie — Thu, 19 Jan 2012 22:40:07 +0000

Perfect subject for the prompt :) It almost felt like the narration of a documentary, I could see the spider crafting her web, strand by gooey strand

Comment on A Typical Morning by Xander

Xander — Tue, 13 Dec 2011 16:05:19 +0000

Suzanne: The dog is fine. She and Katja are getting along very well.

xander

Comment on A Typical Morning by Tom Nixon

Tom Nixon — Mon, 12 Dec 2011 23:21:06 +0000

It’s always nice to have a reminder of why I did not procreate (even though I admit to going through the motions several times). Me + offspring would = a severely irrational me. Keep up the good work (somebody’s gotta do it). With love to you both, great-Uncle Tom (cabin optional).

Comment on A Typical Morning by Suzanne Henderson

Suzanne Henderson — Mon, 12 Dec 2011 07:16:04 +0000

Xander, You are very funny and of course I noticed that Katja always comes first and that is the way it is. I am so happy for your new family. How is the doggy by the way?
Auntie, Suzanne

Comment on A Typical Morning by Xander

Xander — Sat, 10 Dec 2011 17:13:56 +0000

She’s in the office with me right now, “helping” with my topology. ;)

xander

Comment on A Typical Morning by Supermaren

Supermaren — Sat, 10 Dec 2011 01:47:10 +0000

She’ll be helping you out with those proofs in no time. Just you wait.