Comments on: Thue-Morse Odiousness

By: Are you game? (Part II – Numbers are evil! … at least some of them are … others are odious!) « Thesquaredcircle

Mon, 25 May 2009 15:53:12 +0000

[…] called this property putridity. On Tanya Khovanova’s Math blog in a blog entry called Thue-Morse Odiousness, Ingrid Daubechies suggested in a comment that the more appropriate name would be perfidy – a […]

By: Tanya Khovanova

Tanya Khovanova — Sat, 23 May 2009 01:45:02 +0000

I received a letter from John Conway in reply to Ingid’s suggestion of “perfidy”:

MUCH BETTER!

In fact it’s better in two ways - closer still to “parity,” AND a better fit with “evil” and “odious”! Gets my vote right off. JHC

By: Ingrid Daubechies

Ingrid Daubechies — Sat, 31 Jan 2009 18:18:14 +0000

Parity of a number tells us whether it is even or odd; why not name “perfidy” the property that describes whether a number is evil or odious?

By: David Wilson

David Wilson — Fri, 30 Jan 2009 14:39:52 +0000

Grudgingly, I have to concede that John Conway’s word “putridity” is indeed a word. However, I much prefer the older and more time-honored synonym which I recall as the closing word of Edgar Allan Poe’s tale “The Facts in the Case of M. Valdemar”:

“Upon the bed, before the whole company, there lay a nearly liquid mass of loathsome——of detestable putrescence.”

The distastefulness of the subject notwithstanding, don’t you find “putrescence” a much more lilting word than “putridity”?

By: Ctibor O. Zizka

Ctibor O. Zizka — Sat, 24 Jan 2009 19:22:47 +0000

Dear Tanya,

the “classic” Thue-Morse seq. can be expressed as a(n)=S2(n)mod 2, where S2(n) = sum of digits of n, n in base-2 notation. A year ago I was looking on the more general case :
Let Sk(n) = sum of digits of n; n in base-k notation. Let F(t) be some arithmetic function.
Then a(n)= F(Sk(n)) mod m is a generalised Thue-Morse sequence.
Nice properties have sequences where F(Sk(n))=floor(Q*Sk(n)); Q is a positive rational number.

Partial sums of a generalized Thue-Morse sequence a(n)=F(Sk(n)) mod m are fractal –> they consist of series of the generalized batrachion Blancmange function (similarly to Hofstadter’s Q-Sequence, Hofstadter-Conway 10000$ and Mallows seq. etc.). A good article is http://arxiv.org/PS_cache/math/pdf/0406/0406078v1.pdf which mirrors my computational results from an ergodic/Pascal-adic transformation point of view. Partial sums of such generalized Thue-Morse seq. also points to the Minkowski Question Mark function and relates the continued-fraction representation of the real numbers to their k-ary expansion http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.1265v2.pdf.

I do not know if my note is of interest for you,

Have a nice day,

Ctibor