Re: [math-fun] binary roulette wheels
I wonder if analogous continuous objects can exist. For instance, can there be a continuous function on the circle R/Z f : R/Z —> R and some fixed L > 0, such that f contains all continuous functions g : [0, L] —> R as represented on an arc [c, c+L] ⊂ R/Z of the circle, via g(x) = f(x+c) (addition modulo 1) ??? Or something along these lines, maybe for a restricted class of functions f and g ? —Dan Adam Goucher wrote: ----- de Bruijn sequences Veit Elser wrote: ----- Is there a name for cyclic sequences (necklaces) of length 2^n that contain all the integers 0, … , 2^n-1 expressed in binary in the 2^n subsequences of length n? For example, for n=4 the sequence 0000100110101111 contains 0, 1, 2, 4, 9, 3, 6, 13, 10, 5, 11, 7, 15, 14, 12, 8. ----- -----
I am feeling strongly that Dan's universal circular function is impossible. For any L, let N be ceil(1/L). Consider just the family of N+1 functions f[i](x) = ix, i ranging from 0 to N. It seems obvious to me that R/Z isn't big enough to fit them all. On Sat, Oct 3, 2020 at 12:24 PM Dan Asimov <dasimov@earthlink.net> wrote:
I wonder if analogous continuous objects can exist.
For instance, can there be a continuous function on the circle R/Z
f : R/Z —> R
and some fixed L > 0, such that f contains all continuous functions
g : [0, L] —> R
as represented on an arc [c, c+L] ⊂ R/Z of the circle, via
g(x) = f(x+c)
(addition modulo 1) ???
Or something along these lines, maybe for a restricted class of functions f and g ?
—Dan
Adam Goucher wrote: ----- de Bruijn sequences
Veit Elser wrote: ----- Is there a name for cyclic sequences (necklaces) of length 2^n that contain all the integers 0, … , 2^n-1 expressed in binary in the 2^n subsequences of length n? For example, for n=4 the sequence
0000100110101111
contains 0, 1, 2, 4, 9, 3, 6, 13, 10, 5, 11, 7, 15, 14, 12, 8. ----- -----
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Or even simpler than that, only 1/L constant functions, out of uncountably many, are possible. Andy On Sat, Oct 3, 2020 at 1:29 PM Allan Wechsler <acwacw@gmail.com> wrote:
I am feeling strongly that Dan's universal circular function is impossible. For any L, let N be ceil(1/L). Consider just the family of N+1 functions f[i](x) = ix, i ranging from 0 to N. It seems obvious to me that R/Z isn't big enough to fit them all.
On Sat, Oct 3, 2020 at 12:24 PM Dan Asimov <dasimov@earthlink.net> wrote:
I wonder if analogous continuous objects can exist.
For instance, can there be a continuous function on the circle R/Z
f : R/Z —> R
and some fixed L > 0, such that f contains all continuous functions
g : [0, L] —> R
as represented on an arc [c, c+L] ⊂ R/Z of the circle, via
g(x) = f(x+c)
(addition modulo 1) ???
Or something along these lines, maybe for a restricted class of functions f and g ?
—Dan
Adam Goucher wrote: ----- de Bruijn sequences
Veit Elser wrote: ----- Is there a name for cyclic sequences (necklaces) of length 2^n that contain all the integers 0, … , 2^n-1 expressed in binary in the 2^n subsequences of length n? For example, for n=4 the sequence
0000100110101111
contains 0, 1, 2, 4, 9, 3, 6, 13, 10, 5, 11, 7, 15, 14, 12, 8. ----- -----
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-- Andy.Latto@pobox.com
participants (3)
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Allan Wechsler -
Andy Latto -
Dan Asimov