- mathfun@mailman.xmission.com - mailman.xmission.com

Re: [math-fun] Interesting limit question (sort of spoiler alert)
by Dan Asimov 06 Sep '20

06 Sep '20

I'm just guessing, but the pictures so far look to me like fractal sets that are in fact connected. Much like how the Mandelbrot set itself seemed to be disconnected in early pictures of it, until higher resolution computer graphics became possible. —Dan Gareth McCaughan wrote: ----- ... ... here's c=0.773+0.57i (which doesn't exhibit such instabilities): https://i.imgur.com/sn6Qk4v.png It seems to be disconnected but _not_ dust-like, which you never get with ordinary Julia sets. -----

2 1

Re: [math-fun] Interesting limit question (sort of spoiler alert)
by Dan Asimov 06 Sep '20

06 Sep '20

Now I'm wondering what all the complex solutions z are of ----- f(1) = z, f(n+1) = (1 + 1/f(n))^n, lim_{n —> oo} f(n) = oo ----- —Dan Kerry Mitchell wrote: ----- If you allow for f(1) < 0 and if I did my analysis correctly, then I see a point similar to George's 1.187452351 at f(1) ~ -0.71774114214285. Taking this problem as a fractal iteration: z(n+1) = (1+1/z(n))^n with z(1) = pixel, then there is a fractal contour connecting the points 1.187452351+0i and -0.71774114214285+0i. -----

2 2

Re: [math-fun] Interesting limit question (sort of spoiler alert)
by Dan Asimov 06 Sep '20

06 Sep '20

Wait — how is f(n+1) = (1 + 1/f(n))^n a "single" iterated function? —Dan ----- ... it is asking about the divergence of a *single* iterated function. -----

2 1

[math-fun] Two Extra Scattering Experiments
by Brad Klee 06 Sep '20

06 Sep '20

"Yellow Trick Shot": https://www.youtube.com/watch?v=N-j50ANlHgI "Blue Wave Bombardment": https://www.youtube.com/watch?v=35fk2iWITwI Plenty of food for thought (at least). --Brad

1 0

Re: [math-fun] |A|+|B|=|C|
by James Propp 05 Sep '20

05 Sep '20

Is there a mathematical grift along the lines of the missing dollar riddle?: https://en.wikipedia.org/wiki/Missing_dollar_riddle I get the feeling I’ve seen such a thing in a movie, or maybe an Abbot and Costello routine. I thought I might find what I’m looking for in “Paper Moon”, but I could only find a clip showing a two-person con involving a marked $20 bill. Jim Propp

2 3

Re: [math-fun] Interesting limit question (sort of spoiler alert)
by Dan Asimov 05 Sep '20

05 Sep '20

Kerry, that's incredibly interesting and strongly suggests that it could lead to a paper worth publishing. —Dan ----- If anyone is interested, here is the fractal boundary that I found looking at the Foias' constant iteration using complex numbers. http://www.kerrymitchellart.com/temp/foias-constant.htm Kerry On Fri, Sep 4, 2020 at 11:17 AM Kerry Mitchell <lkmitch(a)gmail.com> wrote: > If you allow for f(1) < 0 and if I did my analysis correctly, then I see a > point similar to George's 1.187452351 at f(1) ~ -0.71774114214285. > > Taking this problem as a fractal iteration: > > z(n+1) = (1+1/z(n))^n > > with z(1) = pixel, then there is a fractal contour connecting the points > 1.187452351+0i and -0.71774114214285+0i. > -----

1 0

[math-fun] Interesting limit question
by Dan Asimov 05 Sep '20

05 Sep '20

I found this interesting question in an undisclosed location: Let f : Z+ —> R be a function such that a) f(1) > 0 and b) f(n+1) = (1 + 1/f(n))^n for all n >= 1. Is it possible that f(n) —> oo as n —> oo ??? —Dan

6 6

[math-fun] funny to some, boring to many
by Wouter Meeussen 05 Sep '20

05 Sep '20

musing on https://math.stackexchange.com/questions/3760125/on-the-value-of-a-skew-sch… and playing a bit, I was happy to find a fast implementation of the principal specialisation of the Schur function in https://dspace.mit.edu/bitstream/handle/1721.1/107965/26_2016_Article_312.p…. Isn’t the web great? Using that, I looked at what would happen to the high degree polynomial in q if I further specialised it. For prime n, say n=11, set q to exp(2 i pi/n) or, equivalently, transform q^k to q^mod(k,n). For \lambda = {4,3,3,1} this transforms q^12 + 3 q^13 + 8 q^14 + ... + 8 q^96 +3 q^97 + q^98 to 679536 (1 + q + q^2 + ... + q^10) Generally, for n prime, we get as coefficient : c(\lambda , n) /n where c is the content of \lambda in n variables. Here, ‘generally’ means ‘for all partitions with an empty n-core’ alias ‘all partitions that are not hook-shaped’. For these cases only, the coefficient is increased by (-1)^l /n with l being the length of the partition and an extra –(-1)^l is added (to the constant term). The fun was mainly in figuring out how to identify and correct these ‘abnormal’ cases. That’s ‘playing’. The real math would of course start with trying to see the how and why of it, aka as ‘proving it’. Apologies to those who dislike partitions into more than 1 part. Wouter.

1 0

[math-fun] |A|+|B|=|C|
by James Propp 04 Sep '20

04 Sep '20

Does anyone know of any widely-promulgated bogus assertions of the form “The cardinality of set A plus the cardinality of set B equals the cardinality of set C” (where the C is something like the union of A and B) that sounds convincing until you step back and realize either (a) there are elements of C that are elements of neither A nor B, or (b) there are elements of C that are elements of both A and B? Here I’m talking about flimflam at the interface between math and culture (not mistakes made by mathematicians in a mathematical context); e.g., fallacious stuff said by public figures. Jim Propp

3 3

[math-fun] Mersenne conjecture solved?
by Neil Sloane 01 Sep '20

01 Sep '20

There is a claim that there are infinitely many Mersenne primes: Fengsui Liu, <a href=" https://research.journalspress.com/index.php/science/article/view/567">There are Infinitely Many Mersenne Primes</a>, London Journal of Research in Science: Natural and Formal (2020) Vol. 20, Issue 3, Compilation 1.0. The link has been added to sequence A000668, but if the proof is invalid I should add a comment saying so. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane(a)gmail.com

4 4