July 2016 - mathfun@mailman.xmission.com

Re: [math-fun] 9*99 etc
by James Propp 14 Jul '16

14 Jul '16

Thanks, Neil! I don't know why I didn't think to check the OEIS. Jim, that is https://oeis.org/A027828. > Actually it's https://oeis.org/A027878. > If you find out something interesting, please add a comment there. There > is a certain amount of info there already, of course. > I doesn't appear that anyone has published anything about the middle digits. When I get a chance I might tally the hundred middle digits for progressively larger n, to see if some sort of asymptotic equidistribution appears to be going on there. Jim

3 3

[math-fun] Interesting field of ratios of formal power series
by Dan Asimov 14 Jul '16

14 Jul '16

Let K be a field. Then K[[x]] denotes the ring of formal power series over K. I.e., {Sum_0<=j<oo} c_j x^j} for all sequences c = (c_0, c_1, c_2, c_3, …) of elements of K. Two elements of K[[x]] are defined as equal if an only if their coefficients of each power x^j of x are equal, j = 0,1,2,3,…. Addition and multiplication make sense (since only finitely many terms need be added to get the coefficient of x^j of the sum or product). So K[[x]] is a ring. It is easy to check that K[[x]] has no zero-divisors, so it is an integral domain, so has a field of quotients. I don't know the standard notation for this, but we can write Quo((x)) = {f(x)/g(x) | f, g in K[[x]]}, where we define f(x)/g(x) = h(x)/k(x) precisely when f(x) k(x) = g(x) h(x), as you might expect. * * * Interesting fact (exercise): The field Quo((x)) is identical to the formal Laurent series over K, namely Quo((x)) = {Sum_L<=j<oo} c_j x^j} for all sequences c = (c_L, c_(L+1), c_(L+2), c_(L+3), …) of elements of K where L is any integer (possibly negative). ------------------------------------------- QUESTIONS: --------- 1) In the simplest case — where the base field K = {0,1} — what is Quo((x)) ??? In other words, can we decompose Quo((x)) = quotients of formal power series in x over {0,1} into some kind of nice structure? It would in this vein be nice to know the prime elements of K[[x]] while we're at it. 2) And what can we say about the algebraic closure of Quo((x)) ??? —Dan

1 0

[math-fun] 9 * 99 * 999 * ...
by James Propp 14 Jul '16

14 Jul '16

One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition. I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply. Jim Propp

5 9

Re: [math-fun] Draft of my July 2016 blog post
by Henry Baker 13 Jul '16

13 Jul '16

I just heard lecture last Sunday about the *Jain* religion, and was astonished to hear that it incorporates the concepts of both "infinity" and "countable" v. "uncountable" in its vocabulary. I did a quick Google search, but couldn't find out very much about the intersection of the Jain religion and Indian mathematics. I don't think that Ramanujan was Jain, but he certainly would have known something of the Jain religion and ideas. If this connection holds up, then the Indian/South Asian origin of "infinity" becomes firmly established, and the Greeks must have picked it up from them. So this begs the question: which infinity did Ramanujan know? Aleph-0, Aleph-1, omega, ??? At 08:33 PM 7/12/2016, James Propp wrote: >Hi, I started writing a new draft titled "The Man Who Knew Infinity: what the film will teach you (and what it won't)" and would love to get your feedback.

2 1

[math-fun] Amazon primes in OEIS ?
by Henry Baker 13 Jul '16

13 Jul '16

Are Amazon primes in OEIS ? ;-) They look to be highly composite, to me....

1 0

[math-fun] Karatsuba Multiplikation (history) question and Wikipedia
by Joerg Arndt 13 Jul '16

13 Jul '16

More (and worse) Wiki-weirdness: The equivalent of https://en.wikipedia.org/wiki/Karatsuba_algorithm in German is https://de.wikipedia.org/wiki/Karazuba-Algorithmus which states that "Die Methode von Karazuba wurde zum Vorbild fÃ¼r das Teile-und-herrsche-Prinzip in der Informatik." which (roughly) translates as "Karatsuba's [multiplication] method became the prototype for the divide and conquer principle in computer science." ...and that is _exactly_ what his daughter wants the world to believe. Can anybody refute that statement? Thanks and best regards, jj

4 8

[math-fun] In[84]:= Limit[HarmonicNumber[E^n,E^(1/n)]/n,n->∞]
by Bill Gosper 13 Jul '16

13 Jul '16

Out[84]= Limit[HarmonicNumber[E^n,E^(1/n)]/n,n->\[Infinity]] looks suspiciously like In[83]:= 1-1./E Out[83]= 0.6321205588285577 In[85]:= Table[HarmonicNumber[E^n,E^(1/n)]/n/.n->9.`44^k,{k,11}]//Column Out[85]= 0.682225592958289245702463034317220139455191 0.637623065384041617378550215129083728051 0.6327312065329068054767860013575408919 0.63218839948584639769227186418818269 0.632128096567272612533554169955621 0.6321213963536978066178286646303 0.63212065188688949834913479179 0.632120569168372114161664887 0.63212055997742594644041940 0.63212055895620970815410 0.632120558842741237265 --rwg (HarmonicNumber[a, s] == Zeta[s] - HurwitzZeta[s, 1 + a])

1 1

Re: [math-fun] Checkerboard effect
by Dan Asimov 13 Jul '16

13 Jul '16

I can't believe we're actually discussing this. The "squares" near the big X visible in the mod 2 picture are misshapen squares. This is immediate upon magnifying the image. That's more than enough to make the X visible. —Dan -----Original Message----- >From: James Propp <jamespropp(a)gmail.com> >Sent: Jul 12, 2016 9:06 PM >To: math-fun <math-fun(a)mailman.xmission.com> >Subject: Re: [math-fun] Checkerboard effect > >"Processing issue"? Meaning (computer) image processing, or (human) visual >processing, or what? > >Jim > >On Tuesday, July 12, 2016, Richard Howard <rich(a)richardehoward.com> wrote: > >> Try relaxing your eyes as if looking at a 3-D image--I see lots of patterns >> and multiple diagonal lines. >> I suspect it is a processing issue. >> --R >> >> On Tue, Jul 12, 2016 at 11:41 AM Christian Lawson-Perfect < >> christianperfect(a)gmail.com <javascript:;>> wrote: >> >> > To make life easier, here's a direct link to the image in question: >> > >> > >> http://www.madandmoonly.com/doctormatt/mathematics/ulamSpirals/mod2Spiral.p… >> > >> > On Tue, 12 Jul 2016 at 15:56 James Propp <jamespropp(a)gmail.com >> <javascript:;>> wrote: >> > >> > > Go to >> > > >> > > >> > >> http://www.madandmoonly.com/doctormatt/mathematics/ulamSpirals/ulamSpirals.… >> > > , hover your mouse over "modulo 2 (low res) >> > > < >> > > >> > >> http://www.madandmoonly.com/doctormatt/mathematics/ulamSpirals/ulamSpirals.… >> > > >", >> > > and step back. You should see a large "X" in the image. (With my >> glasses >> > > on, I see the "X" when I'm about five feet away from my laptop; with my >> > > glasss off, a foot will do.) >> > > >> > > What's happening here? >> > > >> > > That's really two questions. What is the image made of, on the pixel >> > level? >> > > (I'm guessing that some of the "squares" are actually rectangles with >> > > aspect ratio close to 1.) And: What are my eyes and brain doing? (Why >> is >> > > the effect so salient from far away but invisible up close?) >> > > >> > > Jim Propp >> > > _______________________________________________ >> > > math-fun mailing list >> > > math-fun(a)mailman.xmission.com <javascript:;> >> > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > >> > _______________________________________________ >> > math-fun mailing list >> > math-fun(a)mailman.xmission.com <javascript:;> >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com <javascript:;> >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

2 1

[math-fun] Checkerboard effect
by James Propp 13 Jul '16

13 Jul '16

Go to http://www.madandmoonly.com/doctormatt/mathematics/ulamSpirals/ulamSpirals.… , hover your mouse over "modulo 2 (low res) <http://www.madandmoonly.com/doctormatt/mathematics/ulamSpirals/ulamSpirals.…>", and step back. You should see a large "X" in the image. (With my glasses on, I see the "X" when I'm about five feet away from my laptop; with my glasss off, a foot will do.) What's happening here? That's really two questions. What is the image made of, on the pixel level? (I'm guessing that some of the "squares" are actually rectangles with aspect ratio close to 1.) And: What are my eyes and brain doing? (Why is the effect so salient from far away but invisible up close?) Jim Propp

4 5

Re: [math-fun] Math or mind-reading?
by Henry Baker 13 Jul '16

13 Jul '16

"Angelina hiked *along* a 2250-foot mountain trail". Perhaps she didn't finish it. Perhaps she didn't even start at the beginning. If you recall, many people have "hiked along" the Appalachian Trail (including at least one U.S. governor who was looking for the Trail in South America), but relatively few have actually hiked it from end to end. At 06:22 PM 7/12/2016, Bill Gosper wrote: >17. Angelina hiked along a 2250-foot mountain trail. She stopped 5 times >along the way to rest, walking the same distance between each stop. How far >did Angelina hike before the first stop?

1 0