December 2012 - mathfun@mailman.xmission.com

Re: [math-fun] J Borwein's sig
by Henry Baker 26 Dec '12

26 Dec '12

Ok, I think I see a potential definitional problem. "Long strands of DNA floating in a cells nucleus can easily become tangled, just as a long extension cord does when left in a heap." <I>. Let's suppose that one end of the extension cord is plugged into the inside wall of a rigid sphere, and that the other end is attached to another point on the inside wall of a sphere, and the entire length of the cord lies within the sphere and is (of course) restricted to remain within the sphere. Assuming that the extension cord was originally "untangled", meaning that it could slowly contract like a rubber band until it becomes a chord on the sphere, then no normal "tangling" you can do to the cord (so long as you never, ever unplug either end) will change this "untangled" property. <II>. However, if you picked up two ends of an extension cord tangled at random and then attached them to the inside of a rigid sphere as above, there is no guarantee that the tangle could ever be untangled without unplugging at least one end of the cord. I would define situation <II> where it is actually impossible to untangle the cord as "knotted", whereas situation <I> is merely "tangled". In real life, of course, even untangling a cord in situation <I> may be extremely difficult w/o unplugging it, but it isn't impossible. At 12:32 PM 12/25/2012, W. Edwin Clark wrote: >In this article it is explained how some "circular" DNA knots are obtained: > >http://physics.aps.org/story/v12/st25

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Re: [math-fun] Puzzle I heard -- spoiler alert
by Adam P. Goucher 26 Dec '12

26 Dec '12

Yes, that is a beautiful puzzle. It actually works if you replace 'open' with 'closed'. For a triangle to be acute, we require x^2, y^2 and z^2 to be the sidelengths of a triangle. So, let X' be a subset of size 12 of the interval [1, 144], obtained by squaring each element of X. Order X' and call the elements x_1 < x_2 < ... < x_12. Assume there are no triangles with sidelengths in X'. This gives us the following set of weak inequalities: x_1 + x_2 <= x_3 x_2 + x_3 <= x_4 x_3 + x_4 <= x_5 ... x_10 + x_11 <= x_12 By substituting the first inequality into the second, and the first two into the third, and so on, we obtain this: 89 x_1 + 55 x_2 <= x_12 But, we still have another strict inequality! x_1 < x_2 --> 144 x_1 < x_12 As x_1 is at least 1, x_12 is strictly greater than 144. Contradiction. Sincerely, Adam P. Goucher http://cp4space.wordpress.com > ----- Original Message ----- > From: Dan Asimov > Sent: 12/26/12 05:10 AM > To: math-fun > Subject: [math-fun] Puzzle I heard > > Not a killer, but cute: > > Suppose X is a subset of size 12 of the open interval (1,12). > > Show there must exist distinct x, y, z in X such that > there is an acute-angled triangle of sides x, y, z. > > --Dan > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] J Borwein's sig
by W. Edwin Clark 25 Dec '12

25 Dec '12

I'm not too clear on where they got their circular DNA in the Summers article. It is mentioned in the article that it was genetically engineered. In this article it is explained how some "circular" DNA knots are obtained:http://physics.aps.org/story/v12/st25 Of course one cannot make a non-trivial (mathematical) knot with linear DNA without joining the two ends. But linear DNA can become tangled and these tangles can be studied mathematically (http://math.sfsu.edu/vazquez/sugakuDNA-English.pdf) On Tue, Dec 25, 2012 at 1:16 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > This AMS article talks exclusively about _circular_ DNA -- e.g., that > found in some bacteria & in mitochondria. I believe that this is because > bacteria/mitochondria have to use DNA itself to make stuff directly, rather > than first translating into RNA. > > It may also have relevance to _RNA_, but this AMS paper doesn't talk about > RNA. > > Once again, I'm not aware of any evidence that non-circular DNA is knotted. > > At 09:19 AM 12/25/2012, W. Edwin Clark wrote: > >You will find pictures of knots and links of DNA in this 1995 Notices of > >AMS article by DeWitt Summers > >one of the first knot theorist to look at DNA > > http://www.ams.org/notices/199505/sumners.pdf > > > >On Tue, Dec 25, 2012 at 11:10 AM, Henry Baker <hbaker1(a)pipeline.com> > wrote: > > > >> DNA does get incredibly _folded_ & kinked, but not mathematically > >> _knotted_, at least according to the lectures I've listened to. > >> > >> There are some wonderful videos of DNA being replicated by a graphics > >> group in Australia, which show how the two different strands are > separated, > >> one of them is broken up into chunks (because it has to be copied the > >> other direction) and then copied & put together. > >> > >> If you believe these videos, there aren't any knots involved. > >> > >> At 07:45 AM 12/25/2012, Michael Kleber wrote: > >> >Not at all! DNA does indeed become knotted; most notably, this is > >> >inevitable when a circular chromosome gets replicated to create two > >> >daughter chromosomes, whose knotting is forced by the helical > structure of > >> >DNA. The cell's solution is a class of enzymes called topoisomerases, > >> >which serve only to break the phosphate backbone of one strand, pass > the > >> >other through it, and then repair the broken one -- a topology-only > >> >chemical reaction. > >> > > >> >--Michael > >> >On Dec 25, 2012 7:08 AM, "Henry Baker" <hbaker1(a)pipeline.com> wrote: > >> > > >> >> The amazing thing about DNA is that even though it is incredibly > long, it > >> >> never seems to "knot". > >> >> > >> >> I guess that if it ever does get knotted, the whole cell commits > suicide > >> >> (or at least tries to -- perhaps a source of cancer?) > >> >> > >> >> Some relatively small bacterial and mitochondrial DNA are circular > (except > >> >> when being copied), which probably aids in keeping them from > knotting. > >> >> > >> >> At 04:47 PM 12/24/2012, Fred lunnon wrote: > >> >> >On 12/25/12, Bill Gosper <billgosper(a)gmail.com> wrote: > >> >> >> Awful thought: What is the expected time at which a 3D random > walk first > >> >> >> becomes "knotted"? (W.r.t. pulling on the ends.) > >> >> > > >> >> >Bearing in mind that the ends may in general lie deep within the > convex hull > >> >> >of the walk, perhaps this question requires rather more careful > >> >> >definition ... WFL > > > >

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Re: [math-fun] J Borwein's sig
by Henry Baker 25 Dec '12

25 Dec '12

DNA does get incredibly _folded_ & kinked, but not mathematically _knotted_, at least according to the lectures I've listened to. There are some wonderful videos of DNA being replicated by a graphics group in Australia, which show how the two different strands are separated, one of them is broken up into chunks (because it has to be copied the other direction) and then copied & put together. If you believe these videos, there aren't any knots involved. At 07:45 AM 12/25/2012, Michael Kleber wrote: >Not at all! DNA does indeed become knotted; most notably, this is >inevitable when a circular chromosome gets replicated to create two >daughter chromosomes, whose knotting is forced by the helical structure of >DNA. The cell's solution is a class of enzymes called topoisomerases, >which serve only to break the phosphate backbone of one strand, pass the >other through it, and then repair the broken one -- a topology-only >chemical reaction. > >--Michael >On Dec 25, 2012 7:08 AM, "Henry Baker" <hbaker1(a)pipeline.com> wrote: > >> The amazing thing about DNA is that even though it is incredibly long, it >> never seems to "knot". >> >> I guess that if it ever does get knotted, the whole cell commits suicide >> (or at least tries to -- perhaps a source of cancer?) >> >> Some relatively small bacterial and mitochondrial DNA are circular (except >> when being copied), which probably aids in keeping them from knotting. >> >> At 04:47 PM 12/24/2012, Fred lunnon wrote: >> >On 12/25/12, Bill Gosper <billgosper(a)gmail.com> wrote: >> >> Awful thought: What is the expected time at which a 3D random walk first >> >> becomes "knotted"? (W.r.t. pulling on the ends.) >> > >> >Bearing in mind that the ends may in general lie deep within the convex hull >> >of the walk, perhaps this question requires rather more careful >> >definition ... WFL

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Re: [math-fun] J Borwein's sig
by Henry Baker 25 Dec '12

25 Dec '12

The amazing thing about DNA is that even though it is incredibly long, it never seems to "knot". I guess that if it ever does get knotted, the whole cell commits suicide (or at least tries to -- perhaps a source of cancer?) Some relatively small bacterial and mitochondrial DNA are circular (except when being copied), which probably aids in keeping them from knotting. At 04:47 PM 12/24/2012, Fred lunnon wrote: >On 12/25/12, Bill Gosper <billgosper(a)gmail.com> wrote: >> Awful thought: What is the expected time at which a 3D random walk first >> becomes "knotted"? (W.r.t. pulling on the ends.) > >Bearing in mind that the ends may in general lie deep within the convex hull >of the walk, perhaps this question requires rather more careful >definition ... WFL

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Re: [math-fun] knotted random walks
by W. Edwin Clark 24 Dec '12

24 Dec '12

Enver Karadayi recently wrote a dissertation <http://scholarcommons.usf.edu/cgi/viewcontent.cgi?article=4707&context=etd&…>in which he generated successively n random points in the unit cube. He connected successive points with straight line segments and then connected the first point to the last point. His simulations (if I understand correctly) show that after n = 40, one almost always get a non-trivial knot. The difficulty is determining when you have a non-trivial knot. One runs up against the unknotting problem<http://en.wikipedia.org/wiki/Unknotting_problem> . Karadayi's method seems to be to compute the determinant of the knot and if it is divisible by a prime p > 2 then it is "p-colorable" which implies that it is not the unknot. If the determinant of the knot is 1 then it may be knotted, but presumably the probability is low. (KnotInfo <http://www.indiana.edu/~knotinfo/> shows 17 prime knots with crossing numbers 10 to 12 which have determinant = 1). It should be easy (well, at least possible) to do similar simulations for random walks. --Edwin On Mon, Dec 24, 2012 at 8:56 PM, <rcs(a)xmission.com> wrote: > Or just draw a straight line from one end of the walk to the other. > Maybe exclude cases where this line intersects the path-so-far. > > Rich > > --- > Quoting meekerdb <meekerdb(a)verizon.net>: > > On 12/24/2012 4:47 PM, Fred lunnon wrote: >> >>> On 12/25/12, Bill Gosper<billgosper(a)gmail.com> wrote: >>> >>>> Awful thought: What is the expected time at which a 3D random walk >>>> first >>>> becomes "knotted"? (W.r.t. pulling on the ends.) >>>> >>> Bearing in mind that the ends may in general lie deep within the convex >>> hull >>> of the walk, perhaps this question requires rather more careful >>> definition ... WFL >>> >> >> You could constrain the question to those random walks that return to >> the starting point (although of measure zero) or you could try defining >> a different kind of 'random walk' that starts with a unit loop and >> randomly expands and rearranges it, with crossovers allowed. >> >> Brent >> >> ______________________________**_________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun> >> > > > > ______________________________**_________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun> >

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[math-fun] J Borwein's sig
by Bill Gosper 24 Dec '12

24 Dec '12

includes "Take a long walk on Pi: http://gigapan.com/gigapans/106803 " which looks sparser and stringier and more symmetrical than I expected. But it must not be qualitatively different from good random, else we'd be hearing more. At least it's not decimal. Correction: 3 EllipticE[1/9] - 8/3 EllipticK[1/9] + 8/3 EllipticPi[-1/3, 1/9] == (1/(54 (-I + 2 Sqrt[2])))(-243 I EllipticE[1/81 (17 - 56 I Sqrt[2])] + 27 Sqrt[-17 + 56 I Sqrt[2]] EllipticE[1/81 (17 + 56 I Sqrt[2])] + 2 (81 Sqrt[2] EllipticK[1/9] - 81 Sqrt[2] EllipticK[1/81 (17 + 56 I Sqrt[2])] + 3 (8 I - 7 Sqrt[2]) EllipticPi[1/27 (7 - 4 I Sqrt[2]), 1/81 (17 - 56 I Sqrt[2])] + (104 I + 71 Sqrt[2]) EllipticPi[ 1/3 (7 - 4 I Sqrt[2]), 1/81 (17 - 56 I Sqrt[2])])) ~3.99291 is ONE QUARTER of "the surface area of the convex hull of two edge-to-edge [perpendicular, with collinear diameters] unit disks. ( http://arxiv.org/abs/1211.4514)". But how the bleep do we get the above simplification? Or the even worse one in Finch's original paper? Are there more elliptic integral transformations like EllipticK[r] == E^(-I tan^-1(Sqrt[r/(1 - r)])) EllipticK[ 4 E^(I tan^-1(1/2 Sqrt[1/r - 4 + 1/(1 - r)])) Sqrt[(1 - r) r]] (which usually holds for two of the eight choices of √, and which I can't even prove)? Does this have anything to do with Complex Multiplication? --rwg Awful thought: What is the expected time at which a 3D random walk first becomes "knotted"? (W.r.t. pulling on the ends.)

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[math-fun] Neutronium
by Warren Smith 24 Dec '12

24 Dec '12

ok... 1. The critic who said it was not right for me to argue about a free neutron has a point. However, nonrigorously+observationally speaking, nuclei with too many neutrons are unstable to beta decay, so that indicates neutronium should also be. (Nuclei with enough protons serving as "stable glue" can be stable, but then if you put even more protons they again become unstable because of inter-proton repulsion.) So, I won't pretend my derivation was rigorous, but I still believe it must be approximately right. 2. Re Chandrasekhar limit for white-dwarf collapse, I guess the question would be: is Chandrasekhar's reasoning reversible? That is, if M>Mchand, the dwarf collapses into a neutron star. If however magic aliens then removed some mass from the neutron star thus causing M<Mchand, will it then explode? I think the answer must be that there is some hysteresis. In fact due to energy release during the collapse, the neutron star will have less mass than the white dwarf, but it does not re-expand/explode. So it is not completely obvious what the "reverse threshold" ought to be, where a neutron star with mass below X, would explode. I think X<Mchand, not X=Mchand, but how much less, is not all that obvious. My derivation only gave what I have no doubt (despite criticism 1 above) is a very weak lower bound on X. 3. As somebody else pointed out, there are no magic alien mass-removal techniques, so who cares? Well, yes. That is why this is "math fun," I guess. Neutronium is not a stable substance in small amounts, and not even metastable enough for any use, so sci-fi writers should fuggedaboudit.

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[math-fun] astro-news (and a trivia question)
by Warren Smith 24 Dec '12

24 Dec '12

1. Tau Ceti, one of the closest stars to us (12 lightyears away) allegedly harbors five planets that weigh between two and six Earth masses and take between 14 and 640 days to orbit the star. One is reported to lie in the "habitable zone": five-Earth-mass planet with a period of 168 days. 2. Although you've probably heard about the flood of extrasolar planets found using ultra-high-res optical doppler and ultra-precise light-intensity measurement to detect planet-transits... the first extrasolar planets were found without needing optical telescopes at all before those techniques were mastered. For brownie points: What were they and how found?

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Re: [math-fun] Mathematician/physicist/inventor Richard Crandall dies at 64
by Bill Gosper 24 Dec '12

24 Dec '12

This is indeed a loss. And a lost opportunity for perhaps a nice little tiff over his last paper (with J. Borwein) “Closed forms: What they are and why we care,” http://www.ams.org/notices/201301/rnoti-p50.pdf p63: Relatedly, the classical Bessel expansion is BesselK[0, z] == Sum[((z/2)^n/n!)^2*Sum[1/k, {k, n - 1}], {n, ∞}] - (Log[z/2] + EulerGamma)*BesselI[0, z] Now K0(z) has a (degenerate) Meijer-G representation—so potentially is superclosed for algebraic z—and I0(z) is accordingly hyperclosed, but the nested harmonic series on the right is again problematic. I have three problems with these two sentences! First, the equation should be BesselK[0, z] == Sum[((z/2)^n/n!)^2*Sum[1/k, {k, n}], {n, ∞}] - (Log[z/2] + EulerGamma)*BesselI[0, z] Next, the nested harmonic series is NOT problematic. You can compute it linearly as the 3by3 product of MProd[{{z^2/(4*k^2), z^2/(4*k^3), 0}, {0, z^2/(4*k^2), 1}, {0, 0, 1}}, {k, Infinity}] It would be a mistake to exclude such products from "hyperclosed" (i.e., simple functions of hypergeometric values.) Lastly, the γ arises because K₀ has confluent poles as a Meijer-G. This "degeneracy" makes it unhypergeometric. Now, we have only Jon to blame. --rwg On 2012-12-22, at 9:59 AM, rcs(a)xmission.com wrote: Bob Baillie passed along this note from the experimental math blog of David Bailey and Jonathan Borwein. --Rich Forwarded message from rjbaillie(a)frii.com Subject: Mathematician/physicist/inventor Richard Crandall dies at 64 http://experimentalmath.info/blog/2012/12/mathematicianphysicistinventor-ri…

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