April 2017 - mathfun@mailman.xmission.com

[math-fun] Big Splat theory
by James Propp 04 Sep '17

04 Sep '17

Has anyone ever proposed in the Annals of Improbable Research or a similar venue that, in addition to undergoing periodic reversals of its magnetic field, Earth is also subject to occasional short-lasting reversals of its gravitational field? Such a phenomenon would handily explain selective extinction of megafauna: a mouse could survive an upward fall for five seconds followed by a downward fall for five seconds followed by an extreme deceleration, but a mastodon would not fare as well. :-) Jim Propp

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Re: [math-fun] How does a dull spoon manage to cut a stringy pieceof meat
by Bill Gosper 03 Jun '17

03 Jun '17

Actually, close questioning by Mike Beeler jogged me to remember what is probably a big clue: Somewhat prior to the creeping, I noticed on that shoe the grommetoidoid (occulted, opposite the circled one) in gosper.org/shoe.png can now freely pivot. After the next creep cycle, I'll try gluing it. What the heck are the correct names for grommetoid and (the simply connected) grommetoidoid? On 2017-04-27 13:33, Hilarie Orman wrote: >> From: Bill Gosper <billgosper(a)gmail.com> >> On Sun, Jan 15, 2017 at 4:57 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> > while sliding back an forth against the inside of a slippery bowl? --rwg >> > > >> No one took this question seriously, but I claim the phenomenon needs >> explaining. >> It definitely requires some friction with the bowl: Just now I had it work >> improbably >> well on a piece of meat, yet fail on a tenderer but slipperier partially >> stewed tomato >> in the same bowl. > > I did express concern about your diet. > > Tomato skins are tough. The gold standard for a perfect knife blade > is one that with a light touch will cut a tomato skin. > > I've not found any standard measurements for muscle fiber strength. > The usual measure of tenderness relates to compression. Presumably > you are twisting and tearing the fibers with rotational force. > >> OK, on to the next quotidian mystery: For years I've worn the same pair of >> https://ll-us-i5.wal.co/asr/fd205dcf-4974-49eb-b2d2-6166402bc2b3_1.ecc95c27… >> (only with five pairs of grommetoids (discontinued?) >> instead of four). Possibly coincident >> with replacing the original laces with Kiwis >> several weeks ago, a strange phenomenon has emerged: Despite two different >> lacing patterns, the right lace >> creeps rightward an inch or two per week, until it is so lopsided as to >> drag on the ground. This time, I'm >> going to mirror-image the disparity. This happens automatically if you simply juxtapose the aglets before redistributing all the slack to the shorter lace. Evening them is harder. You only want to distribute half the slack. >> Prior to this, both shoes were coming >> untied, until I modified the knot. Circle the thumbnail twice. Pull hard. Works great. (Is this Propp's "mathematician's knot"?) >> The left shoe doesn't creep. Nor has any other shoe in my experience. The >> creep is sufficiently insidious >> that I can't tell if it's while I'm walking or tying. Probably tying. My foot is so narrow that the knot is immobilized between the top two grommetoidoids. --rwg >> --rwg Some Web news service recently ran filler claiming that CIA agents signaled each other with binary patterns in their shoelaces. The practice was discontinued after feminist protests alerted the KGB. > > Does the left lace creep forward when you are in Australia? > > Hilarie

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[math-fun] Grandi valuations
by James Propp 05 May '17

05 May '17

If S is an eventually-periodic subset of {0,1,2,...}, then |S|_q, defined as the sum of q^n as n ranges over S, is a rational function. The map from S to |S|_q is a finitely-additive measure ("valuation" is probably a better word to use) taking its values in the ring of rational functions in q. If we order that ring "at 1", we get a way to compare sizes of sets of natural numbers. In fact, it is a total ordering (though not a well-ordering) of the set of eventually-periodic subsets of {0,1,2,...}. This may remind some of you of the ordered fields that you get by adjoining to R a formal infinitesimal or a formal infinity. In both cases, you've got the field R(x), and all that differs is the ordering you put on it. My proposal is similar, except that I'm ordering the field at 1 rather than 0 or infinity. Writing |S|_q as A / (1-q) + B + lower order terms, we can focus on just A and B. A measures the density of S, and B captures finer information. The first thing to notice about B is that it's sensitive to perturbations that don't affect the density; if you add an element to S, you increase B by 1, and if you remove an element from S, you decrease B by 1 (even though A doesn't change). But B conveys subtler information too: For instance, compare {0,2,4,...} with {1,3,5,...}; the former has B = 1/4 while the latter has B = -1/4. Informally, we say that sliding the set of even nonnegative integers over the right by 1 reduces the "size" of the set by 1/2. And that makes sense, because if you slide it over to the right again, "losing" half an element again, you get {2,4,...}, which is missing precisely 1/2 + 1/2 = 1 of the elements of {0,2,4,...}. (Yeah, this way of measuring sets is not translation-invariant. But you gain something nice in return: a proper subset of a set is always strictly smaller than the set.) I recently learned that Ilya Chernykh has been exploring the same circle of ideas (see http://mathoverflow.net/questions/215762/non-standard-numbers-and-exponenti…) but I don't know of anyone else who's fished in these waters. (You could argue that analytic number theorists know all this already and don't think it's interesting.) Anyway, the map that sends S to (A,B) should I believe be called a "Grandi valuation", since it gives a rigorous framework for making sense of Grandi's formula 1 - 1 + 1 - 1 + ... = 1/2. (Indeed, Bernoulli's formula 1 + 0 - 1 + 1 + 0 - 1 + ... = 2/3 can be understood in the same way: {0,3,6,...} exceeds {2,5,8,...} by 2/3 of an element, if we measure size using this valuation.) More precisely one might call the map the "Grandi-Abel valuation", since it uses Abel regularization of the summation of the indicator function of S. Anyway, other ways of taming divergent series exist, so there will be other Grandi valuations. And maybe the term should also be applied to the non-truncated valuation (the one that takes values in the ordered field R(q)), which gives the sets {0,3,4,7,8,11,...} and {1,2,5,6,9,10,..} different "sizes" (the former is infinitesimally larger). In the non-truncated valuation, we get a translation-covariance property that might console us for the loss of translation-invariance: shifting a set corresponds to multiplication by q. One can try to extend the Grandi valuation to sets that aren't eventually periodic, as I do in my MathOverflow post, and to some extent this works. However, one can construct sets that are incomparable, so if you're a fan of number systems that satisfy trichotomy, it's important not to overreach. I haven't given a definition of what a Grandi valuation is in general, but a hallmark is that a Grandi valuation should assign to any eventually periodic set S a value in some module from which the quantity I called "B" can be derived. Anyway, I have a bit of a crush on this notion, and I'd like to put it to some sort of use. My current target is a general uniqueness theorem in the context of one-dimensional packing. Jim P.S. It has occurred to me that when we measure infinite sets in this way, a cute name for this kind of generalized cardinality would be "grandiosity". But I think it's prudent to avoid introducing this term until I have a better idea for how it should be defined!

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[math-fun] Unstoppable Brilliance: Irish Geniuses and Asperger's Syndrome
by Bill Gosper 30 Apr '17

30 Apr '17

Carolyn (Neil B's mom) and Jon (Julian & Corey Z H's dad) highly recommend The Curious Incident of the Dog in the Nighttime (coming to San Francisco in June). --rwg Date: 2017-04-16 16:54 From: Fred Lunnon <fred.lunnon(a)gmail.com> Having spent my professional life around comp. science and maths departments, I noticed decades ago that any "neurotypical" individual happening to stray into such realms stood out like a proverbial sore thumb. The introduction of this book reads like a personality test checklist for their academic denizens --- I found myself nodding in recognition sufficiently often to qualify under "repetitive behaviour" --- in addition, that is, to a substantial proportion of the remainder! WFL On 4/16/17, Douglas Rogers <dgrbgsu(a)gmail.com> wrote: > William Rowan Hamilton and more ... > > > * Michael Fitzgerald, Antoinette Walker > Unstoppable Brilliance: Irish Geniuses and Asperger's Syndrome > (liberties Press <http://www.libertiespress.com/>, Dublin, 2006) > <https://books.google.com/books?id=EU0xBwAAQBAJ> > > Michael Fitzgerald is Henry Marsh Professor of Child and Adolescent > Psychiatry > at Trinity College Dublin and is the leading expert on Asperger's syndrome > and > autism in Ireland. He has written extensively about these subjects and > addressed > national and international symposia on them. >

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[math-fun] Observation on 2^64
by David Wilson 30 Apr '17

30 Apr '17

18446744073709551616 = 1^844674407370955*16^16 There are also large numbers N with 1N72343 = 1^N*7^2343 1N36821221 = 1^N*36821^221 (concatenation on the left).

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[math-fun] Prime question
by David Wilson 30 Apr '17

30 Apr '17

As n grows, what is the limiting probability that n has a prime divisor >= sqrt(n)?

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Re: [math-fun] How does a dull spoon manage to cut a stringy pieceof meat
by Bill Gosper 29 Apr '17

29 Apr '17

> On Apr 28, 2017, at 9:57 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > > What the heck are the correct names for grommetoid and (the simply > connected) grommetoidoid? Date: 2017-04-28 10:03 From: Dan Asimov <asimov(a)msri.org> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> >The word eyelet comes to mind. And to Neil B's mom's mind came eyestay. But that's Pig Latin for an ophthalmic condition. (Damn those esoteric doctors). >Or, check out this picture: http://3.bp.blogspot.com/-en07ThXYeUY/TYArIkKeuAI/AAAAAAAAFGk/K6bTVrUXfTg/s… < http://3.bp.blogspot.com/-en07ThXYeUY/TYArIkKeuAI/AAAAAAAAFGk/K6bTVrUXfTg/s… >. >—Dan Bingo: Four pairs of D-rings plus two pairs of hooks, in combination. Perfect! --rwg Wait a minute, aren't those parts of a cyclotron?

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[math-fun] sinh/asinh numbers & positional notation
by hbaker1 24 Apr '17

24 Apr '17

(Please read the following as a not-very-precise set of intuitions; I haven't been able to work out all of the details.) One thing that I forgot to say about sinh/asinh representation is the potential symmetry in "positional" representation. This symmetry is more easily seen in the "binary" version of sinh: sinh2(x)=(2^x-2^-x)/2 sinh2(k)=(2^k-2^-k)/2, k a positive integer =a long string of 1's in binary positional notation. and cosh2(x)=(2^k+2^-k)/2 So the output of cosh2(x) should -- in general -- be a *palindrome* in a "suitable" positional notation. Knuth discusses "balanced" positional notations, in which the digits 0-9 are replaced by -5..+5; these representations are often utilized in hardware multipliers to reduce the amount of circuitry required. Are there number representations which make the palidromic nature of cosh2(x) obvious? Once we have such a representation, then we can consider sinh2(x), which will have a similar type of anti-palidromic representation. We should be able to decompose *any* number into a sum(?) of elementary anti-palindromes.

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[math-fun] sinh/asinh numbers & positional notation
by hbaker1 24 Apr '17

24 Apr '17

(Please read the following as a not-very-precise set of intuitions; I haven't been able to work out all of the details.) One thing that I forgot to say about sinh/asinh representation is the potential symmetry in "positional" representation. This symmetry is more easily seen in the "binary" version of sinh: sinh2(x)=(2^x-2^-x)/2 sinh2(k)=(2^k-2^-k)/2, k a positive integer =a long string of 1's in binary positional notation. and cosh2(x)=(2^k+2^-k)/2 So the output of cosh2(x) should -- in general -- be a *palindrome* in a "suitable" positional notation. Knuth discusses "balanced" positional notations, in which the digits 0-9 are replaced by -5..+5; these representations are often utilized in hardware multipliers to reduce the amount of circuitry required. Are there number representations which make the palidromic nature of cosh2(x) obvious? Once we have such a representation, then we can consider sinh2(x), which will have a similar type of anti-palidromic representation. We should be able to decompose *any* number into a sum(?) of elementary anti-palindromes.

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Re: [math-fun] Trig question
by Dan Asimov 23 Apr '17

23 Apr '17

Yes, it's very interesting, and if I get any time soon I will write up why. —Dan ----- Is there anything mathematically interesting about the function f(a, b) = arctan(tan(a) + tan(b))? -----

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