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

[math-fun] On sheep and goats
by Fred Lunnon 02 Jul '19

02 Jul '19

Extract from https://rjlipton.wordpress.com/2019/06/29/a-prime-breakthrough/ --- << What GORZ prove is this: Theorem 1 For each {d} and almost all {n}, the polynomial J(n, d)(x) is hyperbolic. Of course, “almost all” means that there at most a finite number of polynomials that are not hyperbolic. >> The quotation raises a number of questions, such as What inspired its author to decorate an initial straightforward, unambiguous quotation with the subsequent elementary howler in the first place? And how did this piffle manage to earn a link ("Gödel’s Lost Letter and P=NP" in the left-hand column) from https://terrytao.wordpress.com/2019/06/29/symmetric-functions-in-a-fraction… in the second place? In contrast TT's own contribution is informative and provocative, even for a reader (like myself) who manages to grasp approximately 10% of its contents. I'm a sucker for investigations that manage to extend what are apparently inherently functions of natural numbers to the complex domain! Fred Lunnon

1 0

[math-fun] Comment on Sengai. Possible Quadruple Miracle?
by Brad Klee 01 Jul '19

01 Jul '19

ITEM 4269604 (History of Science). Circa 1820, Sengai Gibon drew a famous calligraphy often referred to as "Circle, Triangle, Square": https://terebess.hu/zen/sengai.html Compare with the following algebraic plane curves: 1 = X^2 + Y^2 1 = X^2 + Y^2 - sqrt(4/27)*X^3 1 = X^2 + Y^2 - (1/4)*X^4 1 = X^2 + Y^2 - sqrt(4/27)*(3*Y^2*X - X^3) 1 = X^2 + Y^2 - X^2*Y^2 Did Sengai forget to draw a monogon and a digon? ITEM 4269605 (a minor gripe). The title "Circle, Triangle, Square" is not canonical. The same drawing is sometimes referred to as "The Universe". But how could one conceive of a universe without "Hotaru"? Did poor Sengai forget the Kanji for "Hotaru" as well? Or should we interpret Sengai's minimalism as a prognostication of future bio-devastation ( i.e. of "the impermanence of all conditioned phenomena" ) ? ITEM 4269606 ( On Impatience ). First, learn how to expand the following nomes: https://oeis.org/draft/A308835 https://oeis.org/draft/A308836 https://oeis.org/draft/A308837 After: https://oeis.org/A005797 The same technique applies to either of the following differential equations: 0 = (x-1)*(x-2)*T + d/dx( (x-3)*(x-4)*(x-5)*(x-6)*dT/dx) 0 = 15*(84 - 197*x + 78*x^2)*T + d/dx(4*(x-1)*(2*x-1)*(9*x-29) (9*x-2)*dT/dx) One is poorly-motivated, might as well be random, thus analysis of the nomes at any of the four singular points does not lead to much enlightenment. In particular, it does not even seem possible to integerize the expansion coefficients. As opposed, the second differential equation comes from a geometric precursor. It appears that the nome expansions around each of four singular points are integral after a scale transformation of the x variable: 2/9: 0, 1, 11742, 156203784, 2284549324448 ... 1/2: 0, 1, 304, 618436, 584357184 ... 1: 0, 1, 2152, 6629044, 21558540128 ... 29/9: 0, 1, 622056, 365686615476, 208699680038836576 ... The first one hundred terms have been quickly checked. PROBLEM: How to prove integrality? (not obvious to me). PROBLEM: WTF (What's The Function)? (Hint: Try searching through x-variate sphere curve geometries with local dihedral symmetry) PROBLEM: Once you've succeeded in finding something: Is it possible to use the nome variables to define pseudo-elliptic time parameterizations of the disjoint submanifolds? ITEM 4269607 ( On patience ). After many seasons of study in very dim light, a hotaru lantern takes definite form. Even more work remains to be done. PROBLEM: How do we construct useful objects in the material world? If we need to make a new lantern, what type of paper should we use? Cheers, Brad

1 0

Re: [math-fun] optimal advance tournament scheduling
by Keith F. Lynch 30 Jun '19

30 Jun '19

Tomas Rokicki <rokicki(a)gmail.com> wrote: > Graph six is K4,4, the same as the graph we have been discussing > (both the cube with long diagonals and the octagon with horizontal > and 45 degree chords). Both vertex and edge transitive. > Average bits 24 16:1152 19:2304 20:1152 22:4608 23:4608 24:8064 25:3456 > 26:10368 27:3456 28:1152 I get the same numbers. Note that the number of instances are all divisible by 2 * 4!^2 = 1152, the size of the who-plays-whom-preserving symmetry group of players in K4,4. There are 4! permutations of the players on one side of K4,4, all of whom play everyone except each other, another 4! permutations of the players on the other side of K4,4, all of whom also play everyone except each other, and the final factor of 2 comes from the fact that the two sets of players can be swapped. So it's reduced to 16:1 19:2 20:1 22:4 23:4 24:7 25:3 26:9 27:3 28:1, for a total of 35 rather than 40320 cases. (However, I still find it unexpected that the average is an integer.) 8! / (2 * 4!^2) = 35 is the number of different adjacency matrices. I'll show them all if anyone is interested. The players are numbered in order of playing ability, i.e. a lower-numbered player always beats a higher-numbered player. The vertices are then, in each of the 40320 permutations, given the same numbers as the players they represent. The directed matrix, in which A,B is non-zero only if player A played *and beat* player B, is always simply the upper diagonal half of the who-played-whom matrix. I add the identity matrix to it so that instead of having to add all the powers of the directed matrix together to get the score, I just have to look at the highest power. (How high is highest? I stop after the matrix stops changing. For all of K4,4, the square is high enough. The cube always equals the square.) The matrix which does worst is: 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 The directed matrix then is: 1 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 Each of the four best players plays each of the four worst players, so there are no instances of transitivity, i.e. a player being beaten by one player and beating another, hence the score is a miserable 16. Not coincidentally, the square of the directed matrix is the same as the directed matrix. As is its cube, ad infinitum. The number of 1 bits (not counting the diagonal) is 16, hence that's the score. The matrix which does best is: 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 (All black squares on the checkerboard.) Its directed version is: 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 The odd-numbered players played the even-numbered players. The square of that matrix is: 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 The score is 28, the best possible. Of course the players can't be deliberately assigned to their respective vertices like this unless their ranking is already known, so we have to take the average over all 35 matrices. The first matrix I found, the Thue-Morse one, scores 26. > So it looks like the pretty symmetric graph we have been discussing > is the winner; it averages 24 of the 28 "decisions" and also gives > the complete ordering the greatest number of times. Very likely, but that's presumably only out of the six graphs you tested. It's possible that there's a better graph out there somewhere. I'm still trying to figure out an efficient way to test them all. At least the fact that I'm doing most of the calculations on upper-triangular matrices rather than square matrices speeds things up. This took me far longer than I thought it would. I guess that means I really needed the mental exercise. A large part of the problem was that I was convinced that if the matrix is on a checkerboard, the number of black squares with 1 bits is unchanged under a row permutation followed by an identical column permutation, given that any color swaps that the row permutation caused would be undone by the column permutation. That is completely wrong. Can anyone see why?

2 1

Re: [math-fun] There's a guy selling fresh roasted corn
by Bill Gosper 30 Jun '19

30 Jun '19

Great! Martin Gardner was a great magic and mathematics popularizer, in whose honor there is a biennial Gathering. ( https://www.youtube.com/watch?v=Wap2aj_ZhhU , https://www.youtube.com/watch?v=lXsVWwPa7bc) Maybe the only conference to which *both* Koornwinders could get invited! On Sun, Jun 30, 2019 at 5:48 AM Mourad Ismail <mourad.eh.ismail(a)gmail.com> wrote: > Dear Bill > You may not know that Tom has a cousin, Dick Koornwinder who is a > magician. Check him out on the internet. Diaconis knew of Dick before Tom, > of course. Diaconis has a very funny story about the first time he met Tom > at a party. Diaconis did not know about Tom and thought he was the > magician, so he said to Tom some thing like I like your recent work, how > did you do that. Then Tom went ahead and Told Diaconis abut his latest > paper. > Enjoy the story if you have not heard before. > Mourad > > On Sun, Jun 30, 2019 at 6:18 AM Bill Gosper <billgosper(a)gmail.com> wrote: > >> in front of Apna Bazar (sic). I told him about Tom Koornwinder, >> but I think that's more about winding than vending. >> —Bill >> > > > -- > Mourad Ismail > Professor of Mathematics > University of Central Florida > Tel: 407 770-9959 > >

1 0

[math-fun] There's a guy selling fresh roasted corn
by Bill Gosper 30 Jun '19

30 Jun '19

in front of Apna Bazar (sic). I told him about Tom Koornwinder, but I think that's more about winding than vending. —Bill

2 1

Re: [math-fun] More good news: Luciferin Exists!
by Bill Gosper 29 Jun '19

29 Jun '19

When I was a kid in the 50s, rather greenish "lightning bugs" were rife in the Camden & Philadelphia areas, despite massive anti-mosquito DDT spraying. (Are there multiple color variants of luciferin?) Despite the DDT ban, they're gone now. Probably habitat destruction. But they were trivial to catch, so maybe an invasive predator? E.g., northern California has lately become infested with little black phorid(?) flies. They attack your food in restaurants. And they (painfully) attack scratches and cuts. Probably came in on an Asian freighter. Or maybe something is wiping out the dragonflies. —rwg On Sat, Jun 29, 2019 at 12:34 AM rwg <rwg(a)ma.sdf.org> wrote: > > > -------- Original Message -------- > Subject: [math-fun] More good news: Luciferin Exists! > Date: 2019-06-28 12:21 > From: Brad Klee <bradklee(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Reply-To: math-fun <math-fun(a)mailman.xmission.com> > > A very nice young woman from California recently told me > she was floored to witness, for her first time ever, the > beautiful flashing of fireflies in the dusk of Illinois. > > If fireflies are not native to your home land, It's certainly worth > travelling to see. In Kansas and Arkansas these bugs are well > loved by just about everyone. I have never travelled to Japan, > or any other part of Asia, but have heard from travellers that > fireflies also exist in those locales--as well as Brazil. > > In case you can not travel to look and wonder what type of > math and science the yellow lights may be intimating, here > are some nice blog articles I found for starters: > > https://www.inhs.illinois.edu/resources/inhsreports/autumn-01/firefly/ > > http://www.mostlycolor.ch/2012/08/firefly.html > > Cheers, > > Brad >

3 2

[math-fun] More good news: Luciferin Exists!
by Brad Klee 28 Jun '19

28 Jun '19

A very nice young woman from California recently told me she was floored to witness, for her first time ever, the beautiful flashing of fireflies in the dusk of Illinois. If fireflies are not native to your home land, It's certainly worth travelling to see. In Kansas and Arkansas these bugs are well loved by just about everyone. I have never travelled to Japan, or any other part of Asia, but have heard from travellers that fireflies also exist in those locales--as well as Brazil. In case you can not travel to look and wonder what type of math and science the yellow lights may be intimating, here are some nice blog articles I found for starters: https://www.inhs.illinois.edu/resources/inhsreports/autumn-01/firefly/ http://www.mostlycolor.ch/2012/08/firefly.html Cheers, Brad

1 0

Re: [math-fun] ellipsoid from girths: multivariate series reversion
by Brad Klee 28 Jun '19

28 Jun '19

Hi Warren, I'm busy this morning, so don't have much more time to over-explain. Should I also have to say that projection is through the ellipse centroid, that the length variables L1', L2', L3' are also quarter-girths, and that the "flattening" procedure applies only to the 1-boundary, not also to the interior surface area? The meaning should be apparent in context, but without figures maybe not. I would think they do these sort of constructions all the time at the better schools on the east coast (MIT, Princeton, Cornell etc...). Is it rather that all the time and money is for them to teach you how to insult lesser language so that you can then reject perfectly acceptable and insightful work? I looked over the post again, and yes I do think it a "proof" of an "obvious" theorem. However, I am well aware that a real proof depends on an agreement between mathematicians, so hope that you or someone else with the credentials will make a genuine effort to try and figure it out, instead of just nit-picking word choice. If you find any serious faults, please let me know. --Brad On Wed, Jun 26, 2019 at 9:04 AM Warren D Smith <warren.wds(a)gmail.com> wrote: > > I don't understand your "proof." > > E.g what does "can be flattened into plane" mean? > What does "curvilinear projection" mean? > > E.g. at least the way I normally use the word, a spherical > triangle cannot be "flattened." > > -- > Warren D. Smith > http://RangeVoting.org <-- add your endorsement (by clicking > "endorse" as 1st step)

4 5

Re: [math-fun] ellipsoid from girths: multivariate series reversion
by Bill Gosper 28 Jun '19

28 Jun '19

Code words for this in MIT lectures: "It is intuitively obvious to the most casual observer that . . ." On Fri, Jun 28, 2019 at 9:47 AM rwg <rwg(a)ma.sdf.org> wrote: > > > -------- Original Message -------- > Subject: Re: [math-fun] ellipsoid from girths: multivariate series > reversion > Date: 2019-06-28 06:18 > From: Fred Lunnon <fred.lunnon(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Reply-To: math-fun <math-fun(a)mailman.xmission.com> > > "But that's obvious. No, wait --- is it obvious?" (G. H. Hardy, > apocryphally). Is there any trope that reveals insecurity more > nakedly and unintentionally? Though Hardy (if it was he --- > reference, anybody?) was alert to the trap before tumbling in > completely; the anecdote continues with him disappearing to his study > to ponder, returning considerably later to announce triumphantly (and > to a noticeably shrunken audience) "Yes, it is obvious!", before > resuming the lecture (on Number Theory). > > Those readers remaining awake are invited to consider the parallel > with a recent splenetic diatribe in this thread in support of the > conjectural obviousness of a certain lemma; then to compare that with > a subsequently contributed proof of the lemma itself in approximately > one seventh of the same length. Their relative success in > illuminating the topic, while harder to quantify, should be easier > still to assess; obvious, even? > > In the outside world of general social intercourse, "Obviously ..." > may reliably be interpreted to indicate that the speaker either knows > or suspects that the claim introduced is actually false, or at best > vacuous; and aims to forestall dissent by implying that only an > ignoramus would question it. From a mathematician one expects a > diminished tendency to purposeful tergiversation --- that goes with > the Asperger's --- so that in a technical setting, a more accurate > gloss might run something like the following: "I fervantly hope ... is > correct, but have never taken the trouble to establish securely under > what constraints; and besides am wary of doing so for fear of > discovering instead that, all along, I have been pontificating via my > fundament". > > Compare "It is well-known that ...", etc. Hardly a secure basis for > scientific advance one might suppose, however effective such > strategies may prove in other areas of human endeavour! Incidentally, > pause to recall the "obvious" impossibility of (say) --- > > Everting a rubbery, phantom sphere without kinking it: > https://en.wikipedia.org/wiki/Sphere_eversion > Dissecting a unit ball into (five) parts which reassemble into two > unit balls: > https://en.wikipedia.org/wiki/Banach–Tarski_paradox > Electrons that sneak through both slits of a screen only when nobody > is looking: > https://en.wikipedia.org/wiki/Double-slit_experiment > > etc. etc. ad infinitum. Personally, I try hard to avoid the word > (except ironically, obviously). Along with "simple", with respect to > which I more often fail ... > > Fred Lunnon >

1 0

[math-fun] New Jersey woman may have killed identical twin sister over a ‘love triangle’: report
by Bill Gosper 28 Jun '19

28 Jun '19

What word leaps to mind? Spoiler: StringReverse@"!selecsosI" —Bill (This was my home town. As kids on bikes we explored it freely. Almost no crime. (Father's (doctor's) office never burgled.) Now Camden is the crime capital of a crime infested state.)

1 0