July 2017 - mathfun@mailman.xmission.com

[math-fun] I just found this and I should sleep on it before posting,
by Bill Gosper 25 Sep '17

25 Sep '17

but just in case it's still mysterious in the "morning": What is this? QPolyGamma[0, 1/2, q] + QPolyGamma[0, 1/2 - (I π)/Log[q], q] == -Log[(1 - q)/(1 + q)] + QPolyGamma[0, 1/2, q^2] q-Euler's constants? In[1199]:= PolyGamma[0, 1/2] // FunctionExpand Out[1199]= -EulerGamma - Log[4] Empirically true in the 1st quadrant of the unit disk, and in the whole 4th quadrant. Does it generalize? Does q-digamma have a Jacobi imaginary transformation? zzzzzzzzzz.... --rwg

1 1

[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

7 9

[math-fun] q-binomial coefficients are Laurent polynomials in q
by Bill Gosper 29 Jul '17

29 Jul '17

This must be well-known, but not by Wikipedia nor Mathworld. (I've misplaced my BHS again! This all might fall out of the q-Beta function, but not glaringly, due to the reciprocal.) Here is a neighborhood of the apex. Table[qbin[n, k], {n, -3, 6}, {k, -2, 3}] // TableForm 2 2 2 2 2 3 4 1 + q + q (1 + q ) (1 + q + q ) (1 + q ) (1 + q + q + q + q ) -(----------) --------------------- -(-------------------------------) 3 7 12 0 0 1 q q q 2 2 1 + q 1 + q + q (1 + q) (1 + q ) -(-----) ---------- -(----------------) 2 5 9 0 0 1 q q q 1 1 1 -- -(--) -(-) 3 6 0 0 1 q q q 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 + q 1 0 2 2 0 0 1 1 + q + q 1 + q + q 1 2 2 2 2 0 0 1 (1 + q) (1 + q ) (1 + q ) (1 + q + q ) (1 + q) (1 + q ) 2 3 4 2 2 3 4 2 2 3 4 0 0 1 1 + q + q + q + q (1 + q ) (1 + q + q + q + q ) (1 + q ) (1 + q + q + q + q ) 2 2 2 2 2 3 4 2 2 2 3 4 0 0 1 (1 + q) (1 - q + q ) (1 + q + q ) (1 - q + q ) (1 + q + q ) (1 + q + q + q + q ) (1 + q) (1 + q ) (1 - q + q ) (1 + q + q + q + q ) qbin[n_, k_Integer] := Factor[qpochgunch[FunctionExpand@QBinomial[x, k, q], x - k + 1, k] /. x -> n] qpochgunch[xp_, old_, by_] := xp /. QPochhammer[q_^old, q_, n___] :> Product[1 - q^¢, {¢, old, old + by - 1}] QPochhammer[q^(old + by), q, n] (*Preserves the value of QPochhammers if q is atomic.*) (*NB: I have In[461]:= Product[f@k, {k, 0, -2}] Out[461]= 1/f[-1] *) Note q-Pascal's *automatic* asymmetry for negative n. These values are completely consistent with the two recurrences {QBinomial[n, m, q] == QBinomial[n - 1, m - 1, q] + q^m QBinomial[n - 1, m, q], QBinomial[n, m, q] == q^(n - m) QBinomial[n - 1, m - 1, q] + QBinomial[n - 1, m, q]} (From which one can In[445]:= Eliminate[%%, %%[[1, 2, 1]]] , getting the two term recurrence Out[445]= (-q^m + q^n) QBinomial[n, m, q] == q^m (-1 + q^n) QBinomial[-1 + n, m, q] and another one.) (As before, imposing "Bermuda" symmetry breaks the recurrences. I think it would be genuinely interesting to find the actual objection that led to this fiasco.) Mathematica's Limit and FunctionExpand both misbehaved here: In[458]:= FunctionExpand@QBinomial[-2, 1, q] Out[458]= Indeterminate In[459]:= FunctionExpand@QPochhammer[1, q] Out[459]= QPochhammer[1, q] (* Hello? *) In[430]:= Limit[QBinomial[n, 3, q], n -> -2] Out[430]= -(QPochhammer[1/q^4, q]/((-1 + q)^3 (1 + q) (1 + q + q^2) QPochhammer[1/q, q])) This is 0/0! --rwg

1 2

Re: [math-fun] Lots of smaller 'Ronin' (sun-less/ejecta) planets
by Keith F. Lynch 29 Jul '17

29 Jul '17

hbaker1 <hbaker1(a)pipeline.com> wrote: > FYI -- And we thought that Earth's orbits were getting full of space junk... > I can envision 'Star Trek'-like starships ending up like windshield- > splatted bugs in picoseconds... "Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space." -- Douglas Adams A collision in interplanetary space is extremely unlikely. James White wrote a novel titled _Deadly Litter_, in which littering in space is a serious crime, since a collision at interplanetary speeds with a single discard coffee ground would wreck a spaceship. I did the math. Collisions in low Earth orbit are an issue, Collisions in interplanetary space are not. If I recall correctly, I determined that if every person who ever lived had a spaceship and traveled at random in the solar system, spending all their time tossing coffee grounds overboard, and kept doing so until the sun turned into a red giant in a few billion years, there would probably not be a single collision. This was partly because the speeds all exceeded solar system escape velocity, so the litter wouldn't accumulate. Debris in the asteroid belt would be a much bigger concern, since there's more of it and since it doesn't dissipate. But even though there are trillions of unmapped objects there big enough to destroy any space probe, not one of the probes that's been sent through the belt has been damaged, and probably none ever will. And the "ronin" planets aren't in interplanetary space, but in interstellar space, which is a couple dozen orders of magnitude larger. A chance collision in interstellar space, even with a grain of sand, is absurdly unlikely. If there was much stuff there, we couldn't see very far, but in most directions we can see all the way to the cosmic horizon. Mankind has sent five probes into interstellar space. (Whether any of them have reached it yet depends on your definition, but they'll all get there eventually.) It's interesting to think about how any of them could be someday detected by aliens, or, conversely, how our own remote descendants could ever detect similar probes from other solar systems. Suppose every one of the ~10^11 stars launched, over its lifetime, a thousand Voyager-like probes into random interstellar trajectories, all of which will forever remain within the galaxy. How would an advanced civilization go about finding any of them? (Not counting ones that had been launched in the past thousand years or so, hence were still close to their parent star.) Unless there's some completely unknown science, the best way seems to be radar. To have a reasonable chance of success, most of the mass of the galaxy would have to be turned into search ships, and every one of them would have to have a radar so powerful that not only would we have already picked it up if it was operating anywhere in the universe in our past light cone, Marconi would have picked it up too. > A handful of free-floating planetary-mass objects have been > discovered by infrared surveys of young stellar clusters and > star-forming regions as well as wide-field surveys, but these > studies are incomplete for objects below five Jupiter masses. I'd bet that those studies are incomplete for objects above five Jupiter masses, too. What would it be like on a Earth-sized planet far from any star? The obvious answer is "cold and dark." But what if the planet was somehow traveling fast enough that the cosmic microwave background's flux was equal to the sun's flux on our planet? That would be very close to the speed of light. I haven't gotten around to working out what the spectrum of the resulting light would be like. What would its color temperature be? And how wide a visible spot in the sky? Also, how long would it last before it was dimmed either by drag from the light slowing the planet or by cosmic expansion dropping the microwave background's temperature? I expect the latter would get you first, due to the enormous time dilation. And what are the chances of a collision with something? Probably quite small despite the absurd velocity, since even if you didn't start in intergalactic space, you'd end up there pretty quickly and almost certainly remain there forever. (Choose a random direction in space. Chances are the first galaxy in that direction, if any, is so distant that it will have gone past the cosmic horizon long before anything from here could reach it.) Suppose the planet continued to accelerate just enough to counteract the dimming due to cosmic expansion. Could that go on forever? Or would the required acceleration reach infinity as the microwave background asymptotically approaches absolute zero? Remember that half the temperature means 1/16th the flux, all else being equal, and that the temperature is dropping faster than inversely proportional to the age of the universe due to dark energy and accelerating expansion. Finally, what would such a planet look like to us? How far away could we see it? If it was headed roughly toward us, the reflected light would be blue shifted again, and would be enormously bright. Its proper motion would be ridiculously high, since its speed would barely lag the light we're seeing it by.

2 1

Re: [math-fun] Maximal packings of the integers
by Dan Asimov 28 Jul '17

28 Jul '17

I see now that I completely overlooked a general type of "periodic" packing of the finite packing body B: where there is ----- a) a finite set F of group elements of Isom(Z) = the isometry group of the integers and b) an integer n in Z-{0} such that all the sets of the form g(B) + K*n for any g in F and any K in Z *are disjoint*. ----- This is much more general than what I originally This definition depends on what kind of transformations are allowed to be in the finite set F. Can they be all isometries of Z (i.e., all mappings for the form x |—> x + j and of the form x |—> j - x for some integer j), or just the first kind (translations)? Jim, is this approximately what you mean by a "periodic packing of the integers" by a finite packing body B ??? —Dan ---- I don't think the initial set of "states" is finite: It is all periodic packings of the finite set B. I take this to mean all packings pack(B, n) of the form pack(B,n) = {B + K*n | K in Z} for some fixed n in Z-{0}, such that it is in fact a packing, i.e.: B + K*n is disjoint from B + L*n for all K <> L. However, it should be fairly easy to exclude all but a finite number of n as candidates for the *densest* packing. When there is a packing for a given n in Z-{0}, I guess the density is just dens(pack(B,n) = |B| / n . Let diam(B) = max{|i - j| | i,j in B} . Then it seems clear to me that if we let n_0 = diam(B) + 1 then any n > n_0 cannot possibly give rise to a denser packing than pack(B,n_0). (Which is a packing since the translates of B by multiples of n_0 must miss each other.) >From: Allan Wechsler <acwacw(a)gmail.com> >Sent: Jul 27, 2017 3:28 PM >I think this would be approached via the same "finite state machine" >technique that Rich sketched regarding Ed Pegg's "Mrs. Perkins's Quilt" >phenomena. Because the set of states is finite, a search algorithm looking >for an optimal packing will eventually enter a periodic regime. Obviously >what I just said is not a proof, but rather a "reason to believe", with a >possible pointer in the direction of a proof. I would be super-surprised if >the conjecture is false. > >On Thu, Jul 27, 2017 at 6:02 PM, James Propp <jamespropp(a)gmail.com> wrote: > >> How does one show that for a given packing body B (a finite set of >> integers) there is a periodic packing of the integers by disjoint >> translates of B that achieves as its density the supremum of the set of >> densities achieved by all periodic packings? >> >> >> I know (via a compactness argument) that the supremum is achieved by some >> packing, but my argument does not give a packing that is periodic. >> >> >> Surely this result is in the literature? Maybe it's easy but I'm just not >> seeing it. (It can't be completely trivial since corresponding assertions >> in higher dimensions are false, at least if we allow ourselves several >> packing bodies: consider aperiodic tilings, conceived of as packings of >> full density.) >> PS: I just posted this on MathOverflow ( >> https://mathoverflow.net/questions/277426/maximal-packings-of-the-integers >> ), so if you "go with the 'flow", consider posting there. -----

4 4

Re: [math-fun] Maximal packings of the integers
by Dan Asimov 27 Jul '17

27 Jul '17

I don't think the initial set of "states" is finite: It is all periodic packings of the finite set B. I take this to mean all packings pack(B, n) of the form pack(B,n) = {B + K*n | K in Z} for some fixed n in Z-{0}, such that it is in fact a packing, i.e.: B + K*n is disjoint from B + L*n for all K <> L. However, it should be fairly easy to exclude all but a finite number of n as candidates for the *densest* packing. When there is a packing for a given n in Z-{0}, I guess the density is just dens(pack(B,n) = |B| / n . Let diam(B) = max{|i - j| | i,j in B} . Then it seems clear to me that if we let n_0 = diam(B) + 1 then any n > n_0 cannot possibly give rise to a denser packing than pack(B,n_0). (Which is a packing since the translates of B by multiples of n_0 must miss each other.) —Dan >From: Allan Wechsler <acwacw(a)gmail.com> >Sent: Jul 27, 2017 3:28 PM >I think this would be approached via the same "finite state machine" >technique that Rich sketched regarding Ed Pegg's "Mrs. Perkins's Quilt" >phenomena. Because the set of states is finite, a search algorithm looking >for an optimal packing will eventually enter a periodic regime. Obviously >what I just said is not a proof, but rather a "reason to believe", with a >possible pointer in the direction of a proof. I would be super-surprised if >the conjecture is false. > >On Thu, Jul 27, 2017 at 6:02 PM, James Propp <jamespropp(a)gmail.com> wrote: > >> How does one show that for a given packing body B (a finite set of >> integers) there is a periodic packing of the integers by disjoint >> translates of B that achieves as its density the supremum of the set of >> densities achieved by all periodic packings? >> >> >> I know (via a compactness argument) that the supremum is achieved by some >> packing, but my argument does not give a packing that is periodic. >> >> >> Surely this result is in the literature? Maybe it's easy but I'm just not >> seeing it. (It can't be completely trivial since corresponding assertions >> in higher dimensions are false, at least if we allow ourselves several >> packing bodies: consider aperiodic tilings, conceived of as packings of >> full density.) >> PS: I just posted this on MathOverflow ( >> https://mathoverflow.net/questions/277426/maximal-packings-of-the-integers >> ), so if you "go with the 'flow", consider posting there.

1 0

[math-fun] Maximal packings of the integers
by James Propp 27 Jul '17

27 Jul '17

How does one show that for a given packing body B (a finite set of integers) there is a periodic packing of the integers by disjoint translates of B that achieves as its density the supremum of the set of densities achieved by all periodic packings? I know (via a compactness argument) that the supremum is achieved by some packing, but my argument does not give a packing that is periodic. Surely this result is in the literature? Maybe it's easy but I'm just not seeing it. (It can't be completely trivial since corresponding assertions in higher dimensions are false, at least if we allow ourselves several packing bodies: consider aperiodic tilings, conceived of as packings of full density.) Jim Propp PS: I just posted this on MathOverflow ( https://mathoverflow.net/questions/277426/maximal-packings-of-the-integers ), so if you "go with the 'flow", consider posting there.

3 2

[math-fun] 122707 is dark on the RGB sphere
by Ed Pegg Jr 27 Jul '17

27 Jul '17

I had an idea -- 1. Get the sum of 3 squares sols for a number 8n+3, http://oeis.org/A017101 . 2. Get all +/- permutations, divide by 2, add 255/2. 3. Draw a sphere centered at (255/2,255/2,255/2) going though those points, and color all points within the RGB cube with their appropriate RGB value. I was surprised that 122707 led to a "dark" sphere. All of the sum of three squares sols, (1, 165,309), (1, 195, 291), ..., (35, 81, 261), ...(125, 171, 279), (147, 163, 273) had a value greater than 255, with 261 as the minimal maximum. So I looked for the first numbers that have a minimal maximum of 3, 5, 7, 9, and so on. 1099 has sum of 3 square sols (3, 27, 55), (15, 17, 57), (15, 27, 53), min max of 53. 11, 35, 67, 91, 195, 259, 235, 379, 555, 547, 763, 795, 1099, 1131, \ 1667, 1387, 1963, 2251, 2563, 2667, 2515, 3259, 3459, 3355, 4187, \ 3763, 5131, 4939, 5691, 6403, 6907, 7395, 7315, 7627, 7291, 9867, \ 9403, 10587, 10651, 8947, 11067, 11715, 11467, 14547, 14779, 13435, \ 15915, 15691, 15787, 19643, 14947, 18883, 21835, 19947, 20155, 23403, \ 25635, 24355, 27267, 28771, 24907, 24787, 28315, 26251, 34203, 30739, \ 32971, 34707, 33163, 35779, 37947, 40627, 37195, 42915, 43051, 43867, \ 43563, 42883, 46627, 49123, 51947, 39715, 54547, 53475, 51811, 57163, \ 58755, 56947, 58195, 64603, 57643, 60019, 71547, 70411, 69115, 75075, \ 69355, 75835, 77883, 71947, 81363, 72499, 76987, 82219, 89539, 88147, \ 90867, 90955, 93163, 98987, 93307, 99547, 93003, 105163, 103051, \ 107043, 107787, 111643, 112291, 116851, 112315, 120523, 121323, \ 116515, 121627, 124723, 122155, 127659, 133627, 122707 Turns out that 122707 is unusually low, following a number of higher values. 3763 and 8947 are two more like that. --Ed Pegg Jr

1 0

[math-fun] Lots of smaller 'Ronin' (sun-less/ejecta) planets
by Bill Gosper 26 Jul '17

26 Jul '17

On 2017-07-25 17:00, hbaker1 wrote: > FYI -- And we thought that Earth's orbits were getting full of space junk... > > I can envision 'Star Trek'-like starships ending up like > windshield-splatted bugs in picoseconds... > > https://motherboard.vice.com/en_us/article/mbaw5y/rogue-planets-are-roaming… That animation looks wrong. The background galaxy(?) doesn't return to its original position after the planet passes. NeilB has done some serious work here: http://neilbickford.com . Can someone explain how a collapsing primary can eject a planet? Julian found traces of the Dancing Stars web page, but the Archive seems to punt Java. That simulation gave a strong but possibly misleading impression that 3-body systems were unstable. (Confinement to 2D massively raises the "triple collision" frequency.) But in a triple or quadruple system, a brown dwarf is likely first to leave. Shouldn't we also worry about dust clouds? Judging by the "summer" of 1816, an interplanetary cloudy spell could kill almost everything. --rwg > > http://www.nature.com/nature/journal/vaop/ncurrent/full/nature23276.html > > Planet formation theories predict that some planets may be ejected > from their parent systems as result of dynamical interactions and > other processes. Unbound planets can also be formed through > gravitational collapse, in a way similar to that in which stars form. > A handful of free-floating planetary-mass objects have been discovered > by infrared surveys of young stellar clusters and star-forming regions > as well as wide-field surveys, but these studies are incomplete for > objects below five Jupiter masses. >

2 1

[math-fun] Lots of smaller 'Ronin' (sun-less/ejecta) planets
by hbaker1 26 Jul '17

26 Jul '17

FYI -- And we thought that Earth's orbits were getting full of space junk... I can envision 'Star Trek'-like starships ending up like windshield-splatted bugs in picoseconds... https://motherboard.vice.com/en_us/article/mbaw5y/rogue-planets-are-roaming… http://www.nature.com/nature/journal/vaop/ncurrent/full/nature23276.html Planet formation theories predict that some planets may be ejected from their parent systems as result of dynamical interactions and other processes. Unbound planets can also be formed through gravitational collapse, in a way similar to that in which stars form. A handful of free-floating planetary-mass objects have been discovered by infrared surveys of young stellar clusters and star-forming regions as well as wide-field surveys, but these studies are incomplete for objects below five Jupiter masses.

1 0