April 2014 - mathfun@mailman.xmission.com

[math-fun] matrix group with bounded eigenvalues
by Fred Lunnon 03 Apr '14

03 Apr '14

Given a finite set of complex matrices generating an infinite group, I need to establish that every element of the group has all eigenvalues within the unit circle. Such problems must have been well investigated --- perhaps (shudder) by math physicists --- but I have no idea where to start looking. [It's actually a semigroup of integer matrices with constraints on allowed products, and circle radius exponential in the number of product factors --- but those details are almost certainly unimportant.] WFL

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Re: [math-fun] Dating the collision that formed the Moon
by Warren D Smith 03 Apr '14

03 Apr '14

> I've been claiming for a long time that the Earth's atmosphere would have > been a lot more like Venus without an event like this one that removed most > of it. For the same reason, we won't be able to turn Earth back into Venus, > no matter how much carbon we burn. --interesting and important claim, if true. But it might have worked the opposite way,e.g. the collision released more gases causing the Earth atmosphere to become larger, not smaller. I have no idea; I'm just asking if this "claim" can be backed up somehow.

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[math-fun] Arndt bit reverse
by Warren D Smith 03 Apr '14

03 Apr '14

it failed on 9993857301930256704.

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[math-fun] reversing the order of bits in a word
by Joerg Arndt 03 Apr '14

03 Apr '14

ulong jj3_revbin(ulong x) { ulong z; z = (x ^ (x >> 2)) & 0x9249249249249249UL; x ^= z ^ (z << 2); // low 3 bits reversed z = (x ^ (x >> 6)) & 0x01c0e070381c0e07UL; x ^= z ^ (z << 6); // low 9 bits reversed z = (x ^ (x >> 18)) & 0x00001ff0000001ffUL; x ^= z ^ (z << 18); // low 27 bits reversed z = (x ^ (x >> 36)) & 0x0000000007ffffffUL; x ^= z ^ (z << 36); // low 63 bits reversed x = (x>>63) | (x<<1); // rotate left by 1 return x; } Can explain if anybody is interested later, got a visitor now. Did run automated testing, but would appreciate independent "OK"s. Yes, tricky. Best, jj

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[math-fun] Gosper factorial stuff
by Warren D Smith 03 Apr '14

03 Apr '14

Well. First of all, I think whatever Gosper is doing about factorials looks pretty brilliant, as fairly usual, but also as usual I do not know how he did it. Second, basically, there was a line of research started by Chowla & Selberg 1967. Here is a recent paper that cites that (I have no idea what the hell this recent paper is doing, but anyway...) http://www.math.wisc.edu/~thyang/Colmez.pdf C&S found some relations between GAMMA(a/b) values and elliptic/modular functions. Putting those together with the usual reflection & n-tupling formulas, then solving the resulting systems of equations, we find out stuff rather like the stuff Gosper is seeking. A number of later authors tried to do exactly that (in 1967 C&S did not have access to Macsyma, etc, handicapping them vs later authors), but you'd have to search the literature to find those (use cite-search on the C&S paper). Now about the transcendence and/or irrationality and/or linear independence (over the algebraic or irrational numbers) results proven by Nesterenko et al, those could sometimes be used to show that there are no further magic relations among the GAMMA(a/b) values... EXCEPT that you'll only be able to show that lack of further relations WITHIN certain classes of allowed kinds of relations. It isn't going to tell you a thing about there being no relations of the form GAMMA(1/5) = exp(sin(GAMMA(3/5)) -- (which is false), but my point is nonlinear nonalgebraic relations cannot be ruled out in this way. > I believe our previous interest in (n/24)! was their rapid computability > via Dedekind eta. Even assuming it is always possible to convert to etas, > do their modular properties tell all? And, assuming we can canonicalize > the eta expression, can we always convert back to Gammas? --These questions are just new questions... If you were to take the attitude you are not even going to permit elliptic/modular functions to appear, then it might be the reflection and ntupling is all there is and Chowla & Selberg might be irrelevant. -- Warren D. Smith http://RangeVoting.org

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[math-fun] Dating the collision that formed the Moon
by Henry Baker 03 Apr '14

03 Apr '14

FYI -- "The resulting collision would have left the Earth a magma ocean, blown away its atmosphere..." I've been claiming for a long time that the Earth's atmosphere would have been a lot more like Venus without an event like this one that removed most of it. For the same reason, we won't be able to turn Earth back into Venus, no matter how much carbon we burn. http://arstechnica.com/science/2014/04/dating-the-collision-that-formed-the… Dating the collision that formed the Moon using late-arriving debris The magma ocean that resulted acted like a reset button on the age of the Earth. by John Timmer - Apr 2, 2014 5:36 pm UTC When did the Earth actually form? There are a number of ways to approach that question. We can use radioactive dating to look at material that has fallen to Earth after remaining largely undisturbed since the formation of our Solar System, or we can obtain a date for the oldest materials we've found on Earth. But those methods simply provide an upper and lower limit; the Earth formed some time after the smaller material in the Solar System, while the earliest materials on Earth would have been produced some time after its formation. Now, researchers have provided a new date for the formation of the Earth, based on the last time the planet was entirely moltenÂ-an event that was triggered by a collision with a body that ultimately created the Moon. The data is calculated using what we know about the early Solar System combined with the debris that fell to Earth after the big collision. The early Solar System was a violent place, as small particles aggregated into bodies that then grew larger by collisions. These collisions eventually produced planetesimals the size of large asteroids, which merged to form the current collection of planets. So there was no clear start to what would ultimately become the Earth, but there was a clear end to the primary process of its formation. That end came when the proto-Earth was smacked by a Mars-sized body. The resulting collision would have left the Earth a magma ocean, blown away its atmosphere and any volatile liquids on its surface, and put enough debris in orbit to form the Moon. In effect, it acted like a reset button for the timing of the Earth's formation, remixing all its components so that the raw material for radioactive datingÂ-the stable maintenance of isotope differences caused by radioactive decayÂ-was eliminated. Everything started afresh. Figuring out the timing of that collision is important if we want to understand the conditions on the early Earth and in the early Solar System in general. The new study uses a clever way of providing an estimate, based on the fact that the last major collision in the Earth's history didn't mean that the Earth escaped further bombardment from space. The logic is pretty clever. In the wake of the Moon-forming collision, the entire Earth was molten, which allowed the iron to sink to the core. A number of heavy elements that have an affinity for iron sunk to the core with it. This would have left the surface of the Earth completely stripped of these metals. (These elements, which include gold and platinum, are generically referred to as siderophiles.) But it's possible to mine this material from the crust. The elements are there because the Earth's supply was partly refreshed by the arrival of new material in collisions with other planetesimals and smaller asteroids. (The most famous asteroid, the one that killed the dinosaurs, was first identified through the presence of a layer of another siderophile, iridium, which it carried to Earth.) If we total how much of these elements are in the crust and compare that number to the composition of asteroids in our Solar System, we can get an estimate of how much mass must have struck the Earth after the Moon-forming collision. How do you go from that mass to a date? The authors recognized that over time, these bodies were lost through collisions with Earth and the other planets. Thus, early in the Solar System's history, there was an ever-shrinking stock of planetesimals that could have delivered material to Earth. To find out the timing of when the stock was depleted, the researchers relied on models of Solar System formation. In fact, they relied on two different types of models: one that keeps the giant outer planets in their current locations and a second class in which Jupiter and Saturn engage in what's called a "Grand Tack," moving inward early in the Solar System's history before receding back out to their current locations. The Grand Tack simulations were more likely to produce a set of rocky inner planets that look like our Solar System, but in both cases, the planetesimals got depleted pretty rapidly. Their vanishing act sets limits on when the Moon-forming collision could have happened and, thus, when the Earth could have formed. Too early, and the Earth would have seen lots of additional collisions and had its crust loaded with the elements that are now rare. Too late, and there wouldn't be enough around to give us any significant quantities. Consequently, the authors calculate that there's only a 0.1 percent chance that the Moon-forming collision took place prior to 40 million years after material started condensing in our Solar System. Instead, they place the likely date at 95 million years, with a margin of error of 30 million years on either side. The authors suggest that we might want to revisit some of the results that were generated using isotope dataÂ-a few of these put the collision at 30 million years, and understanding why they got the number wrong may provide some details of the mechanics of the collision itself. They also think that the reloading of the Earth with rare metals tells us something about the distribution of mass during the formation of the Solar System, which can constrain our models of planet formation. Right now, those models don't do a very good job of creating many of the tightly packed systems that we're currently discovering, so any improvements to them would be a positive step. Nature, 2014. DOI: 10.1038/nature13172 (About DOIs). John Timmer / John became Ars Technica's science editor in 2007 after spending 15 years doing biology research at places like Berkeley and Cornell.

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[math-fun] (1/3)! (11/30)!/(1/5)!/(1/2)!
by Bill Gosper 03 Apr '14

03 Apr '14

== Root[12443200581724796808856220604832743182872134173724163002314398459\ 5588711423357115661911766670262368377396461800488719686050677816033280\ 0000 - 255314037574112279130454964651587557346031347914070186474281265\ 6444640140595523594291757946335630379032075070677622872398556027473100\ 800000 #1^30 - 1078333940243484667045307174981926384244338820333252227598095472333\ 0686179316001773879585045424972028039680914589825721361024421314560000\ #1^60 - 1517849805983441629703332579833879044367807196164420510036054\ 8288882183927368291728216426697673810979955947991089015800887276953600\ 0 #1^90 + 2354120347120945871478598905818240835818029954927757252309180924825\ 756111530532878473839847484455444153358491092971929 #1^120 &, 3] In[219]:= N[List @@ %, 69] Out[219]= {0.976260099016963600222711219445061742080609027579146697139526393292722, 0.976260099016963600222711219445061742080609027579146697139526393292722} --rwg 1/3 + 11/30 = 1/5 + 1/2

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[math-fun] Gosper factorial stuff
by Bill Gosper 03 Apr '14

03 Apr '14

On 2014-04-02 18:28, Bill Gosper wrote: > On 2014-04-02 14:05, Warren D Smith wrote: >> RWG:My objective, more clearly stated, is >> ultimately to automatically simplify factorials(n/d), >> not just factorials(n/24), by canonicalizing first to minimal >> d and then to minimal n. So far, reduction is possible for >> >> n/d, n>d/2 (by reflection) >> 1/2, >> 1/6, >> 3/8, >> 4/9, >> n/10, >> n/12, >> n/14, >> n/15, n>1 >> 5/16, >> 7/16, >> n/18, >> n/20, n>1 >> n/21, n>2 >> n/22, >> n/24, n>1 >> 11/25, >> 12/25, > > Add n/26, Add 10/27, 11/27, 13/27, 5/28, 9/28, 11/28, 13/28, n/30, > strongly suggesting n/(4d+2) > >> n[/] prime, apparently never for prime>2n >> {1,7,11,13}/30 use only (1/3)! and ((1/5)! or (2/5)!), e.g., (11/30)! = (11 (1 + Sqrt[5])^(11/60) (5 + Sqrt[5])^(5/12) Sqrt[π] Cos[π/30]^(3/10) Cos[(2 π)/15]^(11/30) (1/5)! (Sec[π/15] Sin[π/30])^(4/15) Sin[(2 π)/15]^(49/60))/ (9 2^(53/60) 3^(1/20) 5^(1/6) Cos[(7 π)/30]^(41/60) (1/3)! Sin[(7 π)/30]^(7/20)) Cos[(7 π)/30]^(41/60) ? >> It would be amazing to find a reduction inaccessible >> via reflection and tuplication. But how could we tell? > > WDS> --In fact, Chowla & Selberg *DID* find cases inaccessible by >> reflection and tuplication, all (n/24)! can be expressed in close form >> in terms of elliptic functions. No? There's been followup work by other >> authors. Do you know how they demonstrated inaccessibility? >> >> I've posted about this in the past on math-fun, you should check the > archives, >> I think the cites are there. >> >rwg> how could we tell? >> > wds> wikipedia: >> It has been proved that GAMMA(n+r) is a transcendental number and >> algebraically independent of pi for any integer n and each of the >> fractions r = 1/6, 1/4, 1/3, 2/3, 3/4, and 5/6. >> >> citing: >> http://www.math.jussieu.fr/~miw/articles/pdf/TranscendencePeriods.pdf >> which used the connection to elliptic functions. >> >> more generally, if you had algebraic or rational independence results >> you "could tell." >> >> Other papers: >> http://www-fourier.ujf-grenoble.fr/~rivoal/articles/gammater.pdf >> http://www.mast.queensu.ca/~murty/gun-murty-rath2.pdf >> in particular note Nesterenko's theorem in the latter. > <WDS Wow, I didn't know those transcendences (transcendalities?) had been settled, let alone by the Chudnovskys. > > I believe our previous interest in (n/24)! was their rapid computability > via Dedekind eta. Even assuming it is always possible to convert to etas, > do their modular properties tell all? And, assuming we can canonicalize > the eta expression, can we always convert back to Gammas? I now recall a paper showing the latter. *Large numbers* of Gammas. We ought to be able to prove (n/prime)! inexpressible in terms of (a/prime)!, a<n, or (a/d)!, d<prime, (if prime >2n). And we ought to be able to write (n/(4d+2))! in terms of (a/b)!, b≤2d+1. > > I don't think Selberg & Chowla were thinking about computer algebra. > I'm just trying to simplify piles of Gammas. > > A weird possibility: Allow piles of Beta[rational,rational]. Although > traditionally defined in terms of three Gammas, Beta is a much simpler > function with nice hypergeometric representations. > --rwg > Oh, my aching tush: Mathematica is pretty good about Solving linear > systems which require a symbolic determinant to vanish. > ("Unable to determine if blah=blahblah. Assuming it is ...") > But if you append (1/6)! == (3 Sqrt[3/\[Pi]] ((1/3)!)^2)/(2 2^(1/3)) or > a similar identity in constants to a vanilla nonsingular system, > you instantly get the empty set of solutions! E.g., In[260]:= Solve[{x == 1, y == 2, (1/6)! == (3 Sqrt[3/\[Pi]] ((1/3)!)^2)/(2 2^(1/3))}, {x, y}] Out[260]= {} Yet, In[261]:= Solve[{x == 1, y == 2, x == (3 Sqrt[3/\[Pi]] ((1/3)!)^2)/(2 2^(1/3))/(1/6)!}, {x, y}] During evaluation of In[261]:= Solve::ztest1: Unable to decide whether numeric quantity 1-(3 Sqrt[3] ((1/3)!)^2)/(2 2^(1/3) Sqrt[\[Pi]] (1/6)!) is equal to zero. Assuming it is. >> Out[261]= {{x -> 1, y -> 2}} --rwg

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[math-fun] Gosper factorial stuff
by Bill Gosper 03 Apr '14

03 Apr '14

On 2014-04-02 14:05, Warren D Smith wrote: > RWG:My objective, more clearly stated, is > ultimately to automatically simplify factorials(n/d), > not just factorials(n/24), by canonicalizing first to minimal > d and then to minimal n. So far, reduction is possible for > > n/d, n>d/2 (by reflection) > 1/2, > 1/6, > 3/8, > 4/9, > n/10, > n/12, > n/14, > n/15, n>1 > 5/16, > 7/16, > n/18, > n/20, n>1 > n/21, n>2 > n/22, > n/24, n>1 > 11/25, > 12/25, Add n/26, strongly suggesting n/(4d+2) > n[/] prime, apparently never for prime>2n > > It would be amazing to find a reduction inaccessible > via reflection and tuplication. But how could we tell? WDS> --In fact, Chowla & Selberg *DID* find cases inaccessible by > reflection and tuplication, all (n/24)! can be expressed in close form > in terms of elliptic functions. No? There's been followup work by other > authors. > > I've posted about this in the past on math-fun, you should check the archives, > I think the cites are there. > >rwg> how could we tell? > wds> wikipedia: > It has been proved that GAMMA(n+r) is a transcendental number and > algebraically independent of pi for any integer n and each of the > fractions r = 1/6, 1/4, 1/3, 2/3, 3/4, and 5/6. > > citing: > http://www.math.jussieu.fr/~miw/articles/pdf/TranscendencePeriods.pdf > which used the connection to elliptic functions. > > more generally, if you had algebraic or rational independence results > you "could tell." > > Other papers: > http://www-fourier.ujf-grenoble.fr/~rivoal/articles/gammater.pdf > http://www.mast.queensu.ca/~murty/gun-murty-rath2.pdf > in particular note Nesterenko's theorem in the latter. <WDS I believe our previous interest in (n/24)! was their rapid computability via Dedekind eta. Even assuming it is always possible to convert to etas, do their modular properties tell all? And, assuming we can canonicalize the eta expression, can we always convert back to Gammas? I don't think Selberg & Chowla were thinking about computer algebra. I'm just trying to simplify piles of Gammas. A weird possibility: Allow piles of Beta[rational,rational]. Although traditionally defined in terms of three Gammas, Beta is a much simpler function with nice hypergeometric representations. --rwg Oh, my aching tush: Mathematica is pretty good about Solving linear systems which require a symbolic determinant to vanish. ("Unable to determine if blah=blahblah. Assuming it is ...") But if you append (1/6)! == (3 Sqrt[3/\[Pi]] ((1/3)!)^2)/(2 2^(1/3)) or a similar identity in constants to a vanilla nonsingular system, you instantly get the empty set of solutions!

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[math-fun] Gosper factorial stuff
by Warren D Smith 02 Apr '14

02 Apr '14

RWG:My objective, more clearly stated, is ultimately to automatically simplify factorials(n/d), not just factorials(n/24), by canonicalizing first to minimal d and then to minimal n. So far, reduction is possible for n/d, n>d/2 (by reflection) 1/2, 1/6, 3/8, 4/9, n/10, n/12, n/14, n/15, n>1 5/16, 7/16, n/18, n/20, n>1 n/21, n>2 n/22, n/24, n>1 11/25, 12/25, n prime, apparently never for prime>2n It would be amazing to find a reduction inaccessible via reflection and tuplication. But how could we tell? --In fact, Chowla & Selberg *DID* find cases inaccessible by reflection and tuplication, all (n/24)! can be expressed in close form in terms of elliptic functions. No? There's been followup work by other authors. I've posted about this in the past on math-fun, you should check the archives, I think the cites are there.

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