September 2014 - mathfun@mailman.xmission.com

[math-fun] Teachers (Union?) vs Wikipedia
by Bill Gosper 26 Sep '14

26 Sep '14

Over the years, I have had several grammarschoolers and middleschoolers tell me their teachers discouraged or even forbade the use of Wikipedia because it's "unreliable--anyone can edit it". One of them said all .coms were questionable--stick to .govs ! This is enormous hypocrisy--I'll bet those teachers use Wikipedia several times an evening. Do they feel threatened? I just suggested that the latest such acquaintance ask his teacher for specific examples of Wikipedia bogosity. --rwg

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[math-fun] Ramanujan's notebooks for $1000?
by Warren D Smith 26 Sep '14

26 Sep '14

> I remember when I bought my copy of the Abramowitz and Stegun > book for 12 $ canadian in Montr?al, Indeed. Abramowitz & Stegun now available for free electronically, as well as about $20 hardcopy, even though it represents more work, and also more typesetting and printing, and also more objectively valuable, than the Ramanujan notebooks for over $1000. What's the $980 paying for? Why? Well, by law, Abramowitz & Stegun was not copyrighted or copyrightable since produced largely by US govt employees. So here's an idea. Revise US law so that any publication produced by anybody funded by a US govt grant, is not copyrightable. One result: Ramanujan notebooks will be available for low cost. Also, most scientific papers. Also, then NSF grants will actually be paying for something useful for the taxpayers. Oh, and guess how much of all those $1000s, was ever given to Ramanujan himself? Zilch. The defenders of this price heap great praise upon this wonderful work and seem to think price is justified due to author labor. A different view: the so-called authors are parasites, the true creator is getting zilch.

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[math-fun] Stern-Brocot sum (D.Wilson)
by Warren D Smith 26 Sep '14

26 Sep '14

> From: "David Wilson" <davidwwilson(a)comcast.net> > To: "math-fun" <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Observation on Stern-Brocot tree > > It appears that the sum of the nth row of the Stern-Brocot Tree is > (3*2^(n-2) - 1)/2. > I assume this is known, but I don't know how to prove it. http://en.wikipedia.org/wiki/Stern-Brocot_tree seems to think the first row is the single number 1/1, so n starts at n=2? I would prefer to start at n=0 in which case change the formula to (3*2^n - 1)/2. Next row 2/1+1/2=5/2=(3*2-1)/2, check. Next 1/3+2/3+3/2+3/1=11/2=(3*4-1)/2, check. Next (3*8-1)/2=23/2. "The Stern-Brocot sequence of order i is the sequence formed by inserting a mediant between each consecutive pair of values in the Stern-Brocot sequence of order i-1." The picture in wikipedia http://en.wikipedia.org/wiki/Stern-Brocot_tree#mediaviewer/File:SternBrocot… illustrates this, where the sequences are the horizontal rows of the picture, which include the tree nodes on that level as a subsequence, but also include additional values (on the dotted lines), and for example to obtain 3/5 on the bottom row you "add" 1/2 and 2/3. So Wilson's conjecture is equivalent to: the Nth stern-brocot sequence (ignoring the two endpoints 0/1 and 1/0) would sum to 3*2^N - 2 - N/2 beginning 1, 7/2, 9, 41/2 for N=0,1,2,3. [Because the backward difference of this formula is Wilson's.]

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[math-fun] Ramanujan's notebooks for $1000?
by Warren D Smith 26 Sep '14

26 Sep '14

Here's an idea: fly to Cambridge, get his original notebooks from library, scan them, and fly back home. Probably cost less than $1000...

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[math-fun] Ramanujan's notebooks for $1000?
by Warren D Smith 26 Sep '14

26 Sep '14

>I thought the Ramanujan notebooks are already available as nice printed books from >Springer. --It seems Plouffe's claim they cost $1000 was a slight exaggeration. Amazon will sell you part 5 of the notebooks (printed by Springer) for $183, so assuming the other parts same price, that is $915. If there is sales tax that'll indeed reach just about $1000. > Would the library even let you near the original notebooks, much less handle them with > your greasy fingers (you could wear gloves), and open them up and press them against > a scanner? --my point was you could save money versus Springer by flying to Cambridge and copying them yourself. Of course, my further point was, if "publishers" sell books for more than the cost of flying yourself across the world to scan author's work, the meaning of the world "publication" may require revision, and the entire moral/econo/logical justification for the existence of "copyright law" would seem to be a crock of shit.

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Re: [math-fun] Machinoid arcsin π formulæ
by Bill Gosper 25 Sep '14

25 Sep '14

CGreathouse> Naively we want to minimize -sum 1/log x and by that metric this is worse than Machin: -1/log(1/7) - 1/log(239/4802) = 0.847... -1/log(1/5) - 1/log(1/239) = 0.803... It only gets worse if you expand the model to make unit fractions less costly. (Exercise: what's the right discount here? Depends on the relative speed of "divide by a small integer" and "add two multiprecision numbers".) But I guess it beats it in the sense that 1/7 < 1/5? Charles Greathouse Analyst/Programmer Case Western Reserve University Ouch. That's the sense I was using. Non sense. On Thu, Sep 25, 2014 at 5:38 AM, Bill Gosper <billgosper(a)gmail.com> wrote: Doesn't this, in some sense, beat Machin? In[628]:= TrigExpand[Sin[4*ArcSin[1/7] - ArcSin[239/4802]]] Out[628]= 1/2 Is it trying to tell us something? Anyway, here it is demonstrated. The product matrix for a arcsin x: 1 2 a x asinmat[a_, x_] := {{(1 - ---) x , --------}, {0, 1}} 2 k -1 + 2 k For 22 digits, we'll need In[698]:= Ceiling[22/Log[10, 49]] Out[698]= 14 terms of In[696]:= asinmat[24, 1/7] Out[696]= 1 1 24 {{-- (1 - ---), ------------}, {0, 1}} 49 2 k 7 (-1 + 2 k) and In[699]:= Ceiling[22/Log[10, (4802/239)^2]] Out[699]= 9 terms of In[697]:= asinmat[6, 239/4802] Out[697]= 1 57121 (1 - ---) 2 k 717 {{---------------, ---------------}, {0, 1}} 23059204 2401 (-1 + 2 k) In[702]:= Dot @@ Table[%696, {k, 14}] Out[702]= 5014575 8094675459628407766349207262854411849 {{--------------------------------, -------------------------------------}, {0, 1}} 15434586961405991978278722732032 2352869649110314767878447007714508800 In[703]:= Dot @@ Table[%697, {k, 9}] Out[703]= 78689571474191869819784797472677868659791566555 {{------------------------------------------------------------------------, 120803282051429183719400505971423044572368826431183593390793372660137984 13702798156207017640945057262283907448445615550593235749465375823985275 -----------------------------------------------------------------------}, {0, 1}} 45867260308260778678966466579008135363717505787308514504324035131736064 In[706]:= N[%702[[1, 2]] - %703[[1, 2]], 27] Out[706]= 3.14159265358979323846264335 We got four freebies because convergence is a little faster early on. If we were certain never to desire more digits, these exact fractions are silly. But this way they provide the luxury of continuing the calculation without starting over. --rwg

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[math-fun] Physicist Claims Black Holes Mathematically Don't Exist
by Warren D Smith 25 Sep '14

25 Sep '14

> 1. Physicist Claims Black Holes Mathematically Don't Exist > (Henry Baker) > http://beta.slashdot.org/story/207637 > > Black holes, the stellar phenomena that continue to capture the imagination > of scientists and science fiction authors, may not actually exist. > According to a paper published by physics professor Laura Mersini-Houghton > at the University of North Carolina and Mathematics Professor Harald > Pfeiffer of the University of Toronto, as a collapsing star emits Hawking > radiation, it also sheds mass at a rate that suggests it no longer has the > density necessary to become a black hole ? the singularity and event horizon > never form. While the arXiv paper with the exact solution has not yet been > peer reviewed, the preceding paper by Mersini-Houghton with the approximate > solutions was published in Physics Letters B. > > "I'm still not over the shock," said Mersini-Houghton. "We've been studying > this problem for a more than 50 years and this solution gives us a lot to > think about... Physicists have been trying to merge these two theories ? > Einstein's theory of gravity and quantum mechanics ? for decades, but this > scenario brings these two theories together, into harmony." > ------------------ > http://arxiv.org/abs/1409.1837 > > Back-reaction of the Hawking radiation flux on a gravitationally collapsing > star II: Fireworks instead of firewalls > > A star collapsing gravitationally into a black hole emits a flux of > radiation, knowns as Hawking radiation. When the initial state of a quantum > field on the background of the star, is placed in the Unruh vacuum in the > far past, then Hawking radiation corresponds to a flux of positive energy > radiation travelling outwards to future infinity. The evaporation of the > collapsing star can be equivalently described as a negative energy flux of > radiation travelling radially inwards towards the center of the star. Here, > we are interested in the evolution of the star during its collapse. Thus we > include the backreaction of the negative energy Hawking flux in the interior > geometry of the collapsing star and solve the full 4-dimensional Einstein > and hydrodynamical equations numerically. We find that Hawking radiation > emitted just before the star passes through its Schwarzschild radius slows > down the collapse of the star and substantially reduces its mass thus the > star bounces before reach > ing the horizon. The area radius starts increasing after the bounce. > Beyond this point our program breaks down due to shell crossing. We find > that the star stops collapsing at a finite radius larger than its horizon, > turns around and its core explodes. This study provides a more realistic > investigation of the backreaction of Hawking radiation on the collapsing > star, that was first presented in [1]. --Classically a collapsing star takes a finite amount of time according to clocks on the star surface, to turn into a black hole. But according to external observer, the star's surface never falls inside the event horizon in any finite time, albeit it gets exponentially close to it. In view of this, if we regard the Hawking evaporation as taking some large but finite time as viewed by external observer, it will happen before the black hole forms, hence it never forms, as perceived by external observer. But since it gets "exponentially close to forming" in view of same external observer, it sure looks black for a long time! Meanwhile, the co-collapsing observer thinks he has hit singularity & died in finite time. But he has no way to communicate this experience to external observer. So... all that reasoning has been around for ages, and certainly I have been =awre of it for at least 20 years. So I don't see what is wrong with the view proposed by http://arxiv.org/abs/1409.1837 that "the black hole never forms" and the collapse "bounces." It is just a very long wait before the bounce... during which the non-black-hole bears a very great resemblance to a true black hole. And their numerical work can be viewed as providing further evidence for this picture. But I'm not sure how seriously we should regard this numerical result because as they admit, their numerical methods stop working after a "shell crossing" and cannot follow most or all of the outward bounce. (What do they mean by "shell crossing"? They explain: "The inner layers of the star pick up a larger positive velocity than the outer layers, resulting in shell crossing.") As they remark: "The star never crosses its horizon, so neither unitarity nor causality are violated, thereby solving the longstanding information loss paradox." Also no "firewall paradox." Also no singularity and no horizon form. However. What about this OTHER PARADOX: the infalling observer thinks he has hit a singularity and died in finite time by his own clock. Is this still true within their computer simulation? The paradox would be: we have two observers who think a singularity exists, and who think a singularity never exists. They do not explicitly raise or answer this question. But I think they intended the answer to be: "really, the infalling observer perceives a massive blast of outgoing Hawking radiation just before he hits the horizon. This kills him before hitting that horizon. He never really enters a horizon and never really hits any singularity, and the singularity never forms in his view because of a big influx of negative energy toward where it would have been."

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Re: [math-fun] Machinoid arcsin π formulæ
by Bill Gosper 25 Sep '14

25 Sep '14

Doesn't this, in some sense, beat Machin? In[628]:= TrigExpand[Sin[4*ArcSin[1/7] - ArcSin[239/4802]]] Out[628]= 1/2 Is it trying to tell us something? rcs>The atan formulas can be understood as being about atan(N/D) = imag(log D+iN), and factoring D+iN into complex primes. You combine a few relations, and eventually wind up with log(i) or imag(log 1+i). You can go after new formulas by factoring complex numbers A+Bi with large A and small B. "Fortuitously", 239+i has norm 57122 = 2 * 13^4, while 5+i has norm 26 = 2 * 13, leading to pi/4 = 4 atan(1/5) - atan(1/239). ("Fortuitously" comes from 169, a square, turning up in the solutions of x2-2y2 = +-1 as (239,169). Maybe something similar could be made of 144 being a Fib, and being part of x2-5y2 = +-4 as (321,144)?) Can you do something similar with the arcsins? I notice that the implied base leg for the 11/43 case is sqrt(43^2-11^2) -> sqrt(32 * 54) -> 24 sqrt3. This could also be useful in simplifying your arcsin expressions, if you regard "factoring" as a simplification. Rich ----- Quoting Bill Gosper <billgosper(a)gmail.com>: These must have been done to death. In[526]:= FullSimplify[Sin[-ArcSin[23/361] + 2*ArcSin[11/38]]] Out[526]= 1/2 In[528]:= FullSimplify[Sin[ArcSin[97/961] + 2*ArcSin[13/62]]] Out[528]= 1/2 In[532]:= FullSimplify[Sin[2*ArcSin[11/43] + ArcSin[23/3698]]] Out[532]= 1/2 In[517]:= FullSimplify[Sin[ArcSin[23/98] + 2*ArcSin[1/7]]] Out[517]= 1/2 I.e., ArcSin[97/961] + 2*ArcSin[13/62] = π/6, but Wolfram's nascent artificial intelligence is too stupid to notice. I don't know if this scheme can beat Machin--I only looked at 2 arcsin a + arcsin b. The apparent messiness of the arcsin series vs arctan is illusory if we replace clunky old summation with matrix products. (gosper.org/pathi.pdf) --rwg --rwg

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[math-fun] Machinoid arcsin π formulæ
by Bill Gosper 24 Sep '14

24 Sep '14

These must have been done to death. In[526]:= FullSimplify[Sin[-ArcSin[23/361] + 2*ArcSin[11/38]]] Out[526]= 1/2 In[528]:= FullSimplify[Sin[ArcSin[97/961] + 2*ArcSin[13/62]]] Out[528]= 1/2 In[532]:= FullSimplify[Sin[2*ArcSin[11/43] + ArcSin[23/3698]]] Out[532]= 1/2 In[517]:= FullSimplify[Sin[ArcSin[23/98] + 2*ArcSin[1/7]]] Out[517]= 1/2 I.e., ArcSin[97/961] + 2*ArcSin[13/62] = π/6, but Wolfram's nascent artificial intelligence is too stupid to notice. I don't know if this scheme can beat Machin--I only looked at 2 arcsin a + arcsin b. The apparent messiness of the arcsin series vs arctan is illusory if we replace clunky old summation with matrix products. (gosper.org/pathi.pdf) --rwg

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[math-fun] Physicist Claims Black Holes Mathematically Don't Exist
by Henry Baker 24 Sep '14

24 Sep '14

FYI -- http://beta.slashdot.org/story/207637 Black holes, the stellar phenomena that continue to capture the imagination of scientists and science fiction authors, may not actually exist. According to a paper published by physics professor Laura Mersini-Houghton at the University of North Carolina and Mathematics Professor Harald Pfeiffer of the University of Toronto, as a collapsing star emits Hawking radiation, it also sheds mass at a rate that suggests it no longer has the density necessary to become a black hole Â the singularity and event horizon never form. While the arXiv paper with the exact solution has not yet been peer reviewed, the preceding paper by Mersini-Houghton with the approximate solutions was published in Physics Letters B. "I'm still not over the shock," said Mersini-Houghton. "We've been studying this problem for a more than 50 years and this solution gives us a lot to think about... Physicists have been trying to merge these two theories Â Einstein's theory of gravity and quantum mechanics Â for decades, but this scenario brings these two theories together, into harmony." ------------------ http://arxiv.org/abs/1409.1837 Back-reaction of the Hawking radiation flux on a gravitationally collapsing star II: Fireworks instead of firewalls A star collapsing gravitationally into a black hole emits a flux of radiation, knowns as Hawking radiation. When the initial state of a quantum field on the background of the star, is placed in the Unruh vacuum in the far past, then Hawking radiation corresponds to a flux of positive energy radiation travelling outwards to future infinity. The evaporation of the collapsing star can be equivalently described as a negative energy flux of radiation travelling radially inwards towards the center of the star. Here, we are interested in the evolution of the star during its collapse. Thus we include the backreaction of the negative energy Hawking flux in the interior geometry of the collapsing star and solve the full 4-dimensional Einstein and hydrodynamical equations numerically. We find that Hawking radiation emitted just before the star passes through its Schwarzschild radius slows down the collapse of the star and substantially reduces its mass thus the star bounces before reach ing the horizon. The area radius starts increasing after the bounce. Beyond this point our program breaks down due to shell crossing. We find that the star stops collapsing at a finite radius larger than its horizon, turns around and its core explodes. This study provides a more realistic investigation of the backreaction of Hawking radiation on the collapsing star, that was first presented in [1]. http://www.sciencedirect.com/science/article/pii/S0370269314006686/pdf?md5=… Backreaction of Hawking radiation on a gravitationally collapsing star I: Black holes? Particle creation leading to Hawking radiation is produced by the changing gravitational field of the collapsing star. The two main initial conditions in the far past placed on the quantum field from which particles arise, are the HartleÂHawking vacuum and the Unruh vacuum. The former leads to a time-symmetric thermal bath of radiation, while the latter to a flux of radiation coming out of the collapsing star. The energy of Hawking radiation in the interior of the collapsing star is negative and equal in magnitude to its value at future infinity. This work investigates the backreaction of Hawking radiation on the interior of a gravitationally collapsing star, in a HartleÂHawking initial vacuum. It shows that due to the negative energy Hawking radiation in the interior, the collapse of the star stops at a finite radius, before the singularity and the event horizon of a black hole have a chance to form. That is, the star bounces instead of collapsing to a black hole. A trapped surface near the last stage of the starÂs collapse to its minimum size may still exist temporarily. Its formation depends on the details of collapse. Results for the case of Hawking flux of radiation with the Unruh initial state, will be given in a companion paper II.

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