Gosper has an annoying habit of using non-ASCII characters I cannot read, which come out as "?" on my screen. Or maybe they really are "?". Who knows? Anyhow, product_{(k,n)=1} (z+k/n) clearly equals infinity and/or undefined if z>0. [I assume (k,n) means gcd(k,n)... I assume k>0 and n>0 intended... you know Bill, there is nothing actually wrong with writing math in a way intended to make it clearly easily readable+unambiguous...] To see this, pair up terms with k/n and n/k. On 1/18/13, math-fun-request@mailman.xmission.com <math-fun-request@mailman.xmission.com> wrote:
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Today's Topics:
1. Re: Best explanation of the Higgs particle (Wouter Meeussen) 2. Re: Best explanation of the Higgs particle (Henry Baker) 3. Re: Best explanation of the Higgs particle (Meeussen Wouter (bkarnd)) 4. why string theory can have high-spin particles without inconsistency (allegedly) (Warren Smith) 5. Joseph Smith's favorite function (Bill Gosper) 6. product_{(k,n)=1} ?(z+k/n) ? (Bill Gosper)
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Message: 1 Date: Wed, 16 Jan 2013 23:57:03 +0100 From: "Wouter Meeussen" <wouter.meeussen@telenet.be> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Best explanation of the Higgs particle Message-ID: <35C36383F9584D8B92EBA539D0527D04@Wou> Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original
along similar lines,
playing with http://xaonon.dyndns.org/hawking/hrcalc.js I came unto a surprise: fill in the estimated mass 3.4 10^54 kg of the the known (?) universe into the formula for a black hole, and estimate the diameter (or circumference/Pi for you singularophobics out
there), what diameter do you get?
to be (inside) or not to be (inside), that's the question
Wouter.
-----Original Message----- From: Warren Smith Sent: Wednesday, January 16, 2013 2:39 AM To: math-fun Subject: [math-fun] Best explanation of the Higgs particle
Here's the download link for the full-motion video: http://online.itp.ucsb.edu/download/bblunch/arkanihamed2/snd/Arkanihamed2_Bl...
--I listened to this Nima Arkani-Hamed lecture, my computer was able to handle it although took an hour to download.
One thing N.A-H mentioned as a key building block was that it is impossible for a particle to have spin>2.
However, in "string theory" just spin a string harder. You can get whatever spin you want, no matter how large. No?
So: How are these two claims to be reconciled?
[I have asked this question of physicists before, by the way. So far, I have never got any answer back I considered reasonable.]
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Message: 2 Date: Wed, 16 Jan 2013 15:42:37 -0800 From: Henry Baker <hbaker1@pipeline.com> To: Wouter Meeussen <wouter.meeussen@telenet.be> Cc: math-fun@mailman.xmission.com Subject: Re: [math-fun] Best explanation of the Higgs particle Message-ID: <E1TvceG-0005Tc-LI@elasmtp-banded.atl.sa.earthlink.net> Content-Type: text/plain; charset="us-ascii"
Interesting. I got 530 billion light years for the radius. Is this correct?
This is only ~37x the current estimated size of the universe, so at least the orders of magnitude are close. Perhaps it is the estimate of the mass of the universe that is off?
Perhaps we're inside the black hole, and that microwave background we see is our own event horizon?
BTW, Planck's initial insight into the solution to the "ultraviolet catastrophe" (I would have called it the "purple paradox", but I guess that phrase wasn't dramatic enough for the popular press) was to posit energy in quanta.
Then Rutherford developed the "planetary model" of the atom, but the problem with this is that a classical electron would almost immediately radiate way all of its orbital energy. The solution was that electrons could be in resonant "orbits" which didn't radiate (or more accurately, would radiate in particular energy jumps; for long-lived atoms, these jumps were far larger than normally available).
I suspect that something similar may be happening to black holes at the smallest sizes -- there may be resonances which allow very small black holes to survive far longer than the standard Hawking radiation model would predict.
At 02:57 PM 1/16/2013, Wouter Meeussen wrote:
along similar lines,
playing with http://xaonon.dyndns.org/hawking/hrcalc.js I came unto a surprise: fill in the estimated mass 3.4 10^54 kg of the the known (?) universe into the formula for a black hole, and estimate the diameter (or circumference/Pi for you singularophobics out there), what diameter do you get?
to be (inside) or not to be (inside), that's the question
Wouter.
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Message: 3 Date: Thu, 17 Jan 2013 09:12:36 +0100 From: "Meeussen Wouter (bkarnd)" <wouter.meeussen@vandemoortele.com> To: 'math-fun' <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Best explanation of the Higgs particle Message-ID: <C6CF6AA22326BC4FA796A688AFF8551A17887C0D38@MAIL.vdm.grp> Content-Type: text/plain; charset="us-ascii"
This is only ~37x the current estimated size of the universe
that's what I got too. No surprise in hindsight, since we assume the universe's expansion speed to equal c if we multiply the uniform expansion rate by the estimated universe size.
Some things do no grok though: temperature seen from the inside (3K cosmic background) doen't match that seen 'from the outside';
Wouter. (cosmology from the hip, tongue in cheek)
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Henry Baker Sent: donderdag 17 januari 2013 0:43 To: Wouter Meeussen Cc: math-fun@mailman.xmission.com Subject: Re: [math-fun] Best explanation of the Higgs particle
Interesting. I got 530 billion light years for the radius. Is this correct?
This is only ~37x the current estimated size of the universe, so at least the orders of magnitude are close. Perhaps it is the estimate of the mass of the universe that is off?
Perhaps we're inside the black hole, and that microwave background we see is our own event horizon?
BTW, Planck's initial insight into the solution to the "ultraviolet catastrophe" (I would have called it the "purple paradox", but I guess that phrase wasn't dramatic enough for the popular press) was to posit energy in quanta.
Then Rutherford developed the "planetary model" of the atom, but the problem with this is that a classical electron would almost immediately radiate way all of its orbital energy. The solution was that electrons could be in resonant "orbits" which didn't radiate (or more accurately, would radiate in particular energy jumps; for long-lived atoms, these jumps were far larger than normally available).
I suspect that something similar may be happening to black holes at the smallest sizes -- there may be resonances which allow very small black holes to survive far longer than the standard Hawking radiation model would predict.
At 02:57 PM 1/16/2013, Wouter Meeussen wrote:
along similar lines,
playing with http://xaonon.dyndns.org/hawking/hrcalc.js I came unto a surprise: fill in the estimated mass 3.4 10^54 kg of the the known (?) universe into the formula for a black hole, and estimate the diameter (or circumference/Pi for you singularophobics out there), what diameter do you get?
to be (inside) or not to be (inside), that's the question
Wouter.
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Message: 4 Date: Thu, 17 Jan 2013 16:25:38 -0500 From: Warren Smith <warren.wds@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] why string theory can have high-spin particles without inconsistency (allegedly) Message-ID: <CAAJP7Y0ZCy__+UbPBzz3zsFFjhM2hGxehF3refjvkZQmNXfK2g@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1
There are various arguments, some dating back to the 1960s, which show that quantum field theories cannot allow any fundamental particle to exist with spin>2. But "string theory" has arbitrarily high spin "particles." Contradiction?
I had asked several physicists about this over many years & none provided a good answer. However, allegedly the answer is this.
The inconsistency arguments assume (sometimes implicitly) that the number of particle/field types in the quantum field theory is finite. String theory escapes by having infinite set. Essentially, each time there would be an inconsistency, it is canceled out by putting in an interaction with even-higher spin particles which cancel out the inconsistency. Due to infiniteness, the chain of such fudge-corrections need never end.
Still, I think this represents a very important/frightening way in which string theory is fundamentally different from quantum field theories, deserving of note. This all is not mentioned in most (all?) of the popular books "explaining" string theory to the unwashed.
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Message: 5 Date: Fri, 18 Jan 2013 04:47:06 -0800 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Joseph Smith's favorite function Message-ID: <CAA-4O0GHU3adPJ6L0GjMa3X=WEUQwZ07-w+kZtYmKm40Q505bw@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-7
Can Maple do these? Mma 9.0 and Macsyma can't. Not even half the special cases. Out[1015]= PolyGamma[n, 1/8] + (-1)^n PolyGamma[n, 3/8] == 2^(1 + n) PolyGamma[n, 1/4] - (2 I ?)^(1 + n) PolyLog[-n, -((1 + I)/Sqrt[2])]
n>0.
In[1016]:= Table[%, {n, 0, 4}]
In[1017]:= FullSimplify[%]
Out[1017]= {False, 32 Catalan + 2 Sqrt[2] ?^2 + PolyGamma[1, 3/8] == PolyGamma[1, 1/8], 0 == 12 Sqrt[2] ?^3 + PolyGamma[2, 1/8] + PolyGamma[2, 3/8] + 448 Zeta[3], 8 (-16 + 11 Sqrt[2]) ?^4 + 16 PolyGamma[3, 1/4] + PolyGamma[3, 3/8] == PolyGamma[3, 1/8], 0 == 912 Sqrt[2] ?^5 + PolyGamma[4, 1/8] + PolyGamma[4, 3/8] + 380928 Zeta[5]}
Out[1024]= (-1 + 2^(-1 - n)) PolyGamma[n, 1/6] + ((-1)^n 2^(-1 - n) + 2^(1 + n)) PolyGamma[n, 1/3] == I^(-1 + n) ?^(1 + n) PolyLog[-n, 1/2 (-1 - I Sqrt[3])] - (-1)^ n (-1 + 3^(1 + n)) n! Zeta[1 + n]
n>0.
In[1025]:= Table[%, {n, 0, 4}]
In[1026]:= FullSimplify[%]
Out[1026]= {False, 4 ?^2 + 3 PolyGamma[1, 1/6] == 15 PolyGamma[1, 1/3], True, 16 ?^4 + 3 PolyGamma[3, 1/6] == 51 PolyGamma[3, 1/3], True}
--rwg
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Message: 6 Date: Fri, 18 Jan 2013 15:39:44 -0800 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] product_{(k,n)=1} ?(z+k/n) ? Message-ID: <CAA-4O0GimGZRhYOvwVZ015ntba6J+mmVdZDQ8JOz=5Z-vXQmWQ@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-7
Someone once sent me (us?) a paper deriving (presumably by inclusion-exclusion) this gamma product formula. Does anyone remember where it is? Presumably for n=12 it gives something like
(z - 11/12)!*(z - 7/12)!*(z - 5/12)!*(z - 1/12)!== 4*(2*z)!*(12*z)!*?^2/(2^(12*z)*3^(6*z)*(4*z)!*(6*z)!)
It doesn't seem to be on http://functions.wolfram.com/ --rwg
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