November 2013 - mathfun@mailman.xmission.com

Re: [math-fun] Fwd: [CASE:423927] GCD suggestions
by Henry Baker 28 Nov '13

28 Nov '13

Yes! At 06:39 AM 11/28/2013, Andy Latto wrote: >On Thu, Nov 28, 2013 at 12:47 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> At the very least, gcd(a,b) = m*a+n*b for some _integers_ m,n. >> >> The usual definition of gcd requires both m,n to be non-zero (actually >> gcd(m,n)=1). >> > >It does? Then what is gcd(5,15)? I thought it was 5, but you can't write 5 >as m*5 + n*15, with m and n nonzero. > >Maybe you meant "m and n not *both* 0"? > >Andy

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Re: [math-fun] Fwd: [CASE:423927] GCD suggestions
by Henry Baker 28 Nov '13

28 Nov '13

So gcd(sqrt(2),1)=gcd(%pi,%e)=(+)0, no? At 03:33 PM 11/27/2013, Bill Gosper wrote: >New wording: GCD(a,b):= the limit of the Euclidean process of iteratively subtracting the smaller from the larger.

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[math-fun] Fwd: [CASE:423927] GCD suggestions
by Bill Gosper 27 Nov '13

27 Nov '13

Does math-fun quietly discard BCCs? I BCCed them this suggestion to Support(a)wolfram.com but it never distributed. I also BCCed my list of eavesdroppers, but I don't know if they got it. Then I re-sent it to rcs asking if he knew where it went. Maybe he's off on a Thanksgubble holiday. Anyway, comments (as always) welcome on the following: ---------- Forwarded message ---------- From: Bill Gosper <billgosper(a)gmail.com> Date: Tue, Nov 26, 2013 at 2:05 PM Subject: Re: [CASE:423927] GCD suggestions To: Wolfram Technical Support <support(a)wolfram.com> Hi again Jason! Below, I should have said: Likewise. GCD[0,foo] is |foo|. (If foo ϵ Reals.) If your developers are leery of GCD[1,π]→0, I think I can muster authoritative corroborators, if not corroborative literature. Motivation: The gcd of two real quantities is the largest quantity that goes into each a whole number of times. In[642]:= Convergents[π, 4] Out[642]= {3, 22/7, 333/106, 355/113} In[643]:= GCD[%, 1] Out[643]= {1, 1/7, 1/106, 1/113} The better the approximation, the smaller the GCD. There is no positive GCD[π,1], so it must be 0. This extension isn't likely to save work for anyone. But increasingly, people use Mathematica as a source of expertise and instruction. --Bill Gosper On Tue, Nov 26, 2013 at 11:55 AM, Wolfram Technical Support < support(a)wolfram.com> wrote: > > Hello Bill Gosper, > > Thank you for bringing these issues to our attention. > > I was able to reproduce the many results you found with GCD and I have > filed a note with our development team about them. I will be sure to let > you know if I hear anything back. > > We are always interested in improving Mathematica and I want to thank you > again for taking the time to contact us about this issue. If you have any > further questions, or if you find any further issues, please feel free to > contact us again. > > Sincerely, > > Jason Grigsby > Wolfram Technology Group > Wolfram Research, Inc > www.wolfram.com > > > > > > On: 11/25/2013 08:12pm billgosper(a)gmail.com submitted the following Case: > In[576]:= FullSimplify[GCD[0, n], n \[Element] Integers] > > Out[576]= Abs[n] > > Good! > > In[577]:= FullSimplify[GCD[0, n], n \[Element] Reals] > > Out[577]= GCD[0, n] > > Should be Abs[n] > > In[578]:= FullSimplify[GCD[0, n], n \[Element] Rationals] > > Out[578]= GCD[0, n] > > Likewise. GCD[0,foo] is foo. > Correction: |foo|. (If foo ϵ Reals.) In[575]:= FullSimplify[GCD[\[Pi], 2*\[Pi]]] Out[575]= GCD[\[Pi], 2 \[Pi]] Should be ?. In my original mail, this says "Should be π"! Your mail system needs updating. > > In[580]:= FullSimplify[GCD[Sqrt[2], Sqrt[8]]] > > Out[580]= GCD[Sqrt[2], 2 Sqrt[2]] > > Should be ?2. > This said √2 ! > > In[581]:= FullSimplify[GCD[1/Sqrt[2], 1/Sqrt[8]]] > > Out[581]= GCD[1/(2 Sqrt[2]), 1/Sqrt[2]] > > Should be 1/?8. > Hopefully, your developers got the undamaged version. > > In[582]:= FullSimplify[GCD[1/Sqrt[2], 1]] > > Out[582]= GCD[1, 1/Sqrt[2]] > > Should be 0. Likewise GCD[1,?], GCD[?2,?3], GCD[?2,?]. > I.e., GCD[1,π], GCD[√2,√3], GCD[√2,π] > > And everybody knows GCD[E,?]=0, but nobody can prove it. > At any rate, PossibleZeroQ[GCD[E, π]] should not be False. --Bill Gosper > In[583]:= $LicenseID > > Out[583]= "L3290-7570" > > >

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[math-fun] "amplituhedron" blog post by Dixon
by Warren D Smith 27 Nov '13

27 Nov '13

http://www.preposterousuniverse.com/blog/2013/10/03/guest-post-lance-dixon-…

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Re: [math-fun] how many decimal places exist in the mass of an electron (& other such things)?
by Warren D Smith 27 Nov '13

27 Nov '13

> I usually don't have much to add to these discussions, but this one is > intriguing. > A similar question was asked by Chris Rose here at Rutgers: > > What does the communication channel "look like" between an electron > and proton in a hydrogen atom. > > By "look like", he meant bits/second. > > Somehow, they are "telling" each other where they are and adjusting their > properties accordingly. > Rich --yes, this is an excellent kind of question. I presume "seconds" measured in the center of mass frame of the hydrogen atom. Naively guessing one might try this: if E is the ionization energy of hydrogen (13.6 eV) then the time scale for 1 bit transmit is of the order h/E, where h=Planck constant. This is 3*10^(-16) seconds. > The thing is, no experiment ever directly measures the rest mass of the > electron. It measures something else, and then concludes "If <insert > assumptions about physics> is true, then the rest mass of the electron is > X, with a measurement standard deviation of Y". > > The answer to "how small can Y be?" depends crucially on what assumptions > about physics are made.... without restrictions on the physics used in the calculation, > there can't be any lower bound to Y. --ok, fine, I agree; but just choose your favorite chunks of well accepted contemporary physics then find the best bounds you can. Which physical constant has the most intrinsic uncertainty level?

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Re: [math-fun] how many decimal places exist in the mass of an electron (& other such things)?
by Warren D Smith 26 Nov '13

26 Nov '13

>Dave Dyer: electron never at rest, so has no meaning --and would you contend the mass of anything has no meaning? Bah. I'm trying to get estimates of how much meaning. Anyhow, for a "hydrogen atom" the spectrum is determined by stuff like electron mass & charge, and so if you can measure spectrum, then... and so on. > Nobody so far has addressed Warren's actual question, which seems to raise the problem of how well it is ultimately possible to even define such physical constants. > A more conveniently macroscopic example is the length of a day. Variations in the earth's orbit limit the accuracy to which the length of individual days can be treated as a constant. Averaging over an entire year (itself subject to smaller variations) permits greater accuracy, but involves redefining the meaning of "day" in a more sophisticated fashion. > Digging further down encounters alarming philosophical questions concerning (for example) the definition of measurement, and the universe within which it is legitimate to take an average --- over a sufficiently long period, the "day" is progressively lengthening. >Fred Lunnon --yah; been pretty disappointed with the mathfun responses to my question so far. One could argue the total #electrons in the visible (i.e. that can communicate with us) part of the universe is <10^80, and their total mass cannot be measured more accurately than uncertainty principles using the spatial and/or temporal size of said portion of universe, and hence I guess that means the electron mass has no meaning beyond about 130th decimal place. Also note, the accessible part of the universe will never get any bigger than a certain constant (which we are near reaching already) so this will stay true. The most accurate physics measurements so far that I can think of have about 18 significant figures (atomic clock frequency ratios). Superstring theory allegedly would predict all the physical constants (in Planck units) as infinitely precise real numbers. I actually think this is an important question and deserves more thought. It pertains to the amount of information there is inside physics theories. Including "the final theory."

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[math-fun] Closing the prime gap
by W. Edwin Clark 26 Nov '13

26 Nov '13

An interest article in Quanta Magazine about the work of Yitang Zhang, James Maynard, et al --together with tidbits about offers Zhang has had since his discovery: <https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing…> Together and Alone, Closing the Prime Gap<https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing…>

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[math-fun] "how many newman(step)s does it take
by Bill Gosper 26 Nov '13

26 Nov '13

to tack a 1 on the front of a cf?" Apparently, this question is a thought crime. If there are already six 1s on the front, the answer is 128: In[634]:= cfzewm[{1, 1, 1, 1, 1, 1, 69}, 128] Out[634]= {1, 1, 1, 1, 1, 1, 1, 69} Independent of "69": In[635]:= cfzewm[{1, 1, 1, 1, 1, 1, 105}, 128] Out[635]= {1, 1, 1, 1, 1, 1, 1, 105} But for four initial 1s the answer is only 32: In[637]:= cfzewm[{1, 1, 1, 1, 288}, 32] Out[637]= {1, 1, 1, 1, 1, 288} So clearly for five 1s, we need In[636]:= cfzewm[{1, 1, 1, 1, 1, 69}, 28334198897217871282112] Out[636]= {1, 1, 1, 1, 1, 1, 69} Excuse me kids, but WtF?? And it depends (heavily) on the "69": In[638]:= cfzewm[{1, 1, 1, 1, 1, 9}, 24512] Out[638]= {1, 1, 1, 1, 1, 1, 9} This is paradoxical. The four 1s case *is* dependent on the "288". No, not if we make it 69: In[639]:= cfzewm[{1, 1, 1, 1, 69}, 32] Out[639]= {1, 1, 1, 1, 1, 69} But we can make it a noninteger: 139/70 = {1,1,69} In[641]:= cfzewm[{1, 1, 1, 1, 139/70}, 128] Out[641]= {1, 1, 1, 1, 1, 1, 1, 69} which takes 128 because this is equivalent to the six 1s case. Can someone make sense of this? --Bill Here are Neil's magic functions, in case you want to play with this: ([[2;;-1]] instead of //Rest is a sign of the remarkable haste with which he wrote this.) cfzewm[cf_, n_: 1] := ContinuedFraction[ Divide @@ pos2newman[n + newman2pos[FromContinuedFraction[cf]]]] (*Note: n can be negative.*) calkinwilf[num_, denom_] := Block[{lis = {}, n = num, d = denom}, (While[ n > 1 || d > 1, (lis = Prepend[lis, Boole[n > d]]; If[n > d, n -= d, d -= n])]; lis)] wilfcalkin[bitlis_List] := Block[{n = 1, d = 1}, (Do[ If[bitlis[[i]] == 0, d += n, n += d], {i, 1, Length[bitlis]}]; {n,d})] newman2pos[num_, denom_] := FromDigits[Prepend[calkinwilf[num, denom], 1], 2] newman2pos[frac_] := newman2pos[Numerator[frac], Denominator[frac]] pos2newman[pos_] := wilfcalkin[IntegerDigits[pos, 2][[2 ;; -1]]]

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[math-fun] Some developer at WRI deserves extra credit
by Bill Gosper 25 Nov '13

25 Nov '13

In[564]:= FromDigits["IM", "Roman"] During evaluation of In[564]:= FromDigits::nrom: The expression IM is not a proper string of roman digits. >> Out[564]= 999 Most developers would lazily deem the impropriety a fatal error to simplify their coding. Now watch them get sued: "I lost millions on a programmed trade when Mathematica turned my typo into a ridiculous date instead of aborting the transaction." --rwg

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[math-fun] Cicada 3301
by Henry Baker 25 Nov '13

25 Nov '13

FYI -- http://www.telegraph.co.uk/technology/internet/10468112/The-internet-myster…

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