June 2013 - mathfun@mailman.xmission.com

Re: [math-fun] turkeys and gem polishers
by Henry Baker 29 Jun '13

29 Jun '13

I'm worried about what happens when these same botanists find out that lots of trees (at least lots of trees in California) can't germinate & reproduce until after a major forest fire... At 07:04 AM 6/28/2013, Bill Gosper wrote: >I am guilty of repeating the urban legend that botanists were frantically trying to breed >giant turkeys after realizing that the seeds of a certain endangered tree couldn't germinate >without passing through a dodo's digestive system.

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Re: [math-fun] Atrocious Mma 8&9, WolframAlpha bug:
by Bill Gosper 29 Jun '13

29 Jun '13

Thank you Fred! Seconds out indeed. Two conjectures: Kronenburg is a troll who wrote this to incite flamewars on mathematics fora, or Some other troll wrote this to embarrass Kronenburg. --rwg Captious graffiti: G,K&P, p156 "C'est une chose estrange combien il est fertile en proprietez." | B. Pascal [285] Then on the next page, I just hope I don't fall into this trap during the midterm. Google didn't find the Pascal quote except in a TeX source of GKP errata. Both Google and Google Translate wanted to respell the quote as C'est une chose *étrange* combien il est fertile en *propriété*,<https://www.google.com/search?client=firefox-a&hs=i7q&rls=org.mozilla:en-US…> which Google then found in various Pascal collections. Wondering if the graffito could actually be mistyped, I found on http://www-cs-faculty.stanford.edu/~uno/gkp.html a warning: SPECIAL NOTE TO THE SPEAKERS OF FRENCH AND OTHER EXOTIC LANGUAGES: Numerous quotations and bibliographic citations found in this book have been copied verbatim from the original sources. If you believe you have found a typographic error, you must prove it by showing that the original was incorrectly transcribed; believe it or not, your language has changed over the years, just as English has. Digging deeper, sure enough, http://books.google.fr/books?hl=fr&id=UqgUAAAAQAAJ&q=fertile I wonder how many bogus bug reports this particular graffito collects. I also wonder why Google proper didn't find it in the various pirated GKPs online, nor in books.google.fr . Re turkeys and gem polishers, http://www.reviewjournal.com/news/nevada-and-west/protection-sought-rare-wo… Only you can prevent sanctimonious talking bears. Joerg wrote << By sheer coincidence I just stumbled upon http://arxiv.org/abs/1105.3689 Can anyone comment on validity of the statements given? The first error in the Kronenburg draft appears in paragraph 1 of section 2, where his (1.4) Gamma(s) Gamma(1-s) = pi/sin(pi s) is misapplied with s = 0 . He seems to have been seduced by the same siren call that has lured others (they know who they are) to their doom: assuming nice limiting behaviour at a singularity which --- as Bill eloquently observed --- actually resembles curled up lasagne. The mathworld plot I earlier managed to misinterpret at http://mathworld.wolfram.com/BetaFunction.html provides a passable identikit of the miscreant concerned. Defining the continuation of the binomial coefficient n_C_m for x,y real by z = z(x, y) = Gamma(x+1) / Gamma(y+1) Gamma(x-y+1) and examining the neighbourhood of one of these "lasagne points" in the third wedge region, it appears that for m > 0, n >= 0 integer and r^2 = x^2 + y^2 small z(-m + x, -m-n + y) = c (y/x) + O(r) (*) where c = {-m}_C_n = (-1)^n {n+m-1}_C_n . This expansion makes it obvious that approaching the singularity along direction y = 0 produces zeroes in this region; whereas approaching along y = x produces the deprecated nonzero third wedge. In contrast, an analogous analysis of z(-m, n) for m > 0, n >= 0 in the second wedge finds z(-m + x, n + y) = c (1 - y/x) + O(r) , showing that approaching along y = 0 [along the x-axis, 60 degrees below rightwards in the customary orientation] produces desired Taylor coefficients; whereas approaching along y = x produces only zeroes. Despite an earlier assertion to the contrary, it transpires that supporters of the nuclear option were guilty of wanting to have their cake and eat it! [Better go sweep up those crumbs now, I suppose ...] Fred Lunnon On 6/28/13, Fred lunnon <fred.lunnon(a)gmail.com> wrote: Bill Gosper: << In particular, he must have fudged to get the consistency he claims from his flawed premises >> But he is consistent. Restricted to integer arguments, the discussion we've been having boils down to a choice between: [R] Assuming the recursion n_C_m = {n-1}_C_{m-1} + {n-1}_C_m across the whole plane; and [S} Assuming the symmetry n_C_m = n_C_{n-m} across the plane, in which case recursion must be abandoned at (n,m) = (0,0) . He is adopting [S] , and you've plumped for [R] . While I've finally come round to accepting that you're probably correct, I'm astonished at how delicate the reasoning involved has proved to be. Fred Lunnon On 6/28/13, Bill Gosper < billgosper(a)gmail.com> wrote: JPropp>Carelessness or incompetence seems far more likely than fraud, which requires an intent to deceive. (Am I being over-literal?) Jim Propp On Thursday, June 27, 2013, Bill Gosper <billgosper(a)gmail.com> wrote: Jörg> By sheer coincidence I just stumbled upon http://arxiv.org/abs/1105.3689 Can anyone comment on validity of the statements given? Best, jj This paper says "The symmetry identity [3, 4] and the trinomial revision identity [3, 4] are valid for all complex x,y,z" where [3] is G,K,&P's Concrete Math, which the author does not admit to contradicting. But GKP says "So the equation (-1 choose k) = (-1 choose -1-k) is always false! The symmetry identity fails for all other negative integers n, too. But unfortunately it’s all too easy to forget this restriction, since the expression in the upper index is sometimes negative only for obscure (but legal) values of its variables. Everyone who’s manipulated binomial coefficients much has fall into this trap at least three times." This paper is a fraud. --rwg "Never ascribe to malice that which can be explained by incompetence." I think there is deception here, even if Kronenburg imagines that Knuth et al agree with him. (But how much more explicit could they be?) In particular, he must have fudged to get the consistency he claims from his flawed premises, but I don't have time right now to catch him out. --rwg I haven't (re)read the part where GKP extend to noninteger lower index, where I hope they motivate (r choose negativeinteger) :=0. I recall Knuth in Vol 1 relying heavily on summing from -∞ to greatly simplify binomial sum manipulations, blissfully indifferent to the precise points of leftward termination. _______________________________________________

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Re: [math-fun] Atrocious Mma 8&9, WolframAlpha bug:
by Bill Gosper 29 Jun '13

29 Jun '13

JPropp>Carelessness or incompetence seems far more likely than fraud, which requires an intent to deceive. (Am I being over-literal?) Jim Propp On Thursday, June 27, 2013, Bill Gosper <billgosper(a)gmail.com> wrote: Jörg> By sheer coincidence I just stumbled upon http://arxiv.org/abs/1105.3689 Can anyone comment on validity of the statements given? Best, jj This paper says "The symmetry identity [3, 4] and the trinomial revision identity [3, 4] are valid for all complex x,y,z" where [3] is G,K,&P's Concrete Math, which the author does not admit to contradicting. But GKP says "So the equation (-1 choose k) = (-1 choose -1-k) is always false! The symmetry identity fails for all other negative integers n, too. But unfortunately it’s all too easy to forget this restriction, since the expression in the upper index is sometimes negative only for obscure (but legal) values of its variables. Everyone who’s manipulated binomial coefficients much has fall into this trap at least three times." This paper is a fraud. --rwg "Never ascribe to malice that which can be explained by incompetence." I think there is deception here, even if Kronenburg imagines that Knuth et al agree with him. (But how much more explicit could they be?) In particular, he must have fudged to get the consistency he claims from his flawed premises, but I don't have time right now to catch him out. --rwg I haven't (re)read the part where GKP extend to noninteger lower index, where I hope they motivate (r choose negativeinteger) :=0. I recall Knuth in Vol 1 relying heavily on summing from -∞ to greatly simplify binomial sum manipulations, blissfully indifferent to the precise points of leftward termination.

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Re: [math-fun] turkeys and gem polishers
by Henry Baker 28 Jun '13

28 Jun '13

Many animals (including fish) have camouflage color schemes that work correctly when looking up (for fish, light) or when looking down (for fish, dark). When they die & stop being in the correct orientation, they become very obvious & say to their predators: 'dinner is served'. At 07:04 AM 6/28/2013, Bill Gosper wrote: >And while I am this far off topic, why do nearly all fish bother to swim upright? This >supposedly requires rocks in their heads--"otoliths", of which my father had a collection. >Wouldn't bird|dinosaur-style "gastroliths" (well anyway, belly ballast) be simpler? Do >fish go belly-up the moment they die?

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[math-fun] turkeys and gem polishers
by Bill Gosper 28 Jun '13

28 Jun '13

I am guilty of repeating the urban legend that botanists were frantically trying to breed giant turkeys after realizing that the seeds of a certain endangered tree couldn't germinate without passing through a dodo's digestive system. http://en.wikipedia.org/wiki/Sideroxylon_grandiflorum : "To aid the seed in germination, botanists now use turkeys<http://en.wikipedia.org/wiki/Turkey_%28bird%29>and gem polishers <http://en.wikipedia.org/wiki/Tumble_polishing> to erode the endocarp to allow germination." No mysterious digestive juices required. --rwg And while I am this far off topic, why do nearly all fish bother to swim upright? This supposedly requires rocks in their heads--"otoliths", of which my father had a collection. Wouldn't bird|dinosaur-style "gastroliths" (well anyway, belly ballast) be simpler? Do fish go belly-up the moment they die? --rwg

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Re: [math-fun] Atrocious Mma 8&9, WolframAlpha bug:
by Bill Gosper 27 Jun '13

27 Jun '13

Jörg> By sheer coincidence I just stumbled upon http://arxiv.org/abs/1105.3689 Can anyone comment on validity of the statements given? Best, jj This paper says "The symmetry identity [3, 4] and the trinomial revision identity [3, 4] are valid for all complex x,y,z" where [3] is G,K,&P's Concrete Math, which the author does not admit to contradicting. But GKP says "So the equation (-1 choose k) = (-1 choose -1-k) is always false! The symmetry identity fails for all other negative integers n, too. But unfortunately it’s all too easy to forget this restriction, since the expression in the upper index is sometimes negative only for obscure (but legal) values of its variables. Everyone who’s manipulated binomial coefficients much has fall into this trap at least three times." This paper is a fraud. --rwg

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Re: [math-fun] Atrocious Mma 8&9, WolframAlpha bug:
by Bill Gosper 27 Jun '13

27 Jun '13

DanA> As I see it, there's only one natural way to extend the binomial symbol n_C_k to almost all reals (and also complexes), and that is via the gamma function 𐅃(z), since for positive integers z we have from 𐅃(z) = (z-1)! that (*) n_C_k = 𐅃(n+1) / (𐅃(k+1) 𐅃(n-k+1)) This definition would imply that 0_C_0 is equal to 1, and also that for integer n and k then for either a) n >= 0 AND (k <= -1 OR k >= n+1) we have n_C_k = 0; OR b) n <= -1 AND (k <= -1 AND k >= n+1), --or more simply-- n+1 <= k <= -1 we have n_C_k = 0. (Since, the two simple poles in the denominator force the quotient to equal 0, despite the simple pole in the numerator.) Since the gamma function never takes the value 0 and has poles only at the nonpositive integers, a) and b) should cover all cases where the definition (*) of n_C_k = 0. --Dan Why settle for almost all? Using z! := Gamma[z+1], for the 3rd time, _nC_m := (n choose m) := Limit[Limit[x!/(y!*(x-y)!),y->m],x->n] covers *all* the cases. Picture: Overlay on the symmetrical surface x!/(y!*(x-y)!) a symmetrical grid of arrows pointing (unsymmetrically) leftward at each gridpoint, indicating that if we always approach from the *right*, we always get the *right* answer. JPropp> I can't remember the full story, but here's a summary of part of it (which I learned about from Ira Gessel): Assume that we can define C(m,n) for all integers m and n so that C(m,n)=(-1)^n C(-m+n-1,n) and C(m,n)=C(m,m-n) for all m and n. Then applying these two transformations alternately three times we get C(m,n) = -C(m,n): C(m,n) = (-1)^n C(-m+n-1,n) = (-1)^n C(-m+n-1,-m-1) = (-1)^(m+n+1) C(-n-1,-m-1)= (-1)^(m+n+1) C(-n-1,-n+m) = -C(m,-n+m) = -C(m,n). Jim Propp OK, that just says we can't define C(m,n) for all integers m and n so that C(m,n)=C(m,m-n). Good! (I'm surprised Ira, a CAS guy, would believe C(m,n)=C(m,m-n) for all m and n. Knuth Vol 1 must clarify this.) What does Maple say? Fred> I understand now what Bill is driving at. The conclusion seems to be that the Gamma (or Beta) function definition is essentially useless for defining n_C_m for integers m, n < 0 . rwg> Huh? That limit I keep repeating seems to do it all. Fred> It looks like fans of the radiation symbol option need to come up with some convincing alternative justification for taking (*) as an axiom, to extend to the region m < 0, n < 0 . At any rate when (n,m) = (-1,-1) , there are indeed situations where the value 1 gives a consistent combinatorial result; I'm not sure though whether there may be others where 0 is preferable. Fred Lunnon I'd be interested to see a situation where (-1 Choose -1) := 1 avoids absurdity, barring reckless use of C(m,n)=C(m,m-n)). The radiation symbol is more toxic than radiation. This is (n choose m): 1 1 -5 1 -4 1 -3 6 1 -2 3 1 -1 1 -1 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 This biwedge extension is typical of many number triangles, including Stirling and Eulerian. --rwg

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[math-fun] biwedges [Was: Atrocious Mma 8&9, WolframAlpha bug]
by Bill Gosper 27 Jun '13

27 Jun '13

rwg>This biwedge extension is typical of many number triangles, including Stirling and Eulerian. Had to switch to Macsyma--Mma remains extremely stodgy in these matters. Binomials: (c30) PT(N,FRM) := GENMATRIX(LAMBDA([I,J],SUBST("",0,IF EVENP(I+J) THEN 0 ELSE BINOMIAL(I-1,(I+J+1)/2-FLOOR(N/4)))),N,N,FRM,FRM)$ (c31) PT(4,-3); [ 1 ] [ ] [ 1 ] [ ] [ 1 - 2 ] [ ] [ 1 - 1 ] (d31) [ ] [ 1 ] [ ] [ 1 1 ] [ ] [ 1 2 1 ] [ ] [ 1 3 3 1 ] StirlingS1: (c32) S1T(N,FRM) := GENMATRIX(LAMBDA([I,J],SUBST("",0,IF EVENP(I+J) THEN 0 ELSE STIRLING_S1(I-1,(I+J+1)/2-FLOOR(N/4)))),N,N,FRM,FRM)$ (c33) S1T(5,-7); [ ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 15 1 ] [ ] [ 10 1 ] [ ] [ 25 6 1 ] [ ] (d33) [ 7 3 1 ] [ ] [ 1 1 1 1 ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 1 1 ] [ ] [ 2 3 1 ] [ ] [ 6 11 6 1 ] Mirror image due to argument conventions. StirlingS2: (c34) S2T(N,FRM) := GENMATRIX(LAMBDA([I,J],SUBST("",0,IF EVENP(I+J) THEN 0 ELSE STIRLING_S2(I-1,(I+J+1)/2-FLOOR(N/4)))),N,N,FRM,FRM)$ (c35) S2T(5,-7); [ ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 15 1 ] [ ] [ 10 1 ] [ ] [ 35 6 1 ] [ ] (d35) [ 11 3 1 ] [ ] [ 6 2 1 1 ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 1 1 ] [ ] [ 1 3 1 ] [ ] [ 1 7 6 1 ] Eulerian: (c36) ET(N,FRM) := GENMATRIX(LAMBDA([I,J],SUBST("",0,IF EVENP(I+J) THEN 0 ELSE EULERIAN(I-1,(I+J+1)/2-FLOOR(N/4)))),N,N,FRM,FRM)$ (c37) ET(5,-5); [ ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 17 ] [ 1 -- ] [ 8 ] [ ] [ 5 ] [ 1 - ] [ 4 ] (d37) [ ] [ 1 1 ] [ 1 - - ] [ 2 3 ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 1 1 ] [ ] [ 1 4 1 ] [ ] [ 1 11 11 1 ] See p8 of http://go.helms-net.de/math/binomial_new/01_12_Eulermatrix.pdf --rwg

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[math-fun] Generalization of modified twin primes conjecture
by Keith F. Lynch 27 Jun '13

27 Jun '13

Last month, Zhang proved that there exists a number N such that there are infinitely many primes that differ from another prime by not more than N. (He showed that N is at most 70 million. That upper bound has since been reduced to 12,012. See http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_pr… ) I've wondered if the same is true for any monotonically increasing sequence of positive integers (i.e. no duplicate terms) for which the sum of the reciprocals diverges. Can anyone think of a counterexample?

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Re: [math-fun] Always off topic; always on target
by Henry Baker 26 Jun '13

26 Jun '13

Is '404' a sly pun on 'No Such Agency' ? i.e., 'No Such pdf' ? (At first I thought that the url had been corrupted.) At 03:46 PM 6/26/2013, Whitfield Diffie wrote: >NSA has posted a paper on lightweight block ciphers at: > > http://eprint.iacr.org/2013/404.pdf > > > Whit

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