June 2016 - mathfun@mailman.xmission.com

[math-fun] esoterica -- meeting on Lambert function
by Bill Gosper 06 Jun '16

06 Jun '16

Graphically, how it would look entails the usual hard problem of plotting a complex valued function of a complex argument. But by "how it would look", I meant how you could invoke it and interpret it based on a Riemann surface description vs a branch number. I'm not sure I've ever seen this, even for an elementary function. If you have a workable idea, Wolfram might even *give* you a Mathematica. Meanwhile, if you need something plotted, email it to me. --rwg On 2016-06-03 16:51, Dan Asimov wrote: > I would love to tell y'all how it would look in Mathematica > if only I had Mathematica. > > —Dan > P.S. I have remote access to a text-based Mathematica, but not > to graphics. > >> On Jun 3, 2016, at 4:20 PM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >> On 2016-06-03 07:03, Veit Elser wrote: >>> Two fun W (Lambert function) facts: >>> >>> 1) The Taylor series of W records the number of spanning trees of the >>> complete graphs. >>> >>> 2) If 1/2+i y_n is the nth zero of the Riemann zeta function on the >>> critical line, then asymptotically (large n) >>> >>> y_n ~ 2 pi (n-11/8)/ W((n-11/8)/e) >>> >>> -Veit >> >> Less amazing, but useful: They appear in an inelegant but efficient series >> solution to Kepler's equation: >> http://www.tweedledum.com/rwg/pizza.html >> It's a bit disappointing that W doesn't solve Kepler's more elegantly. But >> suppose >> we posit that E(ε,M) := Kepler(ε,M) solves M = E - ε sin E . Does this >> dyadic function Kepler >> subsume Lambert W as a special case? If so, maybe we should embrace it. >> Failing that, I think "Dilbert Lambda"(y), which solves y = Λ exp Λ², is >> usually nicer than W. >> >>> >>>> On Jun 2, 2016, at 10:28 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >>>> >>>> Thanks. I vastly prefer the Lambert W function when it is not defined >> with a branch cut but is allowed to extend to it full Lambertness on a >> Riemann surface. >>>> >>>> Whether considering the Lambert function or the more simply defined >> function of which it is the inverse function: >>>> >>>> f(z) = z exp(z), >>>> >>>> the aforesaid Riemann surface is just the subset of C^2 defined as >>>> >>>> {(z,w) | z = w exp(w)}. >>>> >>>> This is, in my opinion, the appropriate object of study. >>>> >>>> —Dan >> All we need is for you to tell us how it would look in Mathematica. --rwg

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Re: [math-fun] Raghavendra's algorithm for solving linear systems over finite fields
by Henry Baker 06 Jun '16

06 Jun '16

I can see applications in distributed/parallel implementations. You can do most of the checking & probably a lot of the randomization in parallel. At 09:02 AM 8/15/2012, Warren Smith wrote: >>Preliminary draft: >> http://www.eecs.berkeley.edu/~prasad/linsystems.pdf > >--I don't understand why anybody would want to use this algorithm >for any purpose. It's slower than everything else I can think of and also >randomized i.e. can fail. > >However... perhaps it might be useful if changed to be about finite >RINGS not fields. In rings, you cannot divide, hence gaussian elimination is >not usable. Hence this algorithm then has less competition. >(Even then I'm dubious, but then it'd have at least a hope.)

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Re: [math-fun] Digit product proof
by Keith F. Lynch 05 Jun '16

05 Jun '16

Andy Latto <andy.latto(a)pobox.com> wrote: > The missing step is showing that the greedy-algorithm candidate > doesn't have more digits than the non-greedy one. As I posted yesterday, there are failures where it has the same number of digits. For instance 338 rather than 266 in base 9, and 35C rather than 29A in bases 13, 14, and 15. But there are failures where there are too many digits, also. For instance 33333333888 rather than 266666666, 336666668, or 346666666 in base 9.

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[math-fun] Digit product proof
by David Wilson 05 Jun '16

05 Jun '16

To find smallest base-b number with digit product p. The greedy method would be to repeatedly divide p by the largest base-b digit divisor until 1 is reached, then concatenate the divisor digits in reverse order. (If 1 is not reached, the number is not a product of base-b digits). For example, to find the smallest base 10 number with digit product p = 3628800 3628800/9 = 403200 403200/9 = 44800 44800/8 = 5600 5600/8 = 700 700/7 = 100 100/5 = 20 20/5 = 4 4/4 = 1 So, ostensibly, the smallest base-10 number with digit product 3628800 is 45578899(10). Is the greedy algorithm correct for this problem? Could it be possible that dividing by a smaller digit at some step leads to an smaller result? For example, in base 20, for some large p, might it be better to divide first by digit 16 rather than digit 18?

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Re: [math-fun] Digit product proof
by Keith F. Lynch 04 Jun '16

04 Jun '16

"Keith F. Lynch" <kfl(a)KeithLynch.net> wrote: > I just brute-force tested it for bases 3 through 20 for digit > products 1 through 5000. The first two bases for which any > exceptions were found were 9 and 13. The first exception found was > base 9, digit product 72, 266(9) by brute force, 338(9) by greedy. I've pushed a little further. I've confirmed that David Wilson's greedy algorithm works for all numbers less than B^9 for bases 2 through 8, and for bases 10, 11, 12, and 16. For instance I've confirmed that it works in base 10 for all numbers less than a billion. (By "numbers" I mean the numbers whose digits are to be multiplied, not necessarily the products thereof). (Of course proving that it always works in bases 2 and 3 is easy.) It fails quite rapidly for all other bases through at least base 85. For instance for bases 13, 14, and 15, digit product 180, the greedy algorithm finds 35C, but the correct answer is 29A. (Yes, it's the same pattern of digits in all three bases, but not, of course, the same number.) For each failed base, there are lots of failures, not just the ones I listed. Why should it always either fail rapidly or (apparently) not at all? What's special about 2,3,4,5,6,7,8,10,11,12,16? (Not in OEIS.)

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Re: [math-fun] Digit product proof
by Keith F. Lynch 04 Jun '16

04 Jun '16

"David Wilson" <davidwwilson(a)comcast.net> wrote: > So, ostensibly, the smallest base-10 number with digit product > 3628800 is 45578899(10). Confirmed for that one instance. > Is the greedy algorithm correct for this problem? No. I just brute-force tested it for bases 3 through 20 for digit products 1 through 5000. The first two bases for which any exceptions were found were 9 and 13. The first exception found was base 9, digit product 72, 266(9) by brute force, 338(9) by greedy. Here's my C code if anyone wants to play with it. The numbers output are all base 10, e.g. 222 and 278 rather than 266 and 338. Column 1 is 1 for matches, 0 for mismatches. #include <stdio.h> int main() { int b,d,g,i,i2,j,p,t,t2; for (b=3;b<21;b++) { for (t=1;t<5001;t++) { g = 0; t2 = t; p = 1; for (d=b-1;d>1;d--) { while (t2%d == 0) { t2 /= d; g += d*p; p *= b; } } if (t2 == 1) { j = 1; for (i=1;j!=t;i++) { i2 = i; j = 1; while (i2 && j) { j *= i2%b; i2 /= b; } } printf("%d b=%d t=%d %d %d\n",i-1==g,b,t,i-1,g); } } } return(0); }

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[math-fun] squares beginning with n
by Neil Sloane 03 Jun '16

03 Jun '16

Given n, A018851 and A018796 give the smallest square that begins with n Question: does such a square always exist, and if so how big can the smallest example be?

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[math-fun] sum of binary digits
by Bill Gosper 03 Jun '16

03 Jun '16

I believe the TX-0, originally lacking an add instruction, did xor foo cry foo /carry foo instead --rwg "The TX-0 (“Transistor eXperimental - 0”) is the first general-purpose programmable computer built with transistors. For easy replacement, designers placed each transistor circuit inside a "bottle," similar to a vacuum tube. Constructed at MIT´s Lincoln Laboratory, the TX-0 moved to the MIT Research Laboratory of Electronics, where it hosted some early imaginative tests of programming, including writing a Western movie shown on television, 3-D tic-tac-toe, and a maze in which a mouse found martinis and became increasingly inebriated." I can't easily find a color closeup, but the bottles were plastic and color-coded with a disk divided into quadrants. Conceptual holdovers from resistors as well as vacuum tubes (not yet derided as "fire-bottles"). I think the bottles were clearly visible in a circa 1960 Fortune Magazine article on MIT, featuring a large image of Claude Shannon in front of a TX-0 bay festooned with punched tape no doubt by the dunce photographer. You might wonder why transistors needed easy replacement. The melting point of germanium was way lower than the plastic! Silicon came later. (Thank God it did; I'd hate to live in Germanium Valley.) Hopefully, Ed Fredkin still has (or at least has digitized) his precious poster (from a GE cafeteria?) of an ugly, cowering little transistor about to be crushed by an Art Deco shoe. Caption: "Stamp them out!" Back then Fredkin had a huge TV PR fail when he predicted a nanotech robot barber that lived in your hair and maintained your expensive haircut by memorizing each hair's length. On 2016-06-03 09:16, Joerg Arndt wrote: > * rcs(a)xmission.com <rcs(a)xmission.com> [Jun 03. 2016 08:44]: >> [...] >> Probably we can do something with (a&b) + (a|b) = a+b, and >> a+b = (a^b) + 2(a&b). Maybe v(a+b) <= v(a^b) + v(a&b). > >>From a+b = (a^b) + 2(a&b) we get > (a+b)/2 = (a^b)/2 + (a&b) so > > static inline ulong average(ulong x, ulong y) > // Return floor( (x+y)/2 ) > // Result is correct even if (x+y) wouldn't fit into a ulong > // Use: x+y == ((x&y)<<1) + (x^y) > // that is: sum == carries + sum_without_carries > { > return (x & y) + ((x ^ y) >> 1); > // return y + ((x-y)>>1); // works if x>=y > } > > Similarly > > ulong ceil_average(ulong x, ulong y) > // Return ceil( (x+y)/2 ) > // Result is correct even if (x+y) wouldn't fit into a ulong > // Use: x+y == ((x|y)<<1) - (x^y) > // ceil_average(x,y) == average(x,y) + ((x^y)&1)) > { > return (x | y) - ((x ^ y) >> 1); > } > > Best regards, jj > > >> >> Rich >> >> [...]

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[math-fun] The meteoritic origin of Tutankhamuns iron dagger blade
by Bill Gosper 03 Jun '16

03 Jun '16

On 2016-06-03 09:42, Joerg Arndt wrote: > I learned about the use of iron from meteorites by the Eskimos > in the American Museum of Natural History (New York): > https://en.wikipedia.org/wiki/Cape_York_meteorite > Foto of the display in the Museum here: > https://de.wikipedia.org/wiki/Cape_York_%28Meteorit%29 > > Much to my dismay, I found that there appears to be a lack > of funding for the Museum (prior visit was 14 years ealier). > Still, there are many truly awesome exhibits. > > Best regards, jj > > P.S.: terrestial elementary iron also exists. > It is created essentially when magma says "hi" to coal. > I actually own a small sample, it was expensive > and looks positively boring. > > P.P.S.: > The Sterling Hill Mining Museum (NJ, _highly_ recommended (*)) > has 2 big meteorites (one is 50 kg or so) but didn't know it. > I told them (they were surprised) and asked about the origin, > apparently that information has been lost. > (*) if fluorescence means anything to you: world class! > > * Henry Baker <hbaker1(a)pipeline.com> [Jun 03. 2016 08:45]: >> FYI -- I thought this was kind of cool. So people started using iron prior to figuring out how to make it themselves. >> [...] I've seen the claim that the (justifiably) xenophobic Sentinelese are Stone Age but for the iron they harvest from a shipwreck. And perhaps from the belongings of outsiders they kill.

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The meteoritic origin of Tutankhamuns iron dagger blade
by Henry Baker 03 Jun '16

03 Jun '16

FYI -- I thought this was kind of cool. So people started using iron prior to figuring out how to make it themselves. I personally think that the production of iron was closely related to the production of certain high temperature pottery/ceramics. http://onlinelibrary.wiley.com/store/10.1111/maps.12664/asset/maps12664.pdf… Scholars have long discussed the introduction and spread of iron metallurgy in different civilizations. The sporadic use of iron has been reported in the Eastern Mediterranean area from the late Neolithic period to the Bronze Age. Despite the rare existence of smelted iron, it is generally assumed that early iron objects were produced from meteoritic iron. Nevertheless, the methods of working the metal, its use, and diffusion are contentious issues compromised by lack of detailed analysis. Since its discovery in 1925, the meteoritic origin of the iron dagger blade from the sarcophagus of the ancient Egyptian King Tutankhamun (14th C. BCE) has been the subject of debate and previous analyses yielded controversial results. We show that the composition of the blade (Fe plus 10.8 wt% Ni and 0.58 wt% Co), accurately determined through portable x-ray fluorescence spectrometry, strongly supports its meteoritic origin. In agreement with recent results of metallographic analysis of ancient iron artifacts from Gerzeh, our study confirms that ancient Egyptians attributed great value to meteoritic iron for the production of precious objects. Moreover, the high manufacturing quality of Tutankhamuns dagger blade, in comparison with other simple-shaped meteoritic iron artifacts, suggests a significant mastery of ironworking in Tutankhamuns time. http://www.usatoday.com/story/news/nation-now/2016/06/02/king-tut-ancient-e…

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