April 2015 - mathfun@mailman.xmission.com

Re: [math-fun] An end to leap seconds (was Re: SETI breakthrough: ...)
by Keith F. Lynch 08 Apr '15

08 Apr '15

Dan Asimov <asimov(a)msri.org> wrote: > 1. Suppose Earth is a perfect sphere and that each time zone is a > perfect spherical lune of angle 2?/24 that takes 1 hour to traverse > an hour of time. I'm assuming the question mark was meant to be pi. Non-ASCII characters are turned into question marks on this list, at least in its digest form. This is presumably because there are multiple incompatible character sets which senders might be using, and the digest header can specify at most one of them. > Set a stopwatch to begin when it is first mm/dd/yyyy anywhere on > Earth, and stop the watch when it is no longer mm/dd/yyyy (where > mm/dd/yyyy is an arbitrary day of 24 hours' length). > QUESTION: How long did the stopwatch run for? (Mental math only.) Twenty-five hours. The default International Date Line is exactly opposite the prime meridian, hence is in the middle of a time zone, not on the border between two of them. So some clocks are 12 hours ahead of Greenwich when others are 12 hours behind. > 2. Same question, but with time zones as they actually are. > (Any references and writing implements allowed.) Based on the time zone map in the current World Almanac, 27 hours.

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Re: [math-fun] Unusual ways to count thru all w-bit numbers
by Leo Broukhis 08 Apr '15

08 Apr '15

Warren D Smith wrote: > > To DW: > Thank you for proving C++ sucks beyond belief and it NOT, contrary to > propaganda, a superset of plain C. > And, as a followup question, can C++ tell you whether there was an overflow? Don't believe everything you read about C++ in this mailing list. The C++ committee is not made out of idiots. Unlike signed integers for which overflow is undefined behavior, unsigned integers always behave modulo 2^w. Leo

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[math-fun] The length of the day ...
by Guy Haworth 08 Apr '15

08 Apr '15

http://www.scientificamerican.com/article/bad-weather-makes-day-long/ reminds us that the earth is largely 'fluid' ... ... moving atmosphere, oceans, core and often, solid rock, some 'incoming'. A big earthquake can change the length of the day by a few thousandths of a second. Here at UoReading Meteorology, they even bet on the length of the next day, given the forecast weather. Guy

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[math-fun] Unusual ways to count thru all w-bit numbers
by Warren D Smith 08 Apr '15

08 Apr '15

Usual way (w=machine word length in bits, all operations modulo 2^w): x=0; do{ x+=1; }until(x==0); Some other ways: x=0; do{ x+=C; }until(x==0); /*C is any odd residue modulo 2^w*/ x=0; do{ x*=A; x+=B; }until(x==0); /*B is any odd residue modulo 2^w, and A=1(mod 4)*/ The above ways all feature "unidirectional information flow" from the right toward the left end of x, regarded as binary. If w=16 then this works: x=0; do{ x = 59276 - (x XOR (x>>1)); }until(x==0); For most w, it seems like there exist A with 0<A<w and B with 0<B<2^w such that x=0; do{ x = B - (x XOR (x>>A)); }until(x==0); works, but suitable values of A and B are not necessarily easy to find. [x>>A means x shifted rightward A bits. Also can be written floor(x/(2^A)).] Note this method has information flowing in *both* directions. If w=16 this also works: x=0; do{ x*=3; x = 7080 + (x XOR (x>>1)); }until(x==0); and so does this: x=0; do{ x = 39027 + (x XOR (x>>1)); }until(x==0); and so does this: x=0; do{ x*=5; x = 60985 + (x XOR (x>>1)); }until(x==0); and so does this: x=0; do{ x*=5; x = 31241 + (x XOR (x>>3)); }until(x==0); and so does this: x=0; do{ x*=11415; x = 39110 + (x XOR (x>>8)); }until(x==0); etc, etc. Almost certainly there are plenty of 32-bit counters of the above ilk, but I have not found any. If w=64, then this counts thru all 2^64-1 *nonzero* w-bit numbers: x=1; do{ x = x XOR (x>>9); x = x XOR (x<<7); }until(x==1); But there are no 32- and 16-bit counters of this type (for nonzero words). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Unusual ways to count thru all w-bit numbers
by Warren D Smith 08 Apr '15

08 Apr '15

To FWL: "Unidirectional" info flow implies that the low-signif bits of x, have tiny periods, namely the (<=K)th bits have period <=2^(K-1). This behavior is just like the boring repeated-increment counter. If you want to avoid that boredom, you need bidirectionality. Incidentally, another counter for word length W=16 is x=0; do{ x = (x+1)*75 mod (2^16+1) - 1; }until(x=0); The sorts of "bidirectional bit flow" counters I described before are quite fast, but unfortunately rather expensive to find. (They are fast once found.) In fact the only ways I know to find them take about 4^W operations (heuristically). To DW: Thank you for proving C++ sucks beyond belief and it NOT, contrary to propaganda, a superset of plain C. And, as a followup question, can C++ tell you whether there was an overflow?

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[math-fun] Balanced mass transport in Z
by James Propp 07 Apr '15

07 Apr '15

I call to math-funsters' attention my recent MathOverflow post http://mathoverflow.net/questions/202165/transitivity-of-balanced-mass-tran… which has a math-fun-ish quality. One thing that I know, but that I don't prove in that post (though I do mention it in one of the two affiliated posts to MathOverflow) is that if mu is the measure that gives unit mass to every integer greater than or equal to 1, and nu is the measure that gives unit mass to every integer greater than or equal to 0, then nu cannot be obtained from mu by a single balanced-transport move, or by a finite sequence of such moves, or even by an infinite sequence of such moves convergent in the total variation metric. I pose this as a puzzle for math-fun, not only because it has a cute answer, but because I'm hoping it has more than one cute answer, and that one of you will find a different answer to this puzzle than the one that I know! (And it would also be nice if one of you proved or disproved transitivity, but that's a separate issue.) Jim Propp

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Re: [math-fun] Project Durin Succeeds! (KF Lynch)
by Warren D Smith 06 Apr '15

06 Apr '15

Thinking a bit more, I conjecture that a satellite in Earth orbit coplanar with the moon but in 3:2 resonance with it (i.e. 3 periods per 2 moonperiods) would be a stable orbit (i.e. perturbations to the orbit would tend to be damped away). If so, that'd be Durin-suitable. Also, KF Lynch was complaining about my argument that the moon has a 2cm layer of dust on top, "therefore" if willing to sacrifice a few outer cm of your satellite, it has a good chance of lasting for Gyears. His complaint was, that dust generated by a blast on the moon, probably will land elsewhere on the moon, but dust blasted off your satellite is gone. Therefore, the moon has more protection. This is true. However, it seems to me nevertheless, that the moon dust thickness implies an upper bound on how much meteoritic dust came at you ever. Which still leads me to about the same conclusion, up to a factor of 2 or so? Also, I believe that when they brought back rocks from the moon they almost all were very old, which seems to prove that plenty of rocks on or very near the surface, survived. RW Gosper suggested the aliens could leave a message in junk DNA of some lifeform. That had also occurred to me. You'd need error correction techniques of unusual kinds to protect the message. (Block deletions and insertions, block reversals, in addition to the usual model of single-letter changes.) But... first of all, eukaryotic life containing large amount of junk DNA may not be very old. (Prokaryotes lack junk.) If it is only 1 Gyear old, the aliens could not do it before 1 Gyr ago. Second, you'd have to know more than I to be confident your message would survive. A lot of closely related seeming lifeforms have hugely different amounts (factor 10) of junk. I believe this is because there are mechanisms for quite rapidly generating huge amounts junk, and also mechanisms for quite rapidly removing large amounts junk. The latter could place messages in severe danger. Also, species go extinct. I think burying a time capsule on moon ala Clarke ought to be safer and longer lasting than a satellite, but then it's much harder to find. Can you invent a trick that makes it easy to find it?

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[math-fun] Project Durin Succeeds! (KF Lynch)
by Bill Gosper 06 Apr '15

06 Apr '15

WDS> (Also, are aliens really going to work that hard to leave a Gyr message for us? Why should they want to?) If "Durin" aliens were going to make meter-size titanium balls and land them on Earth, it would seem easier for them to do other stuff instead that'd work better. Stromatolite junk DNA? --rwg

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[math-fun] Michelangelo drew legendary circles.
by Bill Gosper 06 Apr '15

06 Apr '15

Michelangelo^-1: gosper.org/p4109340.png . This from subtracting {0,1/121} from a point near the periphery of that mod 6 boxed Julian star from a couple of weeks ago. --rwg

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Re: [math-fun] Project Durin Succeeds! (KF Lynch)
by Warren D Smith 06 Apr '15

06 Apr '15

>> I kind of liked your idea for searching for ET by "looking down." >> The particular details re tuning forks were stupid, but... > Why? Is there something I'm overlooking? --well, storing over 50 bits in a tuning fork frequency is absurd, for one thing... >> Perhaps a satellite in Earth orbit would be even longer lasting, and >> anyhow very easy to spot. >My understanding is that no Earth orbits last more than about ten >million years, due to air drag in low orbits and perturbations from >the Moon in high orbits. --fascinating. Might even be true, but I'd like more evidence please. >> As far as I understand they claim to be tracking everything in orbit >> larger than X, where X is what? The size of a golf ball? And >> nothing was there? >Right. And that was strong evidence that there aren't any long-term >orbits, since otherwise natural debris would have accumulated there, >as it has around Jupiter and Saturn. --I am unconvinced. It is not mathematically possible for any lone body to be "captured" into a high earth orbit, so any would have to have been there since formation. But if so, then it would have been eaten by the Moon, which has gradually spiraled outward from a formerly much lower orbit (probably initially only a few 1000 km up) due to tidal effects. So I would conjecture that the reason the Earth had no other satellites in the year 1940 was: if there were any initially, the moon ate them all. But if a little green man put an artificial satellite there 1 Gyr ago, it perhaps would still be there since it then would not have been eaten in the fashion I described. But if KFL is correct that no long lasting satellite orbits exist even today, then my idea is dead and would have to be heavily revised. Also if KFL is correct, then I ask him: where are the tiny satellites of Venus? >The Voyager records are expected to >remain playable for at least 10 billion years, which is a respectable >amount of time even by my standards. --but you were claiming my 2 cm thick dust estimate was way too small, which seems like it would contradict this? >Anyhow, below the few centimeters of dust isn't a pristine surface, >but smashed up debris of all sizes, getting gradually larger as you >dig deeper. The Moon is impacted by meteors of all sizes, and is >saturation-cratered. --the moon may have formed for a few km of rocks of all sizes in top, already there at formation, not due to meteors. And then, after that short era of formation and initial blasting ended, then the rest of time (several Gyr) only sufficed to cause 2cm of dust to form on top of all that, in most places. If so, indicating decent chance of multi-Gyr survival in space (if willing to sacrifice a few cm thick outer layer) might be feasible. Or, maybe I'm wrong. How do we know which? >> So... I doubt ET left any messages for us. >Again I ask if anyone has done a search like Project Durin. --Not that I know of, but I suspect I know a lot less than you. It's an interesting idea, but clearly neither one of us has thought it through enough. (Also, are aliens really going to work that hard to leave a Gyr message for us? Why should they want to?) If "Durin" aliens were going to make meter-size titanium balls and land them on Earth, it would seem easier for them to do other stuff instead that'd work better.

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