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[math-fun] Epicycles
by James Propp 09 Sep '16

09 Sep '16

Has anyone gone back to the data compiled by early (pre-Kepler) astronomers and created better epicyclic approximations to planetary orbits using Fourier methods? Jim Propp

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[math-fun] Moore curve drawn with epicycles
by James Propp 09 Sep '16

09 Sep '16

Can Bill Gosper or anyone else explain what's going on with https://twitter.com/algoritmic/status/772699702064254976 ("Moore curve drawn with epicycles")? Jim Propp

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[math-fun] imminent world shortage of Hadamard matrices?
by Veit Elser 08 Sep '16

08 Sep '16

The Hadamard matrix conjecture holds that such matrices exist for all orders that are divisible by 4. After surveying what’s been done on the classification/enumeration of Hadamard matrices (e.g. http://neilsloane.com/hadamard/) I’ve felt that what’s humbling about the conjecture is that we lack the knowledge to prove the existence of even a single Hadamard (at each possible order) when the evidence points to a very rapid growth in their number. Now I’m not so sure. Let x = log_2(N), N = order of Hadamard (a multiple of 4), and y = (1/N^2)log_2(num(N)), where num(N) is the number of Hadamard matrices of order N. Sequence A206711 gives num(N) for N = 1, … , 32. If you plot y vs. x you get a very straight line: y = 0.78785 - 0.09458 x. Taking this literally, there should be a maximum in the number of Hadamard matrices at order N = 196, and beyond that the number plummets, vanishing at around N = 322. The available constructions (beyond this number) would then represent isolated points. -Veit

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Re: [math-fun] Gaa, engine BRAKing!
by Bill Gosper 06 Sep '16

06 Sep '16

On 2016-09-05 07:26, Henry Baker wrote: > You can thank the EPA and their 100,000 mile "no pollution", > "no maintenance" requirement. > > Although aimed at reducing pollution, the EPA requirements > had the effect of tightening up quality all throughout > the engine & air intake & exhaust systems. > > For example, you used to have to replace your muffler > every 15 minutes, and that is no longer required, either. > > Of course, this mandated increase in quality (including > crash tests, etc.) also had the effect of raising the > price of a minimal "automobile" from ~$5k to ~$20k. > > These regulations also had the (perhaps unintended) > effect of *standardizing* engines, drive trains, etc., > and suppressing innovation. Who could afford the cost > of proving compliance? E.g., look what happened to poor > Mazda with their Wankel rotary engine. (I'm not a huge > fan of the Wankel, but it did represent an attempt at > innovation.) > > One can only wonder at why it took the 21st C. & Tesla > before someone finally got serious about electric > vehicles. Perhaps heavy regulation exacts an > enormous toll on innovation? > > --- > > BTW, when was the last time you ever heard of a ball > bearing failing? High quality ball bearings are > now assumed, but prior to & during WWII, high quality > ball bearings were unobtainable. The U.S. put a huge > priority on bombing German ball bearing plants, because > they were so ubiquitous and so important. Don't forget the strategically critical Japanese pachinko industry: https://www.youtube.com/watch?v=CFs4zOC2Yg0 (More seriously: https://www.youtube.com/watch?v=19duYMdiXi0 ) --rwg > > At 08:07 PM 9/4/2016, Richard Howard wrote: >>Of course. >>Had to use special "break-in" oil. >> >>Now engines are machined so tightly when they come off the line that they >>use 0W oil--sort of like the consistency of water. >> >>--R >> >>On Sun, Sep 4, 2016 at 9:51 PM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >>> Who remembers when we had to break engines in? --rwg > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Gaa, engine BRAKing!
by Henry Baker 05 Sep '16

05 Sep '16

You can thank the EPA and their 100,000 mile "no pollution", "no maintenance" requirement. Although aimed at reducing pollution, the EPA requirements had the effect of tightening up quality all throughout the engine & air intake & exhaust systems. For example, you used to have to replace your muffler every 15 minutes, and that is no longer required, either. Of course, this mandated increase in quality (including crash tests, etc.) also had the effect of raising the price of a minimal "automobile" from ~$5k to ~$20k. These regulations also had the (perhaps unintended) effect of *standardizing* engines, drive trains, etc., and suppressing innovation. Who could afford the cost of proving compliance? E.g., look what happened to poor Mazda with their Wankel rotary engine. (I'm not a huge fan of the Wankel, but it did represent an attempt at innovation.) One can only wonder at why it took the 21st C. & Tesla before someone finally got serious about electric vehicles. Perhaps heavy regulation exacts an enormous toll on innovation? --- BTW, when was the last time you ever heard of a ball bearing failing? High quality ball bearings are now assumed, but prior to & during WWII, high quality ball bearings were unobtainable. The U.S. put a huge priority on bombing German ball bearing plants, because they were so ubiquitous and so important. At 08:07 PM 9/4/2016, Richard Howard wrote: >Of course. >Had to use special "break-in" oil. > >Now engines are machined so tightly when they come off the line that they >use 0W oil--sort of like the consistency of water. > >--R > >On Sun, Sep 4, 2016 at 9:51 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> Who remembers when we had to break engines in? --rwg

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[math-fun] Comparing sizes of sets of natural numbers
by James Propp 05 Sep '16

05 Sep '16

I'd welcome math-funsters' comments on http://mathoverflow.net/questions/248994/comparing-sizes-of-sets-of-natural… . Teaser: The size of {0,2,4,6,...} exceeds the size of {1,3,5,7,...} by exactly 1/2. "From a certain point of view." Jim Propp

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[math-fun] Gaa, engine BRAKing!
by Bill Gosper 04 Sep '16

04 Sep '16

Who remembers when we had to break engines in? --rwg

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Re: [math-fun] "Finite" meaning "nonzero"
by Bill Gosper 04 Sep '16

04 Sep '16

Your car's gearbox has the finite gears and neutral. For accelerating, neutral = 0. For engine breaking and roll starting, neutral = ∞. On 2016-09-04 09:11, Tomas Rokicki wrote: > It's an interesting use of the word "finite" in the sense that the > opposite of "finite" probably needs to be "infinite", but here the > opposite appears to be "zero". Further, if we are looking at > a case like this, a "small but non-zero" asymptote implies a > satisfying population for "finite" would have to be, in count, > *infinite*. > > I believe we are in the world of Humpty Dumpty here. > > On Sun, Sep 4, 2016 at 5:54 AM, Veit Elser <ve10(a)cornell.edu> wrote: > >> I raised this point in a Physics Today letter to the editor (1980s) >> and was met with unanimous defense of this usage of “finite”. >> >> -Veit Physicists take logarithms a lot. --rwg >> >> > On Sep 3, 2016, at 9:45 PM, James Propp <jamespropp(a)gmail.com> wrote: >> > >> > Are you using "finite" to mean "nonzero", Bill? >> > >> > I have always disparaged this usage, but maybe I should reconsider my >> > prejudice. What do you all think about this? >> > >> > Jim Propp >> > >> > On Saturday, September 3, 2016, Bill Gosper <billgosper(a)gmail.com >> > <javascript:_e(%7B%7D,'cvml','billgosper(a)gmail.com');>> wrote: >> > >> >> Despite my distaste for decimal digits, I noticed >> >> ... >> >> Is the asymptotic density finite? --rwg

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Re: [math-fun] peculiar sequence
by Keith F. Lynch 04 Sep '16

04 Sep '16

Bill Gosper <billgosper(a)gmail.com> wrote: > Despite my distaste for decimal digits, I noticed (while showing > Mathematica to new user Miles) that 52! ends in twelve 0s both in > decimal and hex. Equality happens ... with gradually declining > frequency: Tomas Rokicki <rokicki(a)gmail.com> wrote: > Maybe it's finite, maybe it's not. Asymptotically of course the > count of zeros in both decimal and hex in n! are n/4; surprising > that. > I could only get to 10^11 quickly: .... Overuse of Mathematica rots the brain. If you want an exact count, the number of 0-digits at the end of n! is sum(floor(n/5^j)) for decimal, and floor(sum(floor(n/2^j))/4) for hexadecimal. In both cases j ranges over all positive integers. Using those formulas, it should be easy to get way past 10^11. "How many zero-digits does one thousand factorial end with" is a fun problem to give to people to solve in their head.

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[math-fun] "Finite" meaning "nonzero"
by James Propp 04 Sep '16

04 Sep '16

Are you using "finite" to mean "nonzero", Bill? I have always disparaged this usage, but maybe I should reconsider my prejudice. What do you all think about this? Jim Propp On Saturday, September 3, 2016, Bill Gosper <billgosper(a)gmail.com <javascript:_e(%7B%7D,'cvml','billgosper(a)gmail.com');>> wrote: > Despite my distaste for decimal digits, I noticed > ... > Is the asymptotic density finite? --rwg > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

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