October 2013 - mathfun@mailman.xmission.com

[math-fun] Publishing new proofs of old results
by Henry Baker 30 Oct '13

30 Oct '13

Rich made the following comment re Science. >From: rcs(a)xmission.com >d) Confirming studies are not normally publishable. Q: How does Mathematics treat papers which are new proofs of old results? We all know about Euler & Gauss & multiple proofs, but how are modern mathematicians treated in the published literature when they attempt to publish new proofs of old results? Has anyone counted the percentage of published math papers of new proofs/old results? BTW, I strongly disagree with Boltzmann's comment re "leaving elegance to the tailors"; part of the progress of modern math/science is to convert PhD thesis results into undergraduate homework or lab exercises. This is the only way to condense the incredible amount of material that a budding mathematician/scientist must learn before getting into new territory.

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[math-fun] Firing Squad Synchronization Problem
by Tom Karzes 30 Oct '13

30 Oct '13

Hi folks, Some of you may recall the Firing Squad Synchronization Problem. The problem is basically this: You have a linear array of cellular automata. Each automaton uses the same rule set, and transitions to the next state based on its current state and the states of its immediate neighbors. The ones on the ends know they're on the end and can take that fact into account when transitioning. The initial configuration has all of the automata in an "idle" state, and one of the end automata (the "general") is forced into a "start" state. The array them begins transitioning in unison. At some point, the entire array must transition into a "fire" state in simultaneously (having never entered that state prior to that point). Also, an automaton in the "idle" state must remain in the idle state as long as all of its immediate neighbors are also in the idle state. (This prevents the entire array from immediately transitioning to the "fire" state.) There is a fairly intuitive O(3n) solution to the problem, and it is easily shown that you cannot do better than 2n - 2 (since that's the fastest the array width information can propagate back to the starting general). I've implemented a simple javascript engine that shows my solutions along with the best known solutions: http://karzes.best.vwh.net/fsquad.html Just set the parameters under the "new" column, then click "reset" to load them and "start" to begin running. You can specify the number of rows to show (retain), either a number or else "all" for everything. The rule sets show the number of states in parentheses. The "tjk-slow" rule set is my O(3n) solution, but is included because I consider it to be the most intuitive solution (find the mid-point, then solve the two half-sized problems). The remaining solutions are all time-optimal (i.e., exactly 2n - 2 transitions). My "tjk-fast" solution uses 9 states. Balzer has a similar solution that uses 8 states, and Gerken has a similar solution that uses 7 states. The record holder is Mazoyer, with a 6-state solution which is interesting in that it not only uses fewer states, but also takes a fundamentally different approach: While the other solutions all find the mid-point, Mazoyer instead finds the point that is 2/3 of the way into the array. It has been shown that no 4-state solution exists, but it is unknown whether a 5-state solution is possible. Beware that if you specify too great a width (> 100 or so), Firefox becomes extremely slow when resetting the display (which is just an HTML table). There's no reason for it to be so slow, but I expect it's recalculating the displayed table over and over when it doesn't need to. IE doesn't appear to suffer from this performance bug. Tom

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Re: [math-fun] [EXTERNAL] More GR fun
by Henry Baker 30 Oct '13

30 Oct '13

In this day & age of real-time photorealistic graphics engines on every laptop computer, it amazes me that there is still no relativistic animation program available to every high school student, so that _every_ student can relatively (!) quickly develop intuition re _special_ relativity, at the very least. I realize that animating _general_ relativity may require more horsepower than most laptops possess, but I hope I'm wrong about this. I don't know how the U.S. Dept of Education or the Dept of Energy decide to spend their money, but a simple program of this sort would seem to be a slam dunk in terms of "bang for the buck". I had hoped that someone would have come up with a "first person shooter" game in general relativistic space by now, but it doesn't seem to have happened. Perhaps Grand Theft Auto VI will allow for cars having relativistic speeds, so students can learn about relativity relatively (!) painlessly. At 06:23 AM 10/28/2013, Cordwell, William R wrote: >Another interesting special rel. "paradox" is a spaceship of rest-length L, travelling at a significant fraction of c, approaches a berth, or dock, of rest-length L. When the nose of the spaceship touches the far end of the berth, the spaceship stops. Just before that instant, a person standing on the ship sees the berth as shorter than the ship (the ship is sticking out); a person on the dock sees the ship as shorter than the dock (the ship is well inside the dock). What is going on?

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[math-fun] Fwd: Re: string theory and the music of the spheres
by meekerdb 29 Oct '13

29 Oct '13

> www.youtube.com/watch?v=2rjbtsX7twc GREAT! :-) Brent

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Re: [math-fun] Julian's dodecahedron
by Bill Gosper 29 Oct '13

29 Oct '13

="Bill Gosper" <billgosper(a)gmail.com> We could recognize the (countable) family of "M" antiprisms and call this the pentagonal one. Better name: pentiprisms--like antiprisms but with equilateral pentagons instead of equilateral triangles. Alternatively, see below. MLB>Heh, I see the faces as W's. Curious: are there analogs with concave nonagons, or anything like that? I doubt it. See gosper.org/pentiprisms.png for orders 3 thru 8 concave and convex. I think Julian (but maybe NeilB) found the extra (squat) hexagonal one. I find no more. --rwg I think all the Google hits on pentiprism are misspellings of pentaprism. Other naming scheme: We could regard the regular dodecahedron as a polar-truncated pentagonal trapezohedron, and similarly for the other figures in the bottom two rows. And we could call Julian's solid a polar-overtruncated pentagonal trapezohedron, etc. for the top row. Unfortunately, this is how to make, rather than how they look. Few people can visualize, let alone recognize, the result of the truncation planes passing through each other, extending faces instead of shortening them.

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[math-fun] Moore-Schulman base-3/2 conjecture and OEIS
by Keith F. Lynch 29 Oct '13

29 Oct '13

Two days ago I wrote: > If it *has* been proven that there's no one-sided counterexample, > you could still do the above, but then look the the differences > between each pair of sums, regarding one sum as representing the > +1 digits and the other sum as representing the -1 digits. (Any > collisions would be regarded as 0 digits. For instance sum > (2/3)^squares minus sum (2/3)^cubes would have (2/3)^sixth-powers > in both sequences.) This would be a lot harder, as there would be > about 20 billion such differences. I realized today that this was stupid of me. After generating the 230621 sums (sum (2/3)^j for j in each OEIS entry), which I did and reported on yesterday, I don't have to try all 230621*230620/2 differences to look for a difference of plus or minus one. I can sort the sums, then simultaneously move down the list from the top and up the list from the bottom, and that will quickly find all differences whose absolute values are close to one. I've done so. The only interesting thing I found was that the difference between the sums for A087156 and for A073536 equaled one to fourteen decimal digits. The former is simply all positive integers except one. (2/3)^A087156 sums to 4/3, or would if the terms went on forever. The latter is "Breaking indices for A058842 (i.e. n such that A058842(n) is not equal to 3*A058842 (n-1) )." and A058842 is "From Renyi's "beta expansion of 1 in base 3/2": sequence gives a(1), a(2), ... where x(n) = a(n)/2^n, with 0 < a(n) < 2^n, a(1) = 1, a(n) = 3*a(n-1) modulo 2^n." So, as with A077468, this series is explicitly about base 3/2. (2/3)^A073536 appears to sum to 1/3. The reason the difference differs from one by a few parts per quadrillion is no doubt because the terms in A087156 (intended to be all integers greater than one) stop at 77. So there don't appear to be any sequences in OEIS with a zero asymptotic density of non-zero terms for which (2/3)^j sums to one. Of course there's always more I could try. For instance I could treat every odd term or every odd-placed term as negative and see what happens. > It's certain that many of these 20 billion differences would be very > close to 1 just by chance. More stupidity on my part. With a mere twenty billion differences, whose absolute values are pretty evenly distributed from zero to two, there should be nothing that's much closer to one than one part in twenty billion by mere chance. And indeed that's just what I see.

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Re: [math-fun] barking up the right angle
by Bill Gosper 28 Oct '13

28 Oct '13

WFL>Alright, I'll buy it --- what's this pleasant autumn scene got to do with pi/2 ? Or with excited canids, come to that. Or have I been sold a pup? WFL On 10/28/13, Bill Gosper <billgosper(a)gmail.com> wrote: gosper.org/IMG_0146.JPG --rwg (Viewed at full scale,) the bark is cracking off the foreground trunk in neat rectangles. Coincidentally, the background sound for this photo featured four fierce Pomeranians, whose quince tree I just raided. --rwg

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Re: [math-fun] [EXTERNAL] More GR fun
by Henry Baker 28 Oct '13

28 Oct '13

This looks very interesting! Thanks for posting! I'm going to try it out. At 12:00 PM 10/28/2013, Mike Stay wrote: >On Mon, Oct 28, 2013 at 12:34 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> In this day & age of real-time photorealistic graphics engines on every laptop computer, it amazes me that there is still no relativistic animation program available to every high school student, so that _every_ student can relatively (!) quickly develop intuition re _special_ relativity, at the very least. > >Like this? http://gamelab.mit.edu/games/a-slower-speed-of-light/ > >-- >Mike Stay - metaweta(a)gmail.com >http://www.cs.auckland.ac.nz/~mike >http://reperiendi.wordpress.com

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[math-fun] More GR fun
by Henry Baker 28 Oct '13

28 Oct '13

After watching Susskind's lectures, I think I've finally understood one problem that has bothered me since undergraduate days almost 50 years ago. When studying special relativity, we learned about 'time dilation', where things moving faster than us have clocks that _appear_ to be going slower. We also learned about getting on a star cruiser after fathering a child & later coming home younger than your child (the so-called "twin paradox"). The question for me was 'what is the cause of the time dilation?' -- is it the speed itself, or is it the acceleration? It appears that the time dilation is related to the acceleration, so -- in effect -- all time dilation is caused by acceleration, whether it be 'real' acceleration due to rocket thrust, or to gravitational acceleration due to some mass in the neighborhood. I.e., the father/child paradox requires the acceleration to get up to speed, as well as the deceleration necessary to get back to meet with the child.

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[math-fun] barking up the right angle
by Bill Gosper 28 Oct '13

28 Oct '13

gosper.org/IMG_0146.JPG --rwg

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