April 2012 - mathfun@mailman.xmission.com

Re: [math-fun] xkcd
by Dan Asimov 26 Apr '12

26 Apr '12

My favorite unit is the millihelen, defined as the quantity of feminine beauty required to launch 1 ship. --Dan ________________________________________________________________________________________ It goes without saying that .

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[math-fun] Ivy Bridge is here
by Eugene Salamin 24 Apr '12

24 Apr '12

The Intel Ivy Bridge CPUs are released. See, for example, http://www.phoronix.com/scan.php?page=article&item=intel_corei7_3770k&num=1 These processors implement hardware random number generation. I don't understand why it took so long to get here, but better late than never. Goodbye pseudo RNG. -- Gene

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[math-fun] superstring theory solves math problems? Moonshine
by Warren Smith 24 Apr '12

24 Apr '12

NJA Sloane: There are actually three sequences in the OEIS that refer to the numbers in the article that Warren mentions: A060041 and A076912, which coincide for 9 terms but probably disagree at the 10th term, and A199878, which is a published but incorrect version. Neil On Tue, Apr 17, 2012 at 3:38 PM, Warren Smith <warren.wds at gmail.com> wrote: > > http://www.ams.org/journals/bull/2000-37-04/S0273-0979-00-00875-2/S0273-097… > > describes a remarkable conjectured formula (arising from generating > functions > ad series reversions) for the number of > rational curves of degree d on a generic "quintic threefold." > The formula gives the right answers for the first 9 cases. > However, it is not even clear that the numbers it outputs, are > always integers, although hundreds have been computed and always came > out integer. --My point was that you don't need to know anything about superstring theory, and you don't need to know anything about algebraic geometry. All you need to know is this paper gives some generating function formulas, and you can try to prove or disprove the coefficients in that generating function are always integer. 5, 2875, 609250, 317206375, 242467530000, 229305888887625, 248249742118022000, ... If you do, you've accomplished something important. How hard can it be to prove or disprove they always are integer????!!?! If Sloane is correct that A060041 and A076912 disagree at the next (uncalculated) term, I gather from the rumor mill that that too would seem very important, it would seem to be a refutation of a major article of faith in superstring theory; but you won't be able to settle that unless you know a lot more than I do. The comparison was made between this and the "monstrous moonshine" conjectures. In the MM conjecture JH Conway (and also some other people independently) noticed some numbers arising in series expansions of elliptic functions... that the same numbers were happening as the number of dimensions needed in matrix representations of the "monster" largest sporadic simple group. This was "clearly" a ridiculously lucky coincidence, since there was no was no way in hell some simple little elliptic functions could possibly know anything about the Monster. However, upon trying a slightly harder, Conway & people found more such numbers, amounting to about 24 digits worth of lucky agreements! http://en.wikipedia.org/wiki/Monstrous_moonshine Eventually Richard Borcherds, a former student of Conway's, was able more or less to understand why monstrous moonshine was happening and got awarded the Fields medal. I personally never understood it. Explanation also has some vague connections to string theory. In the present case, "obviously" superstring theory has nothing whatever to do with counting rational curves on generic rational algebraic surfaces. But... tremendous numerical coincidences, and some vague understanding via physical intuition and so forth. At present, as far as I understand, there is some partial progress but still this all is far from fully understood. Tito Piezas has got his own interesting Moonshine writeup and observations, which by the way seem to me on shallow examination to go beyond what Conway noticed and what Borcherds explained (and they also are a lot easier to read than certain other Moonshine papers...) see http://sites.google.com/site/tpiezas/0022 http://sites.google.com/site/tpiezas/0023 http://sites.google.com/site/tpiezas/0026 http://sites.google.com/site/tpiezas/0025 http://sites.google.com/site/tpiezas/0029 http://sites.google.com/site/tpiezas/0027 http://www.oocities.org/titus_piezas/Pi_formulas1.pdf [the last observed "coincidental" connection between the "baby monster" and Weber and Ramanujan functions] and I guarantee your mind will boggle in amazement if you look at this stuff. -- Warren D. Smith http://RangeVoting.org

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Re: [math-fun] Figure 33
by Bill Gosper 23 Apr '12

23 Apr '12

ACW> I would like to see a version of this plot where period is mapped to grayscale, so that I could see which areas had larger periods. Corey's new plotter uses lightness ~ 1/period, e.g., this cave<http://gosper.org/x=1,y=2,13o20%E2%89%A4%CE%B4%E2%89%A461o60,119o60%E2%89%A…>, which, although it starts out with a simple-looking recursion, must grow arbitrary dark by Julian's theorem. But usually these δ-ε plots are too confusing, even with color. E.g., the ("randomly" colored) http://nbickford.files.wordpress.com/2011/03/neighborhood.png . Note the isolated point (period 634) at (13/6,10/7). I believe the 18.7 trillion region is a very short line segment. --rwg I believe that Corey and Julian have shown that the function Period(x0,y0,δ,ε) partitions R^4 into (countably many?) "congruence regions" (producing Minsky orbits pointwise equivalent under rigid translation), where each region is the Cartesian product of two "n-gons" (3 ≤ n ≤ 6) with hyperbolically curved sides. If this seems simpler than the Mandelbrot set, look at some of the x-y "rug-plots", each of which is unbounded, and consider there's a different one for each δ-ε region.

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[math-fun] Beating a traffic ticket with math & physics
by Henry Baker 23 Apr '12

23 Apr '12

FYI -- http://arxiv.org/abs/1204.0162 Including Youtube animated videos: http://www.wired.com/wiredscience/2012/04/an-analysis-of-the-stopping-car-a…

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[math-fun] superstring theory solves math problems?
by Warren Smith 22 Apr '12

22 Apr '12

http://www.ams.org/journals/bull/2000-37-04/S0273-0979-00-00875-2/S0273-097… describes a remarkable conjectured formula (arising from generating functions ad series reversions) for the number of rational curves of degree d on a generic "quintic threefold." The formula gives the right answers for the first 9 cases. However, it is not even clear that the numbers it outputs, are always integers, although hundreds have been computed and always came out integer.

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Re: [math-fun] Cheshire Cat Dark Matter
by Henry Baker 22 Apr '12

22 Apr '12

Gravitation not tied to any material source is indeed a conundrum, because any such material sources of gravitation are pulled together & condense into big stuff like black holes that can be more readily detected. This eliminates the possibility of large numbers of ronin osmium asteroids/planets wandering about, because they would eventually coalesce into something visible. (Are there theories of solar system evolution that would result in 80% of the material being ejected?? The one paper about panspermia indicated that upwards of 20% of ronin asteroids could be ejected from the solar system. But I guess all of these theories are based on the total amount of material being infinitesimal compared with the mass of Jupiter & the Sun.) Perhaps it is time to dust off theories that allow for negative gravitational bodies, so that polarization, shielding, and other phenomena can be possible ??? At 11:11 AM 4/18/2012, Warren Smith wrote: >The universe is supposedly 4.6% ordinary matter, >22.7% unknown "dark matter" (not made of known particles), >and the rest is Einstein's cosmical constant. > >However, this study of stellar motions in our galaxy claims the dark matter is not >present in our region of the galaxy. How can this incredible discrepancy be explained? > >I have no idea... >(Not as easy to see how they could have made such a huge mistake, unlike >the "faster than light neutrinos" case...): > >http://www.sciencedaily.com/releases/2012/04/120418111923.htm > >-- >Warren D. Smith >http://RangeVoting.org

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[math-fun] Gravitational lensing?
by Henry Baker 22 Apr '12

22 Apr '12

Answer from an expert in astronomical simulations: -------------------------------------------------- >Do the current galaxy gravitational simulations take into account the following phenomenon? > >Gravitational lensing produces distortions when viewing distant objects. Presumably gravitational lensing also distorts the transmission of the gravitational force itself. I am not an expert in lensing (but there are some at UCSC!), but I'll try to answer to your question. Gravitational lensing is due to a distorsion of the space-time continuum because of the effect of gravity, and an exact calculation requires a general relativity framework. Within such framework, everything is treated self-consistently, so the description of the system should be OK. I see that with your question you think to something along the lines of a quantum theory of gravity (where gravity is mediated by a graviton particle that transmits the info). Unfortunately, we have no such theory currently, so no exact answer to offer. But if we were to consider a source of gravitational waves behind a lens (e.g. a BH-BH merger), then I would expect that a GR calculation of the propagation of the wavefront will take into account correctly the distorsion because of the gravitational lens. I can run this question with a friend who is an expert on GR and gravitational waves and in case get back to you if I hear a better insight. >Do current galaxy gravitational simulations take into account the gravitational lensing effect of the black hole(s) in the center of the galaxy? Short answer is no: The direct gravitational effect of the black hole is already limited to only a small portion of the galaxy, that is in the central region that has a mass comparable to the black hole mass (and therefore something of the order of 0.001 of the total galaxy mass). This effect, can be included in the simulations if needed by adding the BH as a point particle. For many problems, a 0.001 effect is already negligible (unless one is interested in the dynamics of the central region, of course). I do not think that even if there were any high-order corrections because of a lensing effect, these would ever go into a regime where they would be higher than the typical numerical issues of finite spatial/temporal/mass resolution and then roundoff accuracy. Hope this helps. Let me know if you have any other question. Michele Trenti Kavli Fellow Institute of Astronomy University of Cambridge Madingley Road Phone: +44 (0)1223 337526 Cambridge CB3 0HA Fax: +44 (0)1223 337523 United Kingdom E-mail: trenti(a)ast.cam.ac.uk

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[math-fun] Fractal globule
by Henry Baker 21 Apr '12

21 Apr '12

I was watching a Princeton video about advances in genomic science & it talked about a linear DNA molecule folding up into a "fractal globule" (see below), which has the property that close together in 3-space after folding implies close together in 1-space along the genome. The picture accompanying the article below says it all: if you color a linear fragment of DNA with a standard color spectrum and then fold it up, the resulting folding 1) has no knots; and 2) the colors (even in 3D) are solid blotches -- i.e., all the reds stick together, all the blues stick together, etc. This got me to thinking: what about 2-space? These appear to be space-filling curves. So what is known about space-filling curves in 3-space? Anything more interesting than in 2-space? --------------- http://en.wikipedia.org/wiki/Fractal_globule http://blogs.discovermagazine.com/notrocketscience/tag/fractal-globule/ What is the difference between the human genome and a pair of headphones? If youÂve ever put a pair of headphones in your pocket, youÂll know how difficult it is to keep a long cord in a bundle without getting it hopelessly tangled and knotted. YouÂll also start to appreciate the monumental challenge that our cells face when packaging our DNA. At 2 metres in length, the human genome is longer than the average human. But in every one of our cells, the genome needs to fit inside the nucleus, a tiny compartment just 6 millionths of a metre long. How does it do it? One of the secrets behind this monumental feat of folding has just been revealed by research that shows the human genomeÂs three-dimensional structure. A team of scientists led by Erez Lieberman-Aiden and Nynke van Berkum found that our genome folds into a shape called a Âfractal globuleÂ, where the long strands of DNA are densely packed but without a single knot. ItÂs an awe-inspiring feat of space-saving and keeps DNA accessible. When a particular gene is needed, the DNA it sits on can be easily unpacked. Lieberman explains, ÂThe best way to think about it is that it looks like a pack of ramen noodles when you just start cooking them: really dense, but totally unentangled, so you can pull out a noodle or a bunch of noodles without disrupting the rest.Â Previously, scientists suggested that the genome folds into a more tangled structure called the Âequilibrium globuleÂ, which is more like ramen noodles post-cooking Â a massive knotted mess from which single noodles are difficult to extract. Until now, the fractal globule was a theoretical shape that existed only in the minds of mathematicians. This is the first time that it has been observed in reality. The shape was first described by a mathematician Guiseppe Peano in 1890 and in 1988, Alexander Grosberg proposed that a long molecule might spontaneously fold into such a shape under the right conditions. Still, it took till this week for anyone to observe a fractal globule in reality. Â[Peano] had no idea that it described any actual object in the universe,Â says Lieberman-Aiden, Âbut it turns out it describes the genome!Â Some of the other tricks that cells use to fold the genome are well documented. At the most basic level, DNA is wrapped around proteins called histones, like a series of beads on a string. These are then twisted around each other to form a wider filament, like the individual strands of a piece of rope. Beyond that, things become less clear but this new study shows what happens at these higher levels. Imagine a series of beads on a string. You gather clumps of beads and crumple them together into a globule, carefully avoiding any knots or crossovers. Every row of, say, five beads gets crumpled into a globule, every row of five globules gets crumpled together, and so on and so forth. The final result is a single ball Â a Âglobule-of-globules-of-globulesÂ. Lieberman-Aiden developed a technique called Hi-C that simultaneously analyses adjacent DNA across the entire genome, in order to reveal its 3-D shape. It relies on formaldehyde to immobilise pieces of DNA that sit next to each other, effectively freezing the genome and forming cross-links between adjacent strands. The DNA is then shredded and the cross-linked fragments are isolated, sequenced and mapped onto the reference copy of the human genome. The result is a library of all the DNA strands that were neighbours in the nucleus, which can be analysed to understand how the genome must be folded. The technique confirmed that parts of the genome that would sit far apart if it was fully stretched out are actually very close to each other in space. Because of the complicated molecular origami that goes on inside the nucleus, around three quarters of the close-contact sequences identified by the Hi-C method are actually distant ones. The research also confirmed that the nucleus is divided into two territories Â an ÂONÂ compartment where DNA is rich in genes, highly active and loosely packed, and an ÂOFFÂ compartment where DNA is gene-poor, largely inactive and densely packed for storage. Individual chromosomes snake in and out of these two compartments and when a given gene is activated, it moves from one to the other. ItÂs not clear what defines the boundaries between these two compartments, but Lieberman-Aiden suspects that these boundaries are very sharp. As an example, Lieberman-Aiden use glow-in-the-dark molecules to tag four stretches of DNA called L1, L2, L3 and L4. They lie one after the other on chromosome 14, but in the nucleus, they pair up differently. L1 and L3 are typically found in the ÂONÂ compartment and are always closer to each other than L2. Meanwhile, L2 and L4 are closer to each other than L3, and are usually found in OFF territory. ÂA huge question in biology is how all the different cells in the body perform totally different functions when all of them have the same genome,Â says Lieberman-Aiden. ÂThis work suggests that the spatial arrangement of the genome in a particular nucleus is a big part of why different cells do different things.Â PPS: You may remember Erez from the irregular verbs paper that I recently reposted. Many thanks to Erez for the heads-up about the paper and the awesome ramen noodle analogy. Reference: Science 10.1126/science.1181369

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[math-fun] n and n^3 have all digits odd
by James Buddenhagen 20 Apr '12

20 Apr '12

Suppose the positive integers n and n^3 both have all digits odd. Such numbers include 1, 11, 15, 33, 39, 71, 91, 173, 175, 179, 335, 3337, 5597, 7353. Is this list complete? This sequence is http://oeis.org/A085597 in OEIS. But there is no key word indicating that it is a finite sequence. Is this known to be a finite sequence?

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