April 2012 - mathfun@mailman.xmission.com

[math-fun] Interplanetary microbes?
by Warren Smith 09 Apr '12

09 Apr '12

> From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] "Ejecta" from planet to planet > Message-ID: <E1SH508-0007PE-OK(a)elasmtp-spurfowl.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > The following video explains in pretty good detail how impact craters can > eject material from the surface of a planet into the solar system & find its > way onto other planets. In fact, approximately 20% of such material > actually gets ejected from the solar system. > > The probability appears overwhelming that over the 4.5 billion years of the > solar system, there has been a relatively significant exchange of material > among the planets, including some meteors that fell onto Earth from > relatively recent impacts on Mars. If there were any microbial creatures in > these rocks, they are highly likely to have survived the original launching > (100,000 G's) and the re-entry into the second planet. > > One Los Alamos scientist placed a steel object on the surface for an > underground nuclear explosion in the early 1950's and this object may have > achieved priority as the first man-made object to escape the Earth's > gravity. > > This scientist estimates that the Earth gets many tons of material ejected > from Mars every year, although much of this material may have been hanging > around the solar system for millions of years before finally landing on > Earth. > > He comments that the meteor that caused so much trouble 65 million years ago > almost certainly ejected much material into space. Since the Earth was > absolutely teeming with life at the time, it would be inconceivable if such > life were not carried away with the ejected material. > > If I understood this video correctly, if an explosion exceeds approximately > 250 megatons, then the fireball itself -- i.e., a portion of the atmosphere > -- may escape from the Earth. This apparently answers a question I posed > here a while back about whether one or more large collisions could have > carried away a significant fraction of the Earth's atmosphere -- they answer > may very well be "yes". > > ------- > Are We All Martians? The Meteoritic Exchange of Life between Planets > Monday, April 20, 2009 10:38 AM > > Dr. H. J. Melosh is Professor in the University of Arizona Lunar and > Planetary Labs. His research interests include theoretical geophysics and > planetary surfaces. Presented Feb. 24, 2009. > > 56 minutes. > > http://deimos3.apple.com/WebObjects/Core.woa/FeedEnclosure/arizona-public-d… --I'm skeptical on some of your/Melosh's claims, but agree re others. Ejecting rocks from a planet via a collision is difficult if that planet has a big atmosphere. Several meteors have been found that are known to have come from Mars, which has a small atmosphere. A large number are known to have come from our moon (about 1 in every 1000 meteors). I would not be terribly surprised to see meteors from Mercury, although none have been found yet, and from other moons. However, as far as I know, no meteor has ever been found that came from Venus, Jupiter, Saturn, Uranus, Neptune (big atmospheres). Earth has an intermediate size atmosphere and I think it would be difficult to eject earth rocks. To get a meteor thru an atmosphere, it needs to be large, otherwise it vaporizes. (Small ones can land, but I think the only way that can happen is if they originally were part of something large.) The threshold size should be of this order: approximately so large that the mass of the meteor is the same as the mass of the column of atmosphere "carved out" by the meteor, which for a spherical rock with density D in grams/cc would be diameter>=1500cm/D in Earth atmosphere (pressure = 1kg/cm^2). With D=3 we get diameter>=500cm=5meters. For Venus the size threshold would be more like 3000 meter diameter! However, one might conjecture that an impact big enough to blast huge rocks to well above escape velocity would shatter those rocks, i.e. not giving us big intact ones. Melosh leaves this whole issue totally un-discussed. If so, then ejection would be very rare or impossible. So here is my conjecture: Fugeddaboudit, you are not going to get meteors from Venus. Deinococcus Radiodurans http://en.wikipedia.org/wiki/Deinococcus_radiodurans is a common microbe with amazing radiation tolerance, it has 37% survival of 15000 Grays. http://www.srl.caltech.edu/ACE/ASC/DATA/bibliography/ICRC2005/usa-mewaldt-R… says unshielded space dose is 10 centiGray per year on average So this dose is probably about what you'd get from a 150000 year sojourn in space, and you'd survive longer the deeper you were shielded inside a rock. So anyhow I agree microbe survival would be possible.

1 0

[math-fun] "Ejecta" from planet to planet
by Henry Baker 09 Apr '12

09 Apr '12

The following video explains in pretty good detail how impact craters can eject material from the surface of a planet into the solar system & find its way onto other planets. In fact, approximately 20% of such material actually gets ejected from the solar system. The probability appears overwhelming that over the 4.5 billion years of the solar system, there has been a relatively significant exchange of material among the planets, including some meteors that fell onto Earth from relatively recent impacts on Mars. If there were any microbial creatures in these rocks, they are highly likely to have survived the original launching (100,000 G's) and the re-entry into the second planet. One Los Alamos scientist placed a steel object on the surface for an underground nuclear explosion in the early 1950's and this object may have achieved priority as the first man-made object to escape the Earth's gravity. This scientist estimates that the Earth gets many tons of material ejected from Mars every year, although much of this material may have been hanging around the solar system for millions of years before finally landing on Earth. He comments that the meteor that caused so much trouble 65 million years ago almost certainly ejected much material into space. Since the Earth was absolutely teeming with life at the time, it would be inconceivable if such life were not carried away with the ejected material. If I understood this video correctly, if an explosion exceeds approximately 250 megatons, then the fireball itself -- i.e., a portion of the atmosphere -- may escape from the Earth. This apparently answers a question I posed here a while back about whether one or more large collisions could have carried away a significant fraction of the Earth's atmosphere -- they answer may very well be "yes". ------- Are We All Martians? The Meteoritic Exchange of Life between Planets Monday, April 20, 2009 10:38 AM Dr. H. J. Melosh is Professor in the University of Arizona Lunar and Planetary Labs. His research interests include theoretical geophysics and planetary surfaces. Presented Feb. 24, 2009. 56 minutes. http://deimos3.apple.com/WebObjects/Core.woa/FeedEnclosure/arizona-public-d… http://deimos3.apple.com/WebObjects/Core.woa/Feed/arizona-public-dz.5774022…

1 0

[math-fun] Wordbox challenge
by Victor Miller 08 Apr '12

08 Apr '12

The following puzzle has been floating around work. It's been called "Word Box" but I've seen with a little googling that that name is used for other kinds of puzzles. In any case here it is: You're given a list of words, and an m by n grid. You want to assign letters to the points on the grid so that you can spell out each of the words by starting at some point in the labeled grid, and tracing a path where each step can go left, right, up or down from the current point. The explicit challenge is the names of the nine (this was before Pluto was demoted) planets. Give word boxes which are 4 x 8, 5 x 7 and 6 x 6, and (much much harder) show that you can't do it for 4 x 7 and 5 x 6. Victor

1 0

[math-fun] ... 2, 4, 20, 100, 620, 4420, 35900, ...
by Bill Gosper 08 Apr '12

08 Apr '12

FindSequenceFunction (modulo some fondling) gives closed forms for A153229 <https://oeis.org/A153229>, A153229 <https://oeis.org/A153229>, and A005165 <https://oeis.org/A005165>: In[317]:= Table[ FullSimplify[-(-1)^n E ExpIntegralEi[-1] + E ExpIntegralE[n, 1] Gamma[n]], {n, 9}] Out[317]= {0, 1, 0, 2, 4, 20, 100, 620, 4420} But strangely, In[313]:= FullSimplify[(-1)^ n E (-(-1)^n ExpIntegralEi[-1] + ExpIntegralE[n, 1] Gamma[n]), n \[Element] Integers && n > 0] Out[313]= (-1)^n E (-(-1)^n ExpIntegralEi[-1] + ExpIntegralE[n, 1] Gamma[n]) instead of In[318]:= Factor[ FullSimplify[#, n \[Element] Integers && n > 0] & /@ Expand[%313]] Out[318]= -E (ExpIntegralEi[-1] - (-1)^n ExpIntegralE[n, 1] Gamma[n]) which it concedes is two LeafCounts smaller. --rwg <https://oeis.org/A005165>

1 0

[math-fun] Fwd: Re: generalized law of cosines
by rcs＠xmission.com 08 Apr '12

08 Apr '12

Gosper encountered a paper (via Askey?) with a generalized law of cosines for polygons and polyhedra. The paper subsumes "my" observation that the pythagorean theorem works for right-angled tetrahedra (cut from a cube corner). The paper doesn't mention higher dimensions, but presumably it's valid. Another nicety is that cos() is even, so you don't need to worry about assigning signs to the angles. Does the law of sines works for tetrahedra? The final comment about splitting up the sides other than 1 vs N-1 should give a very simple 2-2 formula for quadrilaterals, and could lead to an angle-free formula relating the lengths of the sides and the diagonals. Presumably this would give the 2nd diagonal as a quadratic in the 4 sides and the first diagonal. And there's likely something pretty involving the sides of a pentagon and the "inscribed star". Rich ----- Quoting Bill Gosper <billgosper(a)gmail.com>: > for the nth side of an n-gon. Easy to remember. Per Askey. Rich: Math > fun? > > http://www.rowan.edu/open/colleges/las_new/departments/math/facultystaff/os… > --Bill >

3 3

[math-fun] Omni-similarity
by Adam P. Goucher 08 Apr '12

08 Apr '12

Enumerate all images (2-colourings of the n*n grid Z_n^2), for instance by lexicographically ordering them for each value of n, and then ordering the sets in increasing order of n. So, we get something like: [0] [1] [0 0] [0 0] [0 0] [0 1] ... [1 1] [1 0] [1 1] [1 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] ... We can take the nth image of this sequence, and scale it to occupy a square of side length (Â½)^n. These can easily be placed on the diagonal of a unit square. Colour everything else arbitrarily. The resulting 2- colouring of the unit square thus contains scaled-down versions of arbitrarily close approximations to all 2-colourings of the unit square, including itself. I propose this property be called 'omni-similarity'. It is stricter than the quasi-self-similarity found in Julia sets. I have heard that the Mandelbrot Set exhibits omni-similarity. Is this true? It certainly seems implausible. Sincerely, Adam P. Goucher

1 0

[math-fun] spectacular patterns in snow
by Michael Beeler 07 Apr '12

07 Apr '12

The blog post is titled, "Man Walks All Day to Create Spectacular Snow Patterns". But that almost understates it. These are big patterns, on math themes. The URL is http://www.mymodernmet.com/profiles/blogs/simon-beck-snow-art or search for Simon Beck snow. I can't post the photos to math-fun, but here's the text, which gives a sense of what it's about... Artist Simon Beck must really love the cold weather! Along the frozen lakes of Savoie, France, he spends days plodding through the snow in raquettes (snowshoes), creating these sensational patterns of snow art. Working for 5-9 hours a day, each final piece is typically the size of three soccer fields! The geometric forms range in mathematical patterns and shapes that create stunning, sometimes 3D, designs when viewed from higher levels. How long these magnificent geometric forms survive is completely dependent on the weather. Beck designs and redesigns the patterns as new snow falls, sometimes unable to finish a piece due to significant overnight accumulations. Interestingly enough, he said, 'The main reason for making them was because I can no longer run properly due to problems with my feet, so plodding about on level snow is the least painful way of getting exercise. Gradually, the reason has become photographing them, and I am considering buying a better camera." Spectacular art for the sake of exercise! -- Mike Beeler

1 0

[math-fun] New IBM App Presents Nearly 1, 000 Years of Math History
by Henry Baker 07 Apr '12

07 Apr '12

FYI -- http://www.wired.com/gadgetlab/2012/04/new-ibm-app-presents-nearly-1000-yea… New IBM App Presents Nearly 1,000 Years of Math History By Alexandra Chang April 6, 2012 | 6:30 am | Categories: Miscellaneous Math nerds and historians, itÂs time to get excited. Minds of Modern Mathematics, a new iPad app released Thursday by IBM, presents an interactive timeline of the history of mathematics and its impact on society from 1000 to 1960. The app is based on an original, 50-foot-long ÂMen of Modern MathematicsÂ installation created in 1964 by Charles and Ray Eames. Minds of Modern Mathematics users can view a digitized version of the original infographic as well as browse through an interactive timeline with more than 500 biographies, math milestones and images of relevant artifacts. IBM hopes that classes and students will use the app, provoking more people to pursue math, science or technology-related educations and jobs. ÂCareers of the future will rely heavily on creativity, critical thinking, problem solving and collaboration Â all themes that were core to the ÂMinds of Modern MathematicsÂ movement and remain equally relevant today,Â Chid Apte, IBM Director of analytics Research and Mathematical Sciences said in a press release. ÂWhat better way than a mobile app to reintroduce this timeless classic to inspire a new generation of learners?Â Starting from the first French pope to celebrated mathematician John von Neumann, users can see every major math-related event or person in a visually impressive layout. You can tap on an entry to read more about it, and view how math events and math superstars relate in time to other historical events, such as the First Crusade that seizes Jerusalem and the signing of The Declaration of Independence. And in case you (like me) havenÂt taken math since high school, the app also includes an ÂIBM Mathematics Peep ShowÂ of nine 2-minute animated clips that will help you brush up on mathematical concepts like topology and exponents. The Minds of Modern Mathematics app is available for free on the App Store. But if the iPad version isnÂt enough, you can always visit the large-scale timeline on display at the New York Hall of Science in Queens, NY and The Museum of Science in Boston, MA. Alexandra Chang is a Wired.com staff writer. Previously she's worked at Macworld and AllThingsD.com.

1 0

[math-fun] "extremely ingenious analysis" (was: Bessel at High Precision?)
by Bill Gosper 06 Apr '12

06 Apr '12

Apropos Watson's Treatise (I didn't manage to cover this with Neil Wednesday, so now everybody gets it, even though most of you understand it better than I do): Early on, (p4 of 804) Watson recounts Euler's analysis of the motion of a hanging chain, which is further modernized (w/o attribution) at http://mysite.du.edu/~jcalvert/math/hchain.htm . I was a bit disappointed that this classic solution makes *two* small-deflection approximations: The chain particles move only horizontally, and only sinusoidally. In this solution, the amplitude varies along 0 < x < length in proportion to J[0](2 ω sqrt (x/g)) at one of a discrete set of angular velocities (frequencies/2π) ω determined by J[0](2 ω sqrt (length/g)) = 0 (reflecting the fixed top end). Thus, to learn the possible frequencies and nodes, we need the roots of J[0](sqrt X). (No less than) Watson: "By an extremely ingenious analysis, ..., Euler proceeded to shew that the three smallest roots of [J[0](2 sqrt x)] are 1.4457965, 7.6178156, and 18.7217517." Should be 1.445796490736696, 7.617815585915522, 18.7217516976738, but this I had to see. Euler's idea is similar to his zeta(2) conjecture, and works for functions where you have the power series, and the sum of the reciprocated roots σ[1] := Σ_k r[k]^-1 converges. (E.g., sinc(sqrt x).) Briefly: Assume you can write f(x) as a Weierstrass product f(0) prod(1-x/r[n]), with 0 < r[n] < r[n+1]. Expanding the log derivative of the product and power series gives a system of linear equations that can be solved for σ[n] := Σ_k r[k]^-n, n=1,2,... . Then the inequality σ[n]^-n < r[1] < σ[n]/σ[n+1] gets r[1]≈1.445795 when n reaches 7. Writing σ'[n]:=σ[n]-r[1]^-n gives the inequality σ'[n]^-n < r[2] < σ'[n]/σ'[n+1], &c. --rwg PS, a few months ago, under the Subject: Timber!, I sent a faulty formula for a toppling pole, followed by a correction. Here's a graph<http://gosper.org/topple.png>of initial tilt vs toppling time. On Mon, Apr 2, 2012 at 5:11 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > *Quad>* > > Hey all. > > Does anyone have any ideas on computing Bessel functions to high > precision (over the complex plane)? > > Googling gives [1] and [2]. Sorry, [1] is behind a paywall. > > I am wondering if one of you has a secret magical sum or product that > just magically works well for high precision computation. If not, > maybe we can develop some strategies! > > Thanks, > > Robert Smith > > [1] dl.acm.org/citation.cfm?id=1466254 > [2] web.cs.dal.ca/~jborwein/bessel.pdf <http://web.cs.dal.ca/%7Ejborwein/bessel.pdf> > > *------ > [...] At the time, I used (a tiny fraction of) Watson's Treatise, > but now there's dlmf*.nist.gov/10 .[...] > --rwg > >

1 0

[math-fun] Elastic potential energy
by Dan Asimov 06 Apr '12

06 Apr '12

Suppose we have an ideal homogeneous, isotropic sheet S of rubber that we stretch in some given fashion. Using the simplest mathematical model (a 2D version of Hooke's law?), what is the potential energy it stores from this deformation? In case it simplifies things, we can assume that it is stretched over a convex surface K in 3-space, say by a mapping f: S -> K, and in such a way that no distances along S are decreased along K. I.e., for all x,y in S, we have d_K(f(x),f(y)) >= d_S(x,y). ---------------------------------------------------- Or, it may be sufficient to know the elastic potential energy density for a linear mapping L of an infinitesimal square Q in the tangent space T_x(S) onto an infinitesimal parallelogram P in the tangent space T_f(x)(K) for arbitrary x in S. (The non-decreasing distance condition can be enforced by assuming all such L satisfy D \sub L(D), where D is the unit disk in T_x(S).) --Dan ________________________________________________________________________________________ It goes without saying that .

1 0