April 2012 - mathfun@mailman.xmission.com

[math-fun] meteorites
by Bill Gosper 11 Apr '12

11 Apr '12

Warren>Here is the situation: A. meteor falls in at somewhat more than escape velocity, perhaps as much as 1.5X escape. This was Barringer's delusion. Stuff is falling in toward the *Sun*, 42.1km/s =rt2*Earth orbital speed, vs 11.2km/s Earth escape speed. So his impactor was (42.1/11.2)^2 ~ 14 times less massive than he thought, and was largely vaporized on impact. Then he had the great misfortune to discover that angled impacts make round craters, and drilled fruitlessly on. --rwg

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Re: [math-fun] Fwd: Re: generalized law of cosines
by Bill Gosper 11 Apr '12

11 Apr '12

Fred>On 4/7/12, rcs(a)xmission.com <http://gosper.org/webmail/src/compose.php?send_to=rcs%40xmission.com> <rcs(a)xmission.com <http://gosper.org/webmail/src/compose.php?send_to=rcs%40xmission.com>> wrote:> ...> 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> The relation between the sides and diagonals of a plane quadrilateral is that the volume of the tetrahedron with those edge-lengths vanishes. The volume is given by the Cayley-Menger determinant for 3-space, a cubic in the squares of the edge-lengths. See eg. http://en.wikipedia.org/wiki/Distance_geometry [I know most of you have heard all this before several times --- Rich obviously wasn't paying attention.] WFL See also: http://en.wikipedia.org/wiki/Brahmagupta%27s_formula#Extension_to_non-cycli… --rwg Something is wrong with either my math-fun archive or Apple's Finder|Spotlight. I can't find any references to "Brahmag" despite, e.g., my (2008?) subj: any noun can be verbed "Everybody" knows that, given any triangle, the sum of the angle tangents equals their product. This "Brahmagupts" into Given any quadrilateral, the sum of the angle tangents equals their product times the sum of the cotangents. Can't be new. But probably newer than Brahmagupta, since, even for trapezoids, it just says 0=0. --rwg which I found in a scratch file. "extremely ingenious" erratum: I miscopied more modern values for Euler's Bessel roots, which were considerably less accurate. Simply manually evaluating Bessel functions must have been so tedious as to discourage Newton's method.

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[math-fun] Interplanetary microbes?
by Warren Smith 10 Apr '12

10 Apr '12

> From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Fwd: Re: Ejecta from Earth ? Venus ? > Message-ID: <E1SHc69-0007aU-9C(a)elasmtp-curtail.atl.sa.earthlink.net> > Content-Type: text/plain; charset="iso-8859-1" > > FYI -- Here is Dr Melosh's response to my question: > >>From: "Melosh, Henry J" <jmelosh(a)purdue.edu> >>To: Henry Baker <hbaker1(a)pipeline.com> >>Date: Tue, 10 Apr 2012 10:06:12 -0400 >>Subject: Re: Ejecta from Earth ? Venus ? >> >>Dear Dr. Baker: >> >>The atmosphere issue was one of my concerns from the very beginning of >> such considerations--it is not something that I have overlooked! There is >> substantial evidence that impacts that produce craters greater than about >> 10 km diameter on the Earth and about 30 km on Venus breach the atmosphere >> and eject material directly into space. This evidence is not (yet!) based >> on the discovery of Earth or Venus meteorites, but rather on the >> observation of crater ejecta that has traveled immense distances before >> reentering the atmosphere: That is, material that nearly, but not quite, >> left the planet. On Venus, this comes in the form of the dark parabolas >> of ejecta from larger Venusian craters that blanketed the surface with >> ejecta more than about 10 cm deep (equal to the wavelength of the Magellan >> radar that detected these deposits) for distances exceeding 5000 km from >> the impact. On Earth, ejecta from craters like Chicxulub, Popagai, down >> to Bosumptwi can be found at distances ranging > from thousands of km to the antipode of the crater. The 22 km diameter Ries > crater in Germany threw intact rocks more than 300 km into Switzerland. > These ejecta cannot have traveled this far through the atmosphere and so > clearly were thrown above it as part of the plume of vaporized and melted > rock, mixed with intact fragments, that expands out of any high speed impact > crater. >> >>As to nuclear explosions breaching the atmosphere, see Jones, E. M. and J. >> W. Kodis (1982). Atmospheric effects of large body impacts: The first few >> minutes. Geological Implications of Impacts of Large Asteroids and Comets >> on the Earth. L. T. Silver and P. H. Schultz, Geol. Soc. Amer. Sp. Pap. >> 190: 175-186. >> >>I have published on this atmosphere breaching in a variety of media, as a >> google or ADS search would quickly show. >> >>Sincerely, Jay Melosh --this doesn't seem convincing to me. Here is the situation: A. meteor falls in at somewhat more than escape velocity, perhaps as much as 1.5X escape. B. ejects material presumably at large initial speeds but not as large as the impactor's speed. C. ejected material slowed and/or destroyed by atmosphere. D. hopefully still has enough speed to escape planet. Now if an ejected item (B) reaches 100s to 1000s of km from the impact site before landing, that is WAY short of reaching escape velocity. It merely indicates it reached maybe 50% of escape velocity. It is hugely more difficult to gain factor 2 in speed. This can be regarded as evidence in my favor and against Melosh, but he regards it as evidence in his favor. Furthermore, since Melosh cites a boulder in Switzerland that was found 300 km from the impact site as his best on-earth evidence, let me ask this: Where are all the boulders, that were ejected from impact sites all over earth, then fell back down in Antartica? Meteorites are very easy to find in Antartica because highly visible on top of ice, and get carried to glaciers to certain sites. Far more meteorites have been found in Antartica than everyplace else combined. It seems to me, if Melosh were right, there'd be lots of them. But there are zero. By contrast, there are loads of meteorites found that came from the moon, where there was no atmosphere to mess it up! I suggest to you that this combination of evidence and non-evidence supports me, not Melosh. Finally re Jones+Kodis, I have not seen this paper. But I can tell you right now, from the power of pure thought, that any claim (which Melosh made in the video) that a small iron disklike object (I thought he was contending it was about the size of a dinner plate) initially launched at 17 km/sec, would escape Earth, is a false claim. The air would slow it down to well below escape velocity and/or destroy it. It simply could not escape Earth. This calculation is easy and you should make your own calculation independently of me if you do not believe me. Oh wait, somebody else did that already: According to the American Meteor Society Fireball and Meteor FAQ http://www.amsmeteors.org/fireballs/faqf/ item 12, meteors weighing less than 8 tons retain essentially none of their cosmic velocity when passing through the atmosphere, they simply end up as a low speed falling rock. Only objects weighing many times this mass retain a significant fraction of their velocity, e.g 1000 tons retains 70%. So we conclude that ejected matter in (B) to escape earth would need to be a chunk weighing>8 tons and probably 1000 tons. The question is: will such chunks exist, or would they be fragmented by the postulated 100,000g acceleration? However... I found this: http://nuclearweaponarchive.org/Usa/Tests/Brownlee.html which seems to be what Melosh had in mind, although the details differ much from Melosh's oral account. In this account, the object was a 4-inch thick steel plate 4ft in diameter, and even though it was predicted by a simulation the simulator himself did not believe to reach 6X earth escape velocity, and they knew this prediction, and they intentionally tried to measure it with a pre-planned hi-speed camera, they failed massively since only one movie frame had it. (Very poor experimental planning...) And it would appear this result was never published, despite Melosh making it sound like it had been, since presumably the primary source Brownlee would have said so. Anyhow, this is much larger than a dinner plate. Could this escape earth? Such a plate would mass 924 kilograms. This is well below 8 tons. Also a randomly rotating disk is way less aero than a ball per unit mass. So the answer still is NO, it could not have escaped Earth.

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[math-fun] Fwd: Re: Ejecta from Earth ? Venus ?
by Henry Baker 10 Apr '12

10 Apr '12

FYI -- Here is Dr Melosh's response to my question: >From: "Melosh, Henry J" <jmelosh(a)purdue.edu> >To: Henry Baker <hbaker1(a)pipeline.com> >Date: Tue, 10 Apr 2012 10:06:12 -0400 >Subject: Re: Ejecta from Earth ? Venus ? > >Dear Dr. Baker: > >The atmosphere issue was one of my concerns from the very beginning of such considerations--it is not something that I have overlooked! There is substantial evidence that impacts that produce craters greater than about 10 km diameter on the Earth and about 30 km on Venus breach the atmosphere and eject material directly into space. This evidence is not (yet!) based on the discovery of Earth or Venus meteorites, but rather on the observation of crater ejecta that has traveled immense distances before reentering the atmosphere: That is, material that nearly, but not quite, left the planet. On Venus, this comes in the form of the dark parabolas of ejecta from larger Venusian craters that blanketed the surface with ejecta more than about 10 cm deep (equal to the wavelength of the Magellan radar that detected these deposits) for distances exceeding 5000 km from the impact. On Earth, ejecta from craters like Chicxulub, Popagai, down to Bosumptwi can be found at distances ranging from thousands of km to the antipode of the crater. The 22 km diameter Ries crater in Germany threw intact rocks more than 300 km into Switzerland. These ejecta cannot have traveled this far through the atmosphere and so clearly were thrown above it as part of the plume of vaporized and melted rock, mixed with intact fragments, that expands out of any high speed impact crater. > >As to nuclear explosions breaching the atmosphere, see Jones, E. M. and J. W. Kodis (1982). Atmospheric effects of large body impacts: The first few minutes. Geological Implications of Impacts of Large Asteroids and Comets on the Earth. L. T. Silver and P. H. Schultz, Geol. Soc. Amer. Sp. Pap. 190: 175-186. > >I have published on this atmosphere breaching in a variety of media, as a google or ADS search would quickly show. > >Sincerely, Jay Melosh > >######################################################## > >Jay Melosh >Distinguished Professor of Earth and Atmospheric Sciences >Departments of Earth and Atmospheric Sciences, >Physics and Aerospace Engineering >Civil Engineering Building, Room 3237 >550 Stadium Mall Drive Tel: (765) 494-3290 >Purdue University Fax: (765) 496-1210 >West Lafayette IN 47907 email: jmelosh(a)purdue.edu

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[math-fun] simpler CF identity
by Bill Gosper 10 Apr '12

10 Apr '12

I simplified the symmetric identity in http://www.tweedledum.com/rwg/stanfordn3.pdf : a x z + ContinuedFractionK[a ((x + n)^2 + b) z^2, ((a - 1) (x + 1 + n) - (a + 1) y + 1) z, {n,0,∞]}] == a y z + ContinuedFractionK[a ((y + n)^2 + b) z^2, ((a - 1) (y + 1 + n) - (a + 1) x + 1) z, {n,0,∞]}] With {x -> 1, z -> z/Sqrt[a], b -> 0, a -> (2 + z^2 - 2 Sqrt[1 + z^2])/z^2, y -> 1/2} this gives z/ArcTan[z] == 1/2 (1 + Sqrt[1 + z^2]) + ContinuedFractionK[(1/2 + n)^2 z^2, 2 + 2 n + Sqrt[1 + z^2], {n, 0, ∞}] or ContinuedFractionK[1/4 (1 - 2 n)^2 (-1 + z^2), 2 n + z, {n, 1, ∞}] == -1/2 - z/2 + Sqrt[-1 + z^2]/ArcSec[z] And with ContinuedFractionK[c*n^2 + d*n + e, a*n + b, {n, 0, Infinity}] == (2*c*e* Hypergeometric2F1[1 + (2*e)/(d + Sqrt[d^2 - 4*c*e]), 1 + (d + Sqrt[d^2 - 4*c*e])/(2*c), (1/2)*(3 + d/c + (2*b*c - a*(c + d))/(c*Sqrt[a^2 + 4*c])), 1/2 - a/(2*Sqrt[a^2 + 4*c])])/((2*b*c - a*(c + d) + Sqrt[a^2 + 4*c]*(c + d))* Hypergeometric2F1[(2*e)/(d + Sqrt[d^2 - 4*c*e]), (d + Sqrt[d^2 - 4*c*e])/(2*c), (1/2)*(1 + d/c + (2*b*c - a*(c + d))/(c*Sqrt[a^2 + 4*c])), 1/2 - a/(2*Sqrt[a^2 + 4*c])]) we get an identity relating four 2F1s. --rwg And the insane, multiminute delays plaguing Sum and Product now also infest ContinuedFractionK. Since Mma knows so little about ContinuedFractionK anyway, just replace it with the undefined function cfk. (Anybody remember Charles F. Karney?)

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Re: [math-fun] Interplanetary microbes?
by Henry Baker 10 Apr '12

10 Apr '12

At 05:47 PM 4/9/2012, Warren Smith wrote: >>Melosh also suggests that the upward-expanding fireball would actually help >propel the object upwards. I suspect that this effect may be similar to >those Soviet-era "supersonic" torpedoes that had small rocket motors facing >_forwards_ which pushed the water ahead of the torpedo out of the way. > >--??huh? That kind of idea definitely sounds bogus at supersonic speeds. What do you think that sharp pointy object is on the front of supersonic aircraft? It is intended to create a shock wave which bypasses much of the rest of the aircraft. Melosh also shows a brief picture of a computer simulation of such a high-speed ejection through the atmosphere. I know that computer simulations are only as good as the physics that goes into them, but I'm willing to give the researcher who did this the benefit of the doubt for the moment. >>I don't know whether anyone has directly addressed the issue of Venus. > >--you had said any planet to any other was possible. I dispute that. >Sounds like you now also dispute it. You are correct; my language was a bit vague: I said exchange among planets, and from the surface of _a_ planet to other planets. Ejecta from the Moon and from Mars now seems incontrovertible; clearly ejecta from the Earth is somewhat controversial among funsters. Perhaps I should contact this professor and ask him directly. >>I get the impression that this has been a relatively active area of >research over the past 10 years, so there may be later results than >this 2009 video. >>Re bacteria surviving: >>Melosh discusses one of the instruments dropped onto the Moon before >one of the Apollo missions brought it back. This instrument had _not_ >been sterilized prior to being placed onto the Moon, and was kept >sealed when brought back from the Moon. When this instrument was >cultured, it showed the presence of live microbes.

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[math-fun] Interplanetary microbes?
by Warren Smith 10 Apr '12

10 Apr '12

>>--I'm skeptical on some of your/Melosh's claims, but agree re others. > >>(big atmospheres). Earth has an intermediate size atmosphere and I think it would >>be difficult to eject earth rocks. >Melosh specifically talks about how stuff can be ejected from the Earth -- in particular, the meteor event of 65 million years ago. >>To get a meteor thru an atmosphere, it needs to be large, otherwise it >>vaporizes. >Which direction are you talking about? Launching or re-entry? According >to Melosh, re-entry is relatively easy -- it happens all the time. --To correct what I said before, my formula for the "threshold diameter" really pertains not to melting/vaporizing, but to possibility to traverse atmosphere in either direction without a huge slowdown. I.e. if you launch something from ground in some upward direction at escape velocity plus some extra safety margin, you are NOT going to be able to eject it from the earth, UNLESS it is large enough that the air-drag slowdown is small fraction of launch speed. In particular Melosh said some bullshit about a small iron object he thought a nuke explosion had ejected from Earth based on a high speed film frame. I claim that is FALSE -- it was not ejected from Earth. This in turn suggests that Melosh never thought about this whole issue, which if so was a major omission in his thinking. My formula for threshold size of the object, roughly delineates how big it has to be so that atmosphere traversal can happen. Melting and vaporizing are a bit harder to think about because a lot of the heat energy does not go into melting or vaporizing but instead is radiated or convected away. But my argument goes through using the corrected statement just about slowdown (for that purpose, convecting or radiating is fine, that is still an energy-loss mechnaism); that suffices. >>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 specifically addresses this issue. Rocks on the surface near by to the collision, but not directly impacted by the collision, will see a huge acceleration _upwards_. His example is if a meteor crashed through the lecture hall and impacted beside him (or more likely some of his bacteria) would cause his body to accelerate upwards at perhaps 100,000 G's. Although he wouldn't survive, some of his bacteria would certainly survive. --I'm ok with *that*. My problem is, the fast flying resulting stuff, will then get slowed down, melted, and/or vaporized by the atmosphere hence will not escape Earth. It will only be able to escape if object being ejected is large enough. But if all large objects are shattered that will not happen. Will they be shattered by 100,000 Gs? I suspect "yes" for objects of the calculated threshold size, or larger. A mere 20G or so is enough to total your car. >Melosh also suggests that the upward-expanding fireball would actually help propel the object upwards. I suspect that this effect may be similar to those Soviet-era "supersonic" torpedoes that had small rocket motors facing _forwards_ which pushed the water ahead of the torpedo out of the way. --??huh? That kind of idea definitely sounds bogus at supersonic speeds. >I don't know whether anyone has directly addressed the issue of Venus. --you had said any planet to any other was possible. I dispute that. Sounds like you now also dispute it. >I get the impression that this has been a relatively active area of research over the past 10 years, so there may be later results than this 2009 video. >Re bacteria surviving: >Melosh discusses one of the instruments dropped onto the Moon before one of the Apollo missions brought it back. This instrument had _not_ been sterilized prior to being placed onto the Moon, and was kept sealed when brought back from the Moon. When this instrument was cultured, it showed the presence of live microbes.

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[math-fun] Interplanetary microbes?rocks
by Warren Smith 10 Apr '12

10 Apr '12

On 4/9/12, math-fun-request(a)mailman.xmission.com <math-fun-request(a)mailman.xmission.com> wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Interplanetary microbes? (Warren Smith) > 2. Re: Interplanetary microbes? (Henry Baker) > 3. Re: real time wind map (Henry Baker) > 4. Re: real time wind map (Mike Stay) > 5. Re: Disk polyominoes (Allan Wechsler) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Mon, 9 Apr 2012 10:25:32 -0400 > From: Warren Smith <warren.wds(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] Interplanetary microbes? > Message-ID: > <CAAJP7Y2xP+5q9bxP9pNzwetF8JPtsOUFYDEff1MoQ92kBdyEpw(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > >> 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. > > > > ------------------------------ > > Message: 2 > Date: Mon, 09 Apr 2012 07:56:55 -0700 > From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Interplanetary microbes? > Message-ID: <E1SHG2U-0002sS-QY(a)elasmtp-spurfowl.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > At 07:25 AM 4/9/2012, Warren Smith wrote: >>--I'm skeptical on some of your/Melosh's claims, but agree re others. >> >>(big atmospheres). Earth has an intermediate size atmosphere and I think >> it would >>be difficult to eject earth rocks. > > Melosh specifically talks about how stuff can be ejected from the Earth -- > in > particular, the meteor event of 65 million years ago. > >>To get a meteor thru an atmosphere, it needs to be large, otherwise it >>vaporizes. > > Which direction are you talking about? Launching or re-entry? According > to Melosh, re-entry is relatively easy -- it happens all the time. > >>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 specifically addresses this issue. Rocks on the surface near by to > the collision, but not directly impacted by the collision, will see a huge > acceleration _upwards_. His example is if a meteor crashed through the > lecture hall and impacted beside him (or more likely some of his bacteria) > would cause his body to accelerate upwards at perhaps 100,000 G's. Although > he wouldn't survive, some of his bacteria would certainly survive. > > He also suggests that the upward-expanding fireball would actually help > propel the object upwards. I suspect that this effect may be similar to > those Soviet-era "supersonic" torpedoes that had small rocket motors facing > _forwards_ which pushed the water ahead of the torpedo out of the way. > > I don't know whether anyone has directly addressed the issue of Venus. > > I get the impression that this has been a relatively active area of > research over the past 10 years, so there may be later results than > this 2009 video. > > Re bacteria surviving: > > Melosh discusses one of the instruments dropped onto the Moon before > one of the Apollo missions brought it back. This instrument had _not_ > been sterilized prior to being placed onto the Moon, and was kept > sealed when brought back from the Moon. When this instrument was > cultured, it showed the presence of live microbes. > > > > > ------------------------------ > > Message: 3 > Date: Mon, 09 Apr 2012 10:10:56 -0700 > From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] real time wind map > Message-ID: <E1SHI8c-0001ze-Cv(a)elasmtp-spurfowl.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > Very cool! However, it would be nice to also see different > elevations/altitudes, because the current 2D view doesn't satisfy Stokes. > > Also, isn't Wash DC supposed to be a source for this field? > > At 09:12 AM 4/8/2012, rcs(a)xmission.com wrote: >>The site http://hint.fm/wind/ shows a wind map of the USA, as a flow field. >>This display technique would be a good way to teach some kinds of >>differential equations. >> >>Rich > > > > > ------------------------------ > > Message: 4 > Date: Mon, 9 Apr 2012 10:18:15 -0700 > From: Mike Stay <metaweta(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] real time wind map > Message-ID: > <CAKQgqTYfCD2xFmzqaBGyZU+-yfNptDe68i3K+V0f1ZcDf9n+ig(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > On Mon, Apr 9, 2012 at 10:10 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> Very cool! ?However, it would be nice to also see different >> elevations/altitudes, because the current 2D view doesn't satisfy Stokes. > > Wind direction often changes with altitude. I think this only shows > wind at ground level. > >> Also, isn't Wash DC supposed to be a source for this field? > > :D > >> At 09:12 AM 4/8/2012, rcs(a)xmission.com wrote: >>>The site http://hint.fm/wind/ shows a wind map of the USA, as a flow >>> field. >>>This display technique would be a good way to teach some kinds of >>>differential equations. >>> >>>Rich >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > -- > Mike Stay - metaweta(a)gmail.com > http://www.cs.auckland.ac.nz/~mike > http://reperiendi.wordpress.com > > > > ------------------------------ > > Message: 5 > Date: Mon, 9 Apr 2012 20:15:33 -0400 > From: Allan Wechsler <acwacw(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Disk polyominoes > Message-ID: > <CADy-sGHxe0_v9n5Fr+u3wj5msmxPvXF_CNECspkETXM=q02Vdg(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-2 > > A long time ago I proposed a definition for "disk polyominoes", which are > essentially rasterized disks. I was able to enumerate these polyominoes > (as usual, considering congruent polyominoes to be identical) up to order > 11, which was just enough to run off the end of OEIS. Neil was kind enough > to key the new sequence in as A147680. Later I was able to add a[12], > which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, > 2, 2, 3, 3, 4. > > Still later, I managed to calculate a[13] = 4, but apparently I didn't > inform anybody of this. At this point the clunkiness of my search methods > caught up with me, and I didn't have the patience to work out a[14]. > Finally, today, I can announce that a[14] = 4. I have some more powerful > lemmas under my belt now, and in a few days I ought to be able to come up > with a[15] as well. > > The actual polyominoes for order 13, in the same notation described earlier > in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for > that "v": it represents a decimal 31, binary 11111, a column of five > lattice-points. > > The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and > (4eve6). > > Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able > to figure out how to enter this update for myself. What am I missing? > > If there is any interest, I will present my lemmas here, so other people > can join the hunt for disk polyominoes. The sequence seems to grow quite > slowly, and I have conflicting intuitions about whether the growth rate is > polynomial. > > 2009/5/2 Allan Wechsler <acwacw(a)gmail.com> > >> I assume you would then define a[n] as the number of such maximal >> polyominoes containing n lattice points. For small n, starting at n=0, I >> get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly >> have to have d=sqrt(5), but so far the only maximal polyomino I have found >> with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] >> = 1, but I'm not sure yet. >> >> >> 2009/5/2 victor miller <victorsmiller(a)gmail.com> >> >>> A related seqiuence to look at would be the following: >>> >>> For each d>0 look at sets of lattice points maximal with respect to >>> the property that every 2 of them is distance <=d apart. For these >>> type of polyominoes you don't have to specify a center. It's not >>> clear that is identical with the circular polyominoes since ther are >>> non-circular regions of constant diameter. >>> >>> Victor >>> >>> On 5/2/09, Allan Wechsler <acwacw(a)gmail.com> wrote: >>> > Here's a reference I can't consult right away, which seems at least >>> partly >>> > germane: >>> > >>> > Sre?ko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of >>> > Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. >>> > 299-309. Springer, Berlin, 2008. >>> > >>> > On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw(a)gmail.com> >>> wrote: >>> > >>> >> Neil was kind enough to enter the sequence for me; the OEIS sequence >>> >> number >>> >> is A147680. He asks for some pictures, but I confess that what I've >>> got >>> >> doesn't actually show circles. Typically I've drawn a bunch of >>> >> lattice >>> >> points, highlighting the ones that lie on the circumference of the >>> >> smallest >>> >> containing circle. Other marked points need to lie outside that >>> circle in >>> >> order for the disk polyomino to be valid, but I'm not sure how >>> >> illuminating >>> >> my diagrams would be. Perhaps a picture of the lone 7-point example >>> with >>> >> its bounding circle and other points clearly lying outside would >>> >> convey >>> >> the >>> >> idea, but I don't have a nice one. >>> >> >>> >> I should probably list the polyominoes I've shown to be disks. I >>> >> could >>> >> use >>> >> the notation we used to use for small Life patterns, where each row is >>> >> represented by the value of a binary number whose ones show which >>> points >>> >> are >>> >> part of the configuration. These numbers are usually small, and we >>> write >>> >> the different row-descriptors with no delimiter between them, going up >>> to >>> >> letters of the alphabet if we run out of digits. We usually pick a >>> scan >>> >> order that minimizes the maximum descripton. >>> >> >>> >> For order 0, we of course have only (0), and for order 1 only (1). >>> Order >>> >> 2 >>> >> gives (11), and order 3 gives the L-tromino (13). Order 4 has two >>> >> examples, >>> >> the block (33) and the T-tetromino (131). Order 5 gives the >>> P-pentomino >>> >> (133) and the X-pentomino (272). >>> >> >>> >> Order 6: (273), (333). >>> >> >>> >> Order 7: (373). >>> >> >>> >> Order 8: (377), (2772). >>> >> >>> >> Order 9: (777), (2773). >>> >> >>> >> Order 10: (2777), (3773), (27f6). (That "f" means 15, with four >>> adjacent >>> >> points in a row included in the polyomino.) >>> >> >>> >> Order 11: (3777), (27f7), (67f6). >>> >> >>> >> I only have 80% confidence that these lists are exhaustive. I'm 99% >>> >> confident that all the polyominoes listed are in fact of the disk >>> >> type. >>> >> >>> >> I leave you with a puzzle: Is the duodecomino (67f7) a disk >>> >> polyomino? >>> >> (I >>> >> suspect RWG can generate much more fiendish conundra of this variety.) >>> >> >>> >> >>> >> On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw(a)gmail.com> >>> wrote: >>> >> >>> >>> A funster who might or might not wish to remain anonymous has >>> corrected >>> >>> me >>> >>> in private. A3 does not equal 2; it equals 1. The corrected sequence >>> is: >>> >>> >>> >>> 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ... >>> >>> >>> >>> As before, I'm not entirely confident of the last couple of values. >>> >>> Thank >>> >>> you for the correction. >>> >>> >>> >> >>> >> >>> > _______________________________________________ >>> > math-fun mailing list >>> > math-fun(a)mailman.xmission.com >>> > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> > >>> >>> _______________________________________________ >>> math-fun mailing list >>> math-fun(a)mailman.xmission.com >>> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> >> >> > > > ------------------------------ > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > End of math-fun Digest, Vol 110, Issue 7 > **************************************** > -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step) and math.temple.edu/~wds/homepage/works.html

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[math-fun] real time wind map
by rcs＠xmission.com 09 Apr '12

09 Apr '12

The site http://hint.fm/wind/ shows a wind map of the USA, as a flow field. This display technique would be a good way to teach some kinds of differential equations. Rich

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Re: [math-fun] Interplanetary microbes?
by Henry Baker 09 Apr '12

09 Apr '12

At 07:25 AM 4/9/2012, Warren Smith wrote: >--I'm skeptical on some of your/Melosh's claims, but agree re others. > >(big atmospheres). Earth has an intermediate size atmosphere and I think it would >be difficult to eject earth rocks. Melosh specifically talks about how stuff can be ejected from the Earth -- in particular, the meteor event of 65 million years ago. >To get a meteor thru an atmosphere, it needs to be large, otherwise it >vaporizes. Which direction are you talking about? Launching or re-entry? According to Melosh, re-entry is relatively easy -- it happens all the time. >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 specifically addresses this issue. Rocks on the surface near by to the collision, but not directly impacted by the collision, will see a huge acceleration _upwards_. His example is if a meteor crashed through the lecture hall and impacted beside him (or more likely some of his bacteria) would cause his body to accelerate upwards at perhaps 100,000 G's. Although he wouldn't survive, some of his bacteria would certainly survive. He also suggests that the upward-expanding fireball would actually help propel the object upwards. I suspect that this effect may be similar to those Soviet-era "supersonic" torpedoes that had small rocket motors facing _forwards_ which pushed the water ahead of the torpedo out of the way. I don't know whether anyone has directly addressed the issue of Venus. I get the impression that this has been a relatively active area of research over the past 10 years, so there may be later results than this 2009 video. Re bacteria surviving: Melosh discusses one of the instruments dropped onto the Moon before one of the Apollo missions brought it back. This instrument had _not_ been sterilized prior to being placed onto the Moon, and was kept sealed when brought back from the Moon. When this instrument was cultured, it showed the presence of live microbes.

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