March 2011 - mathfun@mailman.xmission.com

[math-fun] e = m.c^2
by Guy Haworth 09 Mar '11

09 Mar '11

I didn't say that 'motivation' and 'concentration' were uncorrelated. Guy :-)

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[math-fun] StackExchange-like website for Computer Algebra
by quad 09 Mar '11

09 Mar '11

Hey all. I posted this to sci.math.symbolic too, but I thought I might as well post here. I made a proposal for a StackExchange (like MathExchange, StackOverflow, MathOverflow) specifically geared at computer algebra and symbolic computation. If it gets enough followers, then it can become an actual Q&A site. http://area51.stackexchange.com/proposals/30094/computer-algebra-symbolic-c… It will not be intended for software support. Rather, it's for the theory and practice of computer algebra and symbolic computation. Let me know what you guys think! -Robert

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[math-fun] Asteroids & Earth's atmospheric density
by Henry Baker 09 Mar '11

09 Mar '11

Several years ago on math-fun, we had a spirited discussion about the possibility of smoke-stacks tall enough to reach orbit -- a la space elevators. During this discussion, I came to the conclusion that the Earth's atmosphere was far from equilibrium -- i.e., the Earth's current atmosphere is far too thin to have outgassed into space during its lifetime. Proof: Venus, which has a far denser atmosphere & due to its closeness to the Sun, should have outgassed much more of its atmosphere during this time than the Earth. So what happened to the Earth's atmosphere? The violent collision(s) that produced the Earth's Moon probably took much of the atmosphere away, but probably not enough to produce the current thin atmosphere. I contend that the K-T event (65.5 million years ago) stripped away a significant amount of the Earth's atmosphere & probably some amount of ocean. I don't think that large asteroids hitting the Earth simply get absorbed into the crust; I think that there are rebound splashes that take some amount of the Earth's material into space, including rock & ocean & atmosphere. The analogy to throwing a rock into a pool of water isn't exact, but has some of these features. It should also be noted that with the amount of life on Earth at the time, a huge amount of organic material would have been splattered into space, including a lot of plant & dinosaur DNA. So if we find organic material on the Moon or on Mars, asteroid impacts on Earth may have been one mechanism for getting it there. We know that there were flying dinosaurs prior to 65 million years ago that are too big to fly today. If the Earth's atmosphere was _denser_ prior to 65 million years ago, then it makes it much easier for these dinosaurs to fly. The higher oxygen content (30-35%) prior to K-T would help: oxygen molecules are 14% heavier than nitrogen molecules, but I guess that the atmospheric density would have to have been even greater than the O2/N2 ratio to allow these big birds to fly. I'm not an aerospace engineer, so without knowing the scaling laws for flight, I can't estimate the appropriate density. The higher oxygen content helps animals of large mass quite significantly: anyone who has attempted to exercise at high altitudes will quickly understand this issue. This higher oxygen partial pressure translates into relatively smaller lungs, arteries, etc., for a given size animal, thus allowing the huge dinosaurs to prosper. If the atmospheric pressure were significantly higher as well as having higher oxygen content, the partial pressure of oxygen could have been 2.5-3x current levels pre-K-T. But simply having more efficient circulatory systems & muscles still wouldn't have allowed these enormous birds to fly unless the atmosphere were also significantly denser. I wonder if this atmospheric pressure/density issue has been considered before? Perhaps asteroid collisions cause a lot more damage than we previously considered? -------- Below are some interesting tidbits from Wikipedia. http://en.wikipedia.org/wiki/K%E2%80%93T_boundary "The KÂT boundary is a geological signature, usually a thin band, dated to (65.5 Â± 0.3) Ma (megaannum, or million years ago).[1] K is the traditional abbreviation for the Cretaceous period, and T is the abbreviation for the Tertiary period. The boundary marks the end of the Mesozoic era and the beginning of the Cenozoic era, and is associated with the CretaceousÂTertiary extinction event, a mass extinction" "the Alvarez team went on to calculate the size of the asteroid. The answer was about 10 km (6.2 mi) in diameter, about the size of Manhattan" "suggest that the oxygen content of the atmosphere was very high (30Â35%) during the late Cretaceous" "the possibility of near simultaneous multiple impacts, perhaps from a fragmented asteroidal object" "The potential Shiva crater, 450Â600 km (280Â370 mi) in diameter, would substantially exceed Chicxulub in size and has also been dated at about 65 mya, an age consistent with the KÂT boundary" http://en.wikipedia.org/wiki/Atmosphere_of_Venus "The temperature at the surface is 740 K (467Â°C, 872Â°F), while the pressure is 93 bar" "The main atmospheric gases are carbon dioxide and nitrogen" http://en.wikipedia.org/wiki/Pterosaur http://en.wikipedia.org/wiki/Pterosaur_size "Very large animals are defined as megafauna. Speculation exists about the reasons for megafauna development in animals of different epochs. The gigantic sizes reached by some species of pterosaurs have puzzled many scientists most specifically, because of their capacity for flight. Although some pterosaurs were as small as a sparrow, the largest had wingspans that exceeded thirty feet and some are estimated to have reached a weight of five hundred and fifty pounds. The albatross has the largest wingspan of modern birds, spanning eleven feet, yet it weights less than twenty pounds so the ratio is quite different for analyzing flight mechanisms. Factors such as the warmer climate of the Mesozoic or higher levels of oxygen existing in the atmosphere back then have been proposed, but it's now generally agreed that even the largest pterosaurs could have flown in the modern skies. Partially, this is due to the presence of air sacs in their wing membranes [1], as well as the assessment that some pterosaurs may have launched into flight with their front limbs in a quadrupedal stance, perhaps as bats do (birds are bipedal). [2] The mechanism for launching their flight remains uncertain." http://en.wikipedia.org/wiki/Quetzalcoatlus "Quetzalcoatlus was a pterodactyloid pterosaur known from the Late Cretaceous of North America (Maastrichtian stage, 70Â65.5 ma)" "More recent estimates based on greater knowledge of azhdarchid proportions place its wingspan at 10Â11 meters (33Â36 ft)." "A 2002 study suggested a body mass of 90Â120 kilograms (200Â260 lb) for Quetzalcoatlus, considerably lower than most other recent estimates.[5] Higher estimates tend toward 200Â250 kilograms (440Â550 lb)." "In 2010, Donald Henderson argued that the mass of Q. northropi has been underestimated, even the highest estimates, and that it was too massive to have achieved powered flight. Henderson argued that it may have been flightless.[13] Paul MacCready undertook an aerodynamics experiment testing the flight of Quetzalcoatlus in 1984. He constructed a model flying machine or ornithopter with a simple computer functioning as an autopilot. The model successfully flew with a combination of soaring and wing flapping;[14] however, the model was half scale based on a then-current weight estimate of around 80 kg, far lower than more modern estimates of over 200 kg." http://en.wikipedia.org/wiki/Hatzegopteryx "Hatzegopteryx hails from the Densu-Ciula Formation of western Romania, which has been dated to the late Maastrichtian stage of the late Cretaceous Period, around 65 million years ago." "The skull of Hatzegopteryx was also unique in its heavy, robust construction. Most pterosaur skulls are made up of very lightweight plates and struts. In Hatzegopteryx, the skull bones are stout and robust, with large-ridged muscle insertion areas. In their 2002 description, Buffetaut and colleagues suggested that in order to fly, the skull weight of this pterosaur must have been reduced in some unconventional way (while they allowed that it could have been flightless, they found this unlikely due to the similarity of its wing bones to flying pterosaurs). The authors theorized that the necessary weight reduction was accomplished by the internal structure of the skull bones, which were full of small pits and hollows (alveoli) up to 10 mm long, separated by a matrix of incredibly thin bony struts (trabeculae), a feature also found in some parts of Hatzegopteryx wing bones. The authors pointed out that this unusual construction, which differed significantly from the irregular intern al structure of other pterosaur skulls, resembles the structure of expanded polystyrene, the substance used to make Styrofoam. They noted that this would allow a sturdy, stress-resistant construction while remaining lightweight, and would have allowed the huge-headed animal to fly" "In 2003 they moderated the estimates to a close to twelve metres wing span and an over 2.5 metres skull length.[3] In 2010 Mark Witton e.a. stated that any appearance that the Hatzegopteryx humerus was bigger than TMM 41450-3 had been caused by a distortion of the bone after deposition and that the species thus likely had no larger wingspan than Quetzalcoatlus, today generally estimated at ten to eleven metres."

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Re: [math-fun] Asteroids & Earth's atmospheric density
by Henry Baker 09 Mar '11

09 Mar '11

CO2 dissolving in water might help remove the CO2 from the atmosphere, but then where did all the Earth's nitrogen go? There's a huge amount of nitrogen in the atmosphere of Venus that was presumably also in the early atmosphere of the Earth. I suppose the early Earth oceans could have been concentrated nitric acid, but the nitrogen would have at some point ended up in some form of nitrates & nitrites, which are highly soluble. Is there enough vegetation on the Earth to have absorbed all that nitrogen? I don't recall coal & oil deposits having large amounts of nitrogen in them, so the nitrogen from plant materials would have washed out & ended up somewhere else. At 09:20 AM 3/9/2011, Joshua Zucker wrote: >I always thought the explanation for the difference between Venus and >Earth is that we never got hot enough to vaporize the oceans, and so >the CO2 released by volcanoes and so on gets dissolved in water and >ultimately ends up carbonate rock (limestone, etc) on Earth while it >ends up atmosphere on Venus. Mars has little atmosphere partly >because of the lower gravity but mostly because it's not so >volcanically active. > >In other words, vulcanism and oceans are the dominant processes. >Losses to space are much slower. > >But I'm far from expert on these issues so I'd love to learn more! > >--Joshua Zucker > >On Tue, Mar 8, 2011 at 9:17 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> Several years ago on math-fun, we had a spirited discussion about the possibility of smoke-stacks tall enough to reach orbit -- a la space elevators. During this discussion, I came to the conclusion that the Earth's atmosphere was far from equilibrium -- i.e., the Earth's current atmosphere is far too thin to have outgassed into space during its lifetime. Proof: Venus, which has a far denser atmosphere & due to its closeness to the Sun, should have outgassed much more of its atmosphere during this time than the Earth.

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[math-fun] e = mc^2
by Guy Haworth 08 Mar '11

08 Mar '11

Before my lecture to the first-year students, we discussed how time seemed to speed up as the year went on. Someone said it was the increased gravitational pull of approaching exams. Yeh, it's relativity, said another ... e = m.c^2 I'm not sure why, but at that point I decided that: Education = Motivation * Concentration squared Guy Haworth :-)

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[math-fun] A rhinoceros' pancreas
by Fred lunnon 07 Mar '11

07 Mar '11

Many years ago, I happened across an account by a biologist --- citation alas long-forgotten --- who as a student had dutifully absorbed the well-known fact, immortalised in all the best textbooks, that the pancreas secretes amylase (or some other enzyme) in every mammalian species --- with the sole exception of the rhinoceros. As time went by, he prospered in his profession, and reached at last that pinnacle at which he could command the acquisition of a genuine rhinoceros pancreas (by then unattached, presumably, to its previous owner) in order to satisfy a persistant though entirely natural curiosity concerning whatever physiological peculiarity might be responsible for such an idiosyncrasy. And whaddya know? It did secrete amylase (or whatever it was) after all! [I don't suppose anyone can remind me of my long-forgotten source for this history?] Since then, I have encountered numerous rhinoceros' pancreases (or whatever the plural is). Here's the latest. In books on (fairly) elementary geometry will be commonly be found a construction credited to Klein, in which the usual hyperbolic, elliptic, Euclidean, etc. flavours are specialised from (say 3-space) projective geometry by choosing an "absolute" quadric K --- say c w^2 + x^2 + y^2 + z^2 = 0 with c constant, [x/w, y/w, z/w] Cartesian coordinates of a general finite point. The desired symmetry subgroup is selected by the constraint that K remains invariant. A standard account then proceeds along the lines of an associated polarity, quadric degenerate when Euclidean, winding up with "when c < 0, hyperbolic; c = 0, Euclidean; c > 0, elliptic". 'Ang abaht --- when c = 0, K represents a single (real) point at the origin --- that's not invariant under translation ??!! It seems that what the original author must have had in mind was rather the polarity, for which the quadric was simply a shorthand. In the Euclidean case the polarity turns out to be [w,x,y,z] -> [0,x,y,z]; that is, actually c = oo --- obviously enough, since the other geometries approach flatness as their radius increases. A recent transcription of this momentary lapse of XIX-th century concentration --- from Klein himself, perhaps? --- turns up in Judith N. Cederberg "A course in modern geometries" (2001), in section 4.12 on page 299 --- there in 2-space rather than 3-space. This particular pachyderm will doubtless still be trotting around, in 2-space and 3-space, for another 100 years. A little depressing, innit? Fred Lunnon

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[math-fun] CF pi addendum
by Bill Gosper 07 Mar '11

07 Mar '11

The absence of any really big terms from Neil's 458M means that, at around m=10^900000000, the sums 2 Tan[n Pi ] Sum[------------, {n, -m,m}] == n (1 + n Pi) 1 Cot[- - k] 2 Sum[-(----------), {k, -m,m}] ~ 9.71628672863 1 2 -(-) + k 4 (take the limit Pi^2 for the n=0 term) will agree to > 600M digits. However, no one has managed to prove that these sums even converge, let alone to 9.7162867286... --rwg

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[math-fun] Nth roots of tautology
by Adam P. Goucher 07 Mar '11

07 Mar '11

Dear all: Negation and tautology are both square-roots of tautology, as: "This statement is true." = "This statement is not not true." Do there exist words (in any language) which act as cube-roots of tautology (excluding the trivial case of tautology itself)? Or, more generally, Nth roots of tautology, for arbitrary natural numbers N? If we did have such words in the English language, what logical effect would they have on the meaning of a sentence? Sincerely, Adam P. Goucher

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[math-fun] Nth roots of tautology
by Robert Munafo 06 Mar '11

06 Mar '11

I was reminded of the recent query by James Propp involving the sentence *The people you know are the people who know all the people* *who know all the people you know* (noting that in the thread it was made clear that "know" was to be defined in such a way that for all A and B : [(A knows B) iff (B knows A)]) but at any rate this is only another example of a square root of tautology: it works out to F(I)=F(F(F(I))), or I=F(F(I)), where I is the self and F is the function of symmetric (reciprocated) familiarity. The next thing I thought of was color science, because color has been shown to be partly inseparable from language (see for example http://www.wired.com/wiredscience/2008/03/babies-see-pure/) and because color has three dimensions -- so you could imagine transforming a color by cycling the three axes on the color space {R->G->B->R}, or similarly by adding 120 degrees to the hue on the color wheel, and then it would take three steps to get back to the original color. But there are no words for "the hue that's 120 degrees further around on the color wheel" (in any natural language, I suspect). Then I hit on it: *Every Tuesday is the eve of the eve of the eve of the eve of the eve of the eve of the eve of a Tuesday.* where of course we define "eve" to mean "the day before". - Robert Munafo On Sun, Mar 6, 2011 at 11:18, Adam P. Goucher <apgoucher(a)gmx.com> wrote: > Dear all: > > Negation and tautology are both square-roots of tautology, as: > > "This statement is true." = "This statement is not not true." > > Do there exist words (in any language) which act as cube-roots of > tautology (excluding the trivial case of tautology itself)? Or, more > generally, Nth roots of tautology, for arbitrary natural numbers N? If we did have such words in the English language, what logical effect would they have on the meaning of a sentence? > > Sincerely, > > Adam P. Goucher > -- Robert Munafo -- mrob.com Follow me at: fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com- youtube.com/user/mrob143 - rilybot.blogspot.com

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[math-fun] There's lot's more fun to be had here.
by Bill Gosper 06 Mar '11

06 Mar '11

Replacing W&W's cross-shaped arrangement of poles with a six-rayed asterisk (replacing plus with times?), i.e., expanding z^2/(z^3 + x^3)/Sin[z*(-1)^(2/3)]/Sin[z]/Sin[z*(-1)^(4/3)]: 2 2 3 12 Pi r x Sum[---------------------------------------------, {r, Infinity}] == 6 6 6 r (Pi r - x ) (-1 + (-1) Cosh[Sqrt[3] Pi r]) 1 2 Csc[x] -- + ------------------------ 3 Cos[x] - Cosh[Sqrt[3] x] x Expanding at x=0, 4 1 Pi {Sum[----------------------------------, {r, 1, Infinity}] == -(-----), 4 r 11340 r (-1 + (-1) Cosh[Sqrt[3] Pi r]) 10 1 703 Pi Sum[-----------------------------------, {r, 1, Infinity}] == -(----------), 10 r 7662154500 r (-1 + (-1) Cosh[Sqrt[3] Pi r]) 1 Sum[-----------------------------------, {r, 1, Infinity}] == 16 r r (-1 + (-1) Cosh[Sqrt[3] Pi r]) 16 223199 Pi -(----------------)}, etc. 2338757728807500 --rwg

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