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RE: [math-fun] A modest proposal for carbon sequestration in space
by Henry Baker 01 Aug '06

01 Aug '06

According to Newton, the weight near the center of the earth is near zero, because most of the mass is around you in equal & opposite directions. You raise an interesting question, which I think may have been asked in my physics class when I was an undergraduate: If one drilled a Verne-like well all the way to the center of the Earth, what would the air pressure there be (assuming that the insulated walls of the well withstood the pressure of the rock/magma/iron/etc.), assuming that the temperature at the bottom was reasonable -- room temperature. At 03:20 PM 8/1/2006, Cordwell, William R wrote: >However, I've been wondering about what happens when one calculates the >pressure near the center of the earth. If one uses a column, one gets a >nice, finite pressure, integrating the weight of the rock/magma/stuff >upward. However, if one considers that the surface area of the sphere >around the center of the earth becomes arbitrarily small, it seems that >all of that weight is borne by a tiny area--the pressure should approach >infinity. This is, I think, equivalent to using the frustrum instead of >the column. So, is there a compensating force (the walls of the >frustrum are radial, so there might be a net upward force from the >pressure around it?), or is there something wrong with the model of >using a frustrum, or is it something else? > >Bill

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[math-fun] physics; GF2; non-consecutive primes
by Schroeppel, Richard 01 Aug '06

01 Aug '06

Some physics thoughts ... a) You can reduce the energy required to move things into space by exchanging with moon rocks. Using a few Moravec- style rotating tethers, a bucket brigade can be established that imports moon rocks, while establishing a CO2 atmosphere on the moon. b) Dan's nit about the gravity field being non-parallel. This was dismissed because of the exponential falloff of gas density with height, "so the extra weight of the stuff at the top is negligible". But the effect of that extra weight is multiplicative, not additive: The extra weight needs extra supporting gas in the layer below, etc., so the overall result could be larger. c) The halving-density height of air is about 3.5 miles. A 35 mile deep well would have air that's a little denser than water at the bottom. (Might be interesting to jump into!) The density near the center of the earth would exceed black hole numbers, except that the gas laws wrt compression and pressure fail. d) If you carve out a small hole for yourself in the center of the earth, it's true that the gravitational force on you will balance out to near 0 (ignoring a non-spherical earth). But the pressure might still be important. (Or we'd expect bubbles to form in the middle of fluid planets like Saturn.) e) I think the halving-height of a gas depends on the molecular weight, and that pure CO2 would have a smaller halving height. ----- and some GF2 thoughts ... f) The sum-of-divisors formula should work in GF2. g) I read MLB's problem as "find sets of GF2 prime polynomials that xor to 0". No biggie: 11111 x 11001 x 1101 x 1011. More generally, there are about 2^D / D prime polys of degree D, and 2^that possible sets. Many of those sets must xor to the same value (only 2^(D+1) possible xor results), and thence many 0s. ----- re 10^3 + 1 = 7.11.13: If you drop the "consecutive" requirement on the primes, I think you can come up with lots more examples: 12^3 + 1 = 1729 = 7.13.19 Rich

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[math-fun] Product of three consecutive primes question
by Daniel Asimov 01 Aug '06

01 Aug '06

Quick -- factor 999,999 in your head. [solution 31 lines below] If you got 7*11*13*27*37 fairly quickly, then you probably know that 1001 = 7*11*13, which always seemed like an unlikely curiosity to me. Let p < q < r denote any 3 consecutive primes. How many such p,q,r satisfy p*q*r = N^3 + 1 for some integer N ? (Above all, is it finite or infinite?) If finite, can you list all solutions, or at least give an upper bound for N ? Etc.

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[math-fun] Re: A sequence puzzle
by Rainer Rosenthal 01 Aug '06

01 Aug '06

Joseph Biberstine wrote: > For fellow decipherers, here are some basic diagnostics. > split into increasing runs > {1, 2, 6, 12, 60}, {3, 21, 168, 504, 2520, 27720}, {4, 52, 364, 5460, > 21840, 371280, 1113840, 21162960}, {5, 105, 2310, 53130, 212520, > 1062600, 13813800, 124324200}, {7, 203, 6090, 188790, 3020640, 33227040, > 564859680}, {35, 1260, 46620, 885780, 11515140} > Within these runs, terms seem to rise exponentially. In the first lot of numbers it seemed as if the records started with 1, 2, 3, 4, 5, ... when sorted as follows: { 1} { 2, 6, 12, 60} { 3, 21, 168, 504, 2520, 27720} { 4, 52, 364, 5460, 21840, 371280, 1113840, 21162960} { 5, ... } but then the next lot didn't fit this idea: { 5, 105, 2310, 53130, 212520, 1062600, 13813800, 124324200} { 7, 203, 6090, 188790, 3020640, 33227040, 564859680} {35, 1260, 46620, 885780, 11515140} > successive ratios I also noticed the successive ratios being integer *within* the runs. > {2, 3, 2, 5, 1/20, 7, 8, 3, 5, 11, 1/6930, 13, 7, 15, 4, 17, 3, 19, > 1/4232592, 21, 22, 23, 4, 5, 13, 9, 1/17760600, 29, 30, 31, 16, 11, 17, > 1/16138848, 36, 37, 19, 13, 1/1151514} The construction principle seems to be related to playing with primes and I will write the above ratios within the runs emphasizing that. Each run becomes a block, where the first line contains the primes which are newest and biggest. The second line contains the other factors which are either smaller primes or composite: block 1: { 1} { } block 2: { 2 3 5} { 2 } block 3: { 7 11} { 3 8 3 5 } block 4: { 13 17 19} { 4 7 15 4 3 } block 5: { 23 } { 5 21 22 4 5 13 9} block 6: { 29 31 } { 7 30 16 11 17} block 7: { 37 } { 35 36 19 13} ================ I get some pleasing pattern by completing the upper lines in the blocks: block 1: { 1} { } block 2: { 2 3 4 5} { 2 } block 3: { 6 7 8 9 10 11} { 3 8 3 5 } block 4: { 12 13 14 15 16 17 18 19} { 4 7 15 4 3 } block 5: { 20 21 22 23 24 25 26 27} { 5 21 22 4 5 13 9} block 6: { 28 29 30 31 32 33 34} { 7 30 16 11 17} block 7: { 35 36 37 38 39} { 35 36 19 13} As one can plainly see: the lower numbers are always a factor of their upper neighbour! The given ratios don't look as random as they did on first sight. I wanted to share this with other "fellow decipherers" and feel encouraged by Joseph's mail. I am quite far from a good explanation, but this order looks good, doesn't it? Best regards, Rainer Rosenthal r.rosenthal(a)web.de

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Re: [math-fun] A modest proposal for carbon sequestration in space
by Daniel Asimov 01 Aug '06

01 Aug '06

<< If I remember my physics correctly, air pressure is approx. 14.7#/sq in, which is the weight of the entire column of air 1 inch square sitting on the square inch. >> Not to be a nitpicker, but since "up" is not given by a parallel vector field, wouldn't the shape of this "air column" be more like a frustum of a 4-sided pyramid -- with a spherical "one-square-inch" top of radius R ~ Earth's radius, and a spherical base, of radius R+h (where h = maximum distance of relevant air from Earth) ? --Dan

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Re: [math-fun] A modest proposal for carbon sequestration in space
by Henry Baker 01 Aug '06

01 Aug '06

You are right, except that it doesn't matter much, since the falloff is exponential (i.e., faster than polynomial). According to one source on the internet, the density is only 50% of sea level density at approx. 4-5 miles, so the lion's share of the mass is within 15 miles. At 03:05 PM 8/1/2006, Daniel Asimov wrote: ><< >If I remember my physics correctly, air pressure is approx. 14.7#/sq in, >which is the weight of the entire column of air 1 inch square sitting on >the square inch. >>> > >Not to be a nitpicker, but since "up" is not given by a parallel vector field, >wouldn't the shape of this "air column" be more like a frustum of a 4-sided >pyramid -- with a spherical "one-square-inch" top of radius R ~ Earth's radius, >and a spherical base, of radius R+h (where h = maximum distance of relevant >air from Earth) ? > >--Dan

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[math-fun] Apostol's Riemann song for your iPod
by Henry Baker 01 Aug '06

01 Aug '06

http://olimu.com/Riemann/Song.htm

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[math-fun] A modest proposal for carbon sequestration in space
by Henry Baker 01 Aug '06

01 Aug '06

When you have too many lemons, make lemonade. Instead of a "carbon tax", why not incentivize something really useful, like a space elevator? How about a tax paid in carbon fiber nanotubules strong enough to make such a space elevator: http://en.wikipedia.org/wiki/Space_elevator If such an elevator were built in the form of a tube, it would be the world's tallest smokestack. Yes, you would probably have to pressurize the tube in order to lift the gasses inside, but that could be done relatively efficiently at ground level. Since the pressure would decrease with increasing height, the highest pressure would occur at the bottom, so there should be not problem with the pressure higher up. Presumably, other types of waste -- perhaps even radioactive waste -- could possibly be "sequestered" in space the same way. --- I noticed that 1) all of the proposals involve starting the elevator cable at the ground or some high tower, and 2) the exponential taper means that even small increases in elevation of the base of the elevator help a lot. So, why not start from a balloon which is itself tethered -- alternatively, include balloons along the lower portions of the elevator to increase the effective height of the beginning of the elevator ? Weather & winds would affect the balloons, but some sort of active station-keeping scheme could be used, because it could now be powered from the ground (most of the other proposals don't have enough lifting capacity to include a power cable from the ground).

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Resend: [math-fun] Kaplansky's Obituary
by Daniel Asimov 01 Aug '06

01 Aug '06

[N.B. The last link wasn't clickable since it glommed on to the right angle bracket. I hope this one works.] I don't think I saw any mention of it here, but Kaplansky passed away little over one month ago, on June 25. Here's a link to a nice obituary . . . but I wasn't able to extract specific results of his from this: http://www.signonsandiego.com/uniontrib/20060714/news_1m14kaplansk.html Actually, I'm curious -- if any cognoscenti would care to explain some theorems he's famous for, I'd be very interested. --Dan

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[math-fun] Kaplansky's Obituary
by Daniel Asimov 31 Jul '06

31 Jul '06

I don't think I saw any mention of it here, but Kaplansky passed away little over one month ago, on June 25. Here's a link to a nice obituary . . . but I wasn't able to extract specific results of his from this: < http://www.signonsandiego.com/uniontrib/20060714/news_1m14kaplansk.html> Actually, I'm curious -- if any cognoscenti would care to explain some theorems he's famous for, I'd be very interested. --Dan

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