July 2007 - mathfun@mailman.xmission.com

[math-fun] Stairway to Heaven?
by Steve Witham 10 Jul '07

10 Jul '07

>From: Eugene Salamin <gene_salamin(a)yahoo.com> > >A stairway to heaven? > >May 31st 2007 | ACAPULCO > >From The Economist print edition > >If implemented, this would be another taxpayer sponsored boondoggle. > >1. Even if this concept works, why will it launch into space CO2 >rather than N2 or O2? There exists far more of these in the >atmosphere. My reading was that the 17 Hz frequency is what selects CO2 ions rather than another ion. >2. Atmospheric CO2 is part of the biosphere. If we throw away >atmospheric CO2, the atmosphere will re-equilibrate by taking away >carbon from life. There will be fewer plants and animals. >Conversely, burning fossil fuels returns to the biosphere carbon >that has been locked up for a hundred million years. The whole global-warming-by-manmade-CO2 hypothesis is that the current concentration of CO2 in the atmosphere is unnaturally high and caused by the rate of burning of fossil fuels, and that the natural correction mechanisms either can't reabsorb it at the current production rate, or will do so by processes catastrophic to us. It's true that this method throws away carbon. To call it "sequestration" stretches the term. I can't tell whether throwing out carbon is a bad or good thing. You might say that fossil fuel is spare carbon. But then, what else are we going to build all those diamond cities and buckycable elevators out of? >3. This expenditure is entirely unnecessary. There already exists >a facility called HAARP that can transmit 3.6 MW of RF power from >the Arctic into the ionosphere. http://www.haarp.alaska.edu/ I believe what doesn't exist is the specific combination of radio transmitters (at 17 Hz) and lasers. The article mentions starting with another existing facility: >The site of the conveyor Dr Wong is proposing to build is the >Arctic. More specifically, he is suggesting it be over one of his >workplaces, the High Power Auroral Stimulation facility near >Fairbanks in Alaska that he set up 20 years ago to stimulate and >study artificial auroras. http://www.hipas.alaska.edu/hipasweb/hipas.htm That page says HIPAS has "8 - 150 kW transmitters," which possibly means 1.2 MW (but it continues, "capable of modulation at an amplitude of 100 Hz - 20 kHz and Phase of 0 -20 kHz," which is so incoherent it leaves the whole sentence in doubt). This map shows both HIPAS and HAARP: http://www.hipas.alaska.edu/hipasweb/alaska_map.htm The article says that ...his lasers and radio transmitters, even if powered by fossil-fuel generated electricity, should cause far less CO2 to be put into the atmosphere than they ship out of it. but it doesn't say what percentage of the world's power production would be needed to make a dent in the CO2 concentration any decade soon. --Steve

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[math-fun] Superduperghost
by Dan Asimov 10 Jul '07

10 Jul '07

My new bride, Karen, and I -- stuck in a ferry queue at the start of our honeymoon -- passed the time playing Superduperghost. This is like the word game Ghost, but each time a person adds a letter to the previous string, the new letter can go anywhere in that string. (So a turn consists of reciting the entire new string.) After Karen recited "HREMS" it was my turn, and I couldn't think of anything but HAREMS, which would be a loss for me, of course, so I gave up. But later I searched a word list and found two more common words (both which would've ended on me anyway). Without using any kind of software, I challenge y'all to find the shorter of these two (Scrabble-eligible) words, which happens to have only 8 letters. (The grep search would be something like .*H.*R.*E.*M.*S.* but no fair doing one!) --Dan

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Re: [math-fun] CO2 sequestration in outer space, take two...
by Eugene Salamin 10 Jul '07

10 Jul '07

----- Original Message ---- From: Henry Baker <hbaker1(a)pipeline.com> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Monday, July 9, 2007 3:27:22 PM Subject: [math-fun] CO2 sequestration in outer space, take two... Obviously, our discussion on math-fun last year must have generated some thinking. I was wrong about utilizing space tethered smokestacks, but Dr. Wong was right on. ----- Economist.com Global warming A stairway to heaven? May 31st 2007 | ACAPULCO >From The Economist print edition Earth has a natural transport system standing ready to get rid of carbon dioxide. Here is how it might be turned on MOST solutions to the problem of global warming are tediously, almost oppressively, quotidian. Switch the lights off. Stop using fossil fuels to make electricity. Run an efficient car. Don't fly. A few grandiose projects have also been suggested, such as giant parasols in space or adding iron to the ocean to encourage planktonic algae to grow and soak up carbon dioxide. On the whole, though, those big ideas are either mad or could have dangerously unpredictable consequences. That does not mean that lateral thinking about the problem has no place. And the idea proposed by Alfred Wong of the University of California, Los Angeles, at last week's meeting of the American Geophysical Union, in Acapulco, is about as lateral as they come. Dr Wong reckons the problem is not so much that CO2 is being thrown away, but that it is not being thrown far enough. According to his calculations, a little helping hand would turn the Earth's magnetic field into a conveyor belt that would vent the gas into outer space, whence it would never return. The site of the conveyor Dr Wong is proposing to build is the Arctic. More specifically, he is suggesting it be over one of his workplaces, the High Power Auroral Stimulation facility near Fairbanks in Alaska that he set up 20 years ago to stimulate and study artificial auroras. The Arctic sky is special because it is one of the two places (the other being the Antarctic) where the magnetic shield of the Earth opens up to outer space. Auroras, such as the one pictured above, pleasingly testify to a stream of particles from the sun that gets through and hits the atmosphere. These particles bring with them many gigawatts of power that Dr Wong wants to harness to reduce the concentration of greenhouse gases in the atmosphere. Going up in the world His idea starts with the fact that CO2 molecules like to team up with loose electrons, to form CO2 ions. A few percent of the CO2 molecules in the air manage to find such electrons. As a result they become negatively charged. The second piece of luck is that all over the Earth there is a constant vertical electrical field. The surface and the atmosphere form a giant battery, as the lightning discharges of thunderstorms demonstrate. This field tends to make negatively charged ions, such as those of CO2, drift upward. At first this happens slowly, because collisions with other molecules keep throwing the drifting ions off course. But after a few days they arrive at an altitude, about 125km up, which is so rarefied that an ion can move freely about. This is when the last stage of their one-way trip into space begins: sailing along the magnetic field of the Earth. High in the polar regions, the lines of magnetic force point almost straight upwards. When a charged particle is in a magnetic field, it tends to travel along that field's lines of force, spiralling as it goes. In the case of a CO2 ion at an altitude of 125km, it spirals round 17 times a second. However, as it travels upwards, it experiences a weakening field. It must then make fewer turns per second, in obedience to a law of physics called the conservation of magnetic moment (this is similar to the law of conservation of angular momentum that slows a spinning ice dancer down as he spreads his arms). And because it cannot just shed its energy of movement, it is forced to travel faster and faster in the direction of the field. The eventual result is that it is ejected into space. That, at least, is the theory. And although CO2 is too rare even in today's atmosphere for the phenomenon to be detected by existing satellites, an equivalent ejection of oxygen, a far more abundant gas, can be detected from space. So it seems more than likely that Dr Wong's analysis of what is going on in nature is right. The question is, can CO2 molecules be given an artificial leg-up into space, so that they leave the atmosphere in sufficient numbers to make a difference to climate change? Dr Wong thinks they can. The leg-up he proposes comes in two stages. First, he has to ionise more CO2. There are many ways this might be done, but for a first experiment Dr Wong proposes zapping dust in the atmosphere with powerful lasers, to release electrons that can then combine with CO2. Having created the ions, he will then nudge those that have drifted upwards to the appropriate height with radio waves of exactly 17 cycles a second, which will give them a nice stock of energy at the beginning of their spiralling phase. Once they are there, Dr Wong expects the incoming stream of charged particles that cause auroras to deliver the bonus that will make the whole thing work, by dumping some of their energy into the spiralling as well. This should happen through a process called stochastic resonance: the spiralling molecules get preferential treatment, so to speak, because they stand out in what is otherwise an environment of random movements. So far, Dr Wong has only rough calculations of the energy needs of his scheme, but these suggest that his lasers and radio transmitters, even if powered by fossil-fuel generated electricity, should cause far less CO2 to be put into the atmosphere than they ship out of it. The key to this efficiency is the free energy arriving by stochastic resonance. If the particles do their bit, he thinks that a few dozen megawatts of additional electrical power is all that will be needed to make a dent in the amount of CO2. Exactly how big that dent would be, he is not yet sure. But he is pretty sure it would be big enough to help. Copyright © 2007 The Economist Newspaper and The Economist Group. All rights reserved. _______________________________________________ If implemented, this would be another taxpayer sponsored boondoggle. 1. Even if this concept works, why will it launch into space CO2 rather than N2 or O2? There exists far more of these in the atmosphere. 2. Atmospheric CO2 is part of the biosphere. If we throw away atmospheric CO2, the atmosphere will re-equilibrate by taking away carbon from life. There will be fewer plants and animals. Conversely, burning fossil fuels returns to the biosphere carbon that has been locked up for a hundred million years. 3. This expenditure is entirely unnecessary. There already exists a facility called HAARP that can transmit 3.6 MW of RF power from the Arctic into the ionosphere. http://www.haarp.alaska.edu/ Gene ____________________________________________________________________________________ Fussy? Opinionated? Impossible to please? Perfect. Join Yahoo!'s user panel and lay it on us. http://surveylink.yahoo.com/gmrs/yahoo_panel_invite.asp?a=7

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[math-fun] CO2 sequestration in outer space, take two...
by Henry Baker 09 Jul '07

09 Jul '07

Obviously, our discussion on math-fun last year must have generated some thinking. I was wrong about utilizing space tethered smokestacks, but Dr. Wong was right on. ----- Economist.com Global warming A stairway to heaven? May 31st 2007 | ACAPULCO >From The Economist print edition Earth has a natural transport system standing ready to get rid of carbon dioxide. Here is how it might be turned on MOST solutions to the problem of global warming are tediously, almost oppressively, quotidian. Switch the lights off. Stop using fossil fuels to make electricity. Run an efficient car. Don't fly. A few grandiose projects have also been suggested, such as giant parasols in space or adding iron to the ocean to encourage planktonic algae to grow and soak up carbon dioxide. On the whole, though, those big ideas are either mad or could have dangerously unpredictable consequences. That does not mean that lateral thinking about the problem has no place. And the idea proposed by Alfred Wong of the University of California, Los Angeles, at last week's meeting of the American Geophysical Union, in Acapulco, is about as lateral as they come. Dr Wong reckons the problem is not so much that CO2 is being thrown away, but that it is not being thrown far enough. According to his calculations, a little helping hand would turn the Earth's magnetic field into a conveyor belt that would vent the gas into outer space, whence it would never return. The site of the conveyor Dr Wong is proposing to build is the Arctic. More specifically, he is suggesting it be over one of his workplaces, the High Power Auroral Stimulation facility near Fairbanks in Alaska that he set up 20 years ago to stimulate and study artificial auroras. The Arctic sky is special because it is one of the two places (the other being the Antarctic) where the magnetic shield of the Earth opens up to outer space. Auroras, such as the one pictured above, pleasingly testify to a stream of particles from the sun that gets through and hits the atmosphere. These particles bring with them many gigawatts of power that Dr Wong wants to harness to reduce the concentration of greenhouse gases in the atmosphere. Going up in the world His idea starts with the fact that CO2 molecules like to team up with loose electrons, to form CO2 ions. A few percent of the CO2 molecules in the air manage to find such electrons. As a result they become negatively charged. The second piece of luck is that all over the Earth there is a constant vertical electrical field. The surface and the atmosphere form a giant battery, as the lightning discharges of thunderstorms demonstrate. This field tends to make negatively charged ions, such as those of CO2, drift upward. At first this happens slowly, because collisions with other molecules keep throwing the drifting ions off course. But after a few days they arrive at an altitude, about 125km up, which is so rarefied that an ion can move freely about. This is when the last stage of their one-way trip into space begins: sailing along the magnetic field of the Earth. High in the polar regions, the lines of magnetic force point almost straight upwards. When a charged particle is in a magnetic field, it tends to travel along that field's lines of force, spiralling as it goes. In the case of a CO2 ion at an altitude of 125km, it spirals round 17 times a second. However, as it travels upwards, it experiences a weakening field. It must then make fewer turns per second, in obedience to a law of physics called the conservation of magnetic moment (this is similar to the law of conservation of angular momentum that slows a spinning ice dancer down as he spreads his arms). And because it cannot just shed its energy of movement, it is forced to travel faster and faster in the direction of the field. The eventual result is that it is ejected into space. That, at least, is the theory. And although CO2 is too rare even in today's atmosphere for the phenomenon to be detected by existing satellites, an equivalent ejection of oxygen, a far more abundant gas, can be detected from space. So it seems more than likely that Dr Wong's analysis of what is going on in nature is right. The question is, can CO2 molecules be given an artificial leg-up into space, so that they leave the atmosphere in sufficient numbers to make a difference to climate change? Dr Wong thinks they can. The leg-up he proposes comes in two stages. First, he has to ionise more CO2. There are many ways this might be done, but for a first experiment Dr Wong proposes zapping dust in the atmosphere with powerful lasers, to release electrons that can then combine with CO2. Having created the ions, he will then nudge those that have drifted upwards to the appropriate height with radio waves of exactly 17 cycles a second, which will give them a nice stock of energy at the beginning of their spiralling phase. Once they are there, Dr Wong expects the incoming stream of charged particles that cause auroras to deliver the bonus that will make the whole thing work, by dumping some of their energy into the spiralling as well. This should happen through a process called stochastic resonance: the spiralling molecules get preferential treatment, so to speak, because they stand out in what is otherwise an environment of random movements. So far, Dr Wong has only rough calculations of the energy needs of his scheme, but these suggest that his lasers and radio transmitters, even if powered by fossil-fuel generated electricity, should cause far less CO2 to be put into the atmosphere than they ship out of it. The key to this efficiency is the free energy arriving by stochastic resonance. If the particles do their bit, he thinks that a few dozen megawatts of additional electrical power is all that will be needed to make a dent in the amount of CO2. Exactly how big that dent would be, he is not yet sure. But he is pretty sure it would be big enough to help. Copyright Â© 2007 The Economist Newspaper and The Economist Group. All rights reserved.

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[math-fun] Open problem #6 now solved
by Christian Boyer 09 Jul '07

09 Jul '07

In 2005, I published 10 open problems on magic squares in my paper of The Mathematical Intelligencer. One of them (open problem #6) was: Construct a magic square of cubes of prime numbers I am please to announce that this very difficult problem is now solved by a joint work of Jaroslaw Wroblewski (Poland) and Hugo Pfoertner (Germany). Their square is a 42x42 square, and use 1764 consecutive prime numbers: from 11 to 15137. Magic sum = 33151363766492. It remains now 8 unsolved problems on 10. And my €100 prize + bottle of Champagne are still to win! (open problem #2). Christian.

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[math-fun] Four-star Hanoi: a number-theoretic oddity.
by Fred lunnon 08 Jul '07

08 Jul '07

Sometime in the mid-1990's, Paul Stockmeyer proposed a variation of the Reve's puzzle (Tower of Hanoi with 4 pins) which I propose to christen "four-star": one of the intermediate pins is distinguished, and every move is obliged to involve this pin either as origin or as destination. [ See "Variations on the Four-Post Tower of Hanoi Puzzle" at http://www.cs.wm.edu/~pkstoc/toh.html ] The analysis proceeds much as for the Reve's puzzle. To transfer n discs from origin pin to destination pin: (1) transfer n-k smaller discs from origin to (other) intermediate; (2) transfer k larger discs from origin to destination; (3) transfer n-k smaller discs from intermediate to destination; To choose the optimum value for k, Stockmeyer constructs an integer sequence a_n by arranging the set {2^i 3^j} in ascending order of magnitude, then sets k = k_n = 1 + [log(a_n)/log3]; e.g. (suitably fudged at n = 0) n = 0 1 2 3 4 5 6 7 8 9 10 11 12 a_n = ? 1 2 3 4 6 8 9 12 16 18 24 27 k_n = 0 1 1 2 2 2 2 3 3 3 3 3 4 The number of moves is then optimised at 2 (a_0 + ... + a_n), However, a much simpler algorithm would be simply to set k_n = [sqrt(2 n log2/log3) + 0.1809] ?? giving the same result for n < 1737 [which should satisfy the most enthusiastic apocryphal Brahmin!] It's not hard to see why such a formula might provide a good approximation; what I find surprising is that it fails only after such a protracted run. Fred Lunnon

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[math-fun] Re: <rational>!
by Bill Gosper 05 Jul '07

05 Jul '07

____________________________________________________________________________________ Boardwalk for $500? In 2007? Ha! Play Monopoly Here and Now (it's updated for today's economy) at Yahoo! Games. http://get.games.yahoo.com/proddesc?gamekey=monopolyherenow (Piece of sh*t Yahoo indeed linebroke my attachment. Hopefully, you were able to repair it. And hopefully, this is the last email they'll pollute for me.) The matrix product I gave yesterday can be unspecialized to give the oft-needed Gamma quotient for 2F1[a,b,c]: [ Gamma(c) Gamma(c - b - a) ] [ 0 ------------------------- ] [ Gamma(c - a) Gamma(c - b) ] = [ ] [ 0 1 ] [ (k+c-a) (k+c-b) ] [ --------------- + k ] oo [ c-b-a+1 ] /===\ [ 2 (k + -------) ] | | [ 1 k-a+1 k-b+1 k+c-a k+c-b 2 ] | | [ - ----- ----- ------------- ----------- ------------------- ] | | [ c-b-a c-b-a+1 c-b-a ] k = 0 [ 4 k+1 k+c k + ----- + 1 k + ------- ] [ 2 2 ] [ ] [ 0 1 ] (The matrix input form is [(k-a+1)*(k-b+1)*(k+c-a)*(k+c-b)/(4*(k+1)*(k+c)*(k+(c-b-a)/2+1)*(k+(c-b-a+1)/2)), ((k+c-a)*(k+c-b)/(2*(k+(c-b-a+1)/2))+k)/(c-b-a);0,1]) --rwg

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[math-fun] <rational>!
by Bill Gosper 04 Jul '07

04 Jul '07

(Some of my eavesdroppers' ISPs seem to be blacklisting osots.com, so I'm sending from Yahoo, and setting Rich another chore.) Does anyone know how to widen Yahoo's composition window to a usable width? Meanwhile, I'll try an attachment. If it linebreaks that too, I'll switch to gmail. (Sorry, Rich!) --rwg ____________________________________________________________________________________ Pinpoint customers who are looking for what you sell. http://searchmarketing.yahoo.com/ Rich's old Hakmem (http://www.inwap.com/pdp10/hbaker/hakmem/geometry.html) ####### ITEM 1 (Schroeppel): (1/3)! and (2/3)! are interexpressible. (1/4)! and (3/4)! are interexpressible. Thus these two pairs are of dimensionality one. (1/10)! and (2/10)! are sufficient to express (N/10)! for all N. (1/12)! and (2/12)! are sufficient to express (N/12)! for all N. (1/3)! and (1/4)! are sufficient to express (N/12)! for all N. Thus the three cases above are of dimensionality two. PROBLEM: Find some order to this dimensionality business. The reflection and multiplication formulas: pi Z Z! (-Z)! = --------- sin(pi Z) (N-1)/2 -NZ-1/2 (2 pi) N (NZ)! = Z! (Z-1/N)! (Z-2/N)! ... (Z-(N-1)/N)! ######## can now be rephrased as an applied math problem: Write a simplfier that recognizes when a monomial of fractional factorials is elementary. E.g., 1 2 7/8 (sqrt(6) + sqrt(2)) %pi (--)! (-)! 4 3 sqrt(-----------------------) 12 3 2 ---------- = ------------------------------------. 1 27 (-)! 4 I know of no CAS that can do this. (But haven't tried Mma 6.0). Presumably from its lack of a D.E., we "know" Gamma has no nice series. But the above implies 1/8 %pi 1 1 3 sqrt(-----------------) Beta(-, -) sqrt(6) + sqrt(2) 4 3 sqrt(---------------------------------------) 1 2 (-)! = ---------------------------------------------, 4 2 further evidence that <rational>! = algebraic(hypergeometrics(rationals)). Lastly, I have a 105 page Table of the Gamma Function for Complex Arguments, National Bureau of Standards Applied Mathematics Series : 34 (Issued Agust 6, 1954). >From the preface: The table of the complex gamma function presented in this volume is of fundamental importance in both pure and applied mathematics. The tabulation was prompted by urgent and specific needs in the fields of atomic and nuclear research. [...] Does anyone know of a physics formula with a naked Gamma? I'm guessing that they really needed only Betas and similar Gamma quotients that are merely hypergeometric sums, but involving multiple arguments, making them impractical to tabulate. "Scientist don't need computers-- all we need are a few computers to make them tables." Refutation: oo [ 2 2 2 ] /===\ [ (k + a) (k + b) (2 k + a) (2 k + b) - k ] | | [ ------------------------------------------ ------------------------ ] | | [ 2 b + a b + a + 1 b + a ] | | [ 4 (k + 1) (k + ----- + 1) (k + ---------) ] k = 0 [ 2 2 ] [ ] [ 0 1 ] [ 1 ] [ 0 ---------- ] = [ Beta(a, b) ] [ ] [ 0 1 ] (To hell with hypergeometric notation, but this is essentially a 4F3[1/4] that improves on some F[1/8]/F[1/8]s in Gosper, Strip mining in the abandoned orefields of Nineteenth Century mathematics, in Computers in Mathematics (Lecture Notes in Pure and Applied Mathematics, vol. 125), D. Chudnovsky & R. Jenks, eds. pp 261-283, m. dekker, N. Y.) I.e., a decently convergent series, but of two arguments, so you can't tabulate it for complex a,b. --rwg Video games are a feminist plot to render boys academically noncompetitive.

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