September 2005 - mathfun@mailman.xmission.com

Re: [math-fun] 177 million tons of ice???
by Henry Baker 21 Sep '05

21 Sep '05

Yes, I think that this is on purpose, since the original defn of a ton was used for the displacement of water in a ship. According to www.thefreedictionary.com/ton , a "ton" is also a unit of _volume_ in shipping -- either 35, 40 or 100 cubic feet. 35 cubic feet is approx one cubic meter. I recall from high school chemistry that a cubic foot of water is 64#, so 64*35 = 2240#, which is the definition of a "long ton". At 10:04 AM 9/20/2005, wrote: >Henry wrote: > ><< >177 million tons should be the weight of a small comet or a decent >size meteor (see <http://www.meteorcrater.com/eventsfun/exptheimp.htm>www.meteorcrater.com/eventsfun/exptheimp.htm) I'm >not sure that whether the meteor was iron or water would make much >difference in the resulting crater. >>> > >In discussions about this, a surprising factoid I learned is that >the volume of 1 ton of ice is almost exactly 1 cubic meter. >This is an easy rough estimate, but I'd never thought of doing it >before. At first blush, that a ton of ice should be a mere cubic meter >was amazing to me. (Either a ton is a lot smaller than it seems, >or a cubic meter is much bigger.) > >--Dan

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Re: [math-fun] 177 million tons of ice???
by Steve Rowley 21 Sep '05

21 Sep '05

Just put a magnet in your water bottle and you're ready to sell "ortho-water bottles", since the magnetic field will make the ortho polarization energetically favored. Slap a little sermon on the side about statistical thermodynamics and the magical incantation "exp(-uH/kT)", and it might even become trendy. "Refrigerate for higher polarization", that sort of thing. You could even sell "after-market ortho-water bottle adapters" (aka magnets), to tape onto existing water bottles. Maybe people will think they can only drink it while facing north. [Ok, that's enough flirting with The Dark Side for me this month. :-] >Date: Wed, 21 Sep 2005 13:58:09 -0400 >From: Tom Knight <tk(a)csail.mit.edu> > >I see a market opportunity. Think about the ad campaign for *pure* >ortho water. >You *do* want all of your water be straight, don't you! > >On Sep 21, 2005, at 1:53 PM, Eugene Salamin wrote: >> I found a great web site on water: http://www.lsbu.ac.uk/water/ . [...] >> In addition to the isotopic molecular species due to H, D, and the >> radioactive T, and also the stable O-16, O-17, and O-18, there is >> another class of distinct molecules that differ in the alignment of >> their nuclear spins. Considering only H2O, the H nuclei (protons) have >> spin 1/2. In ortho-water, the two spins are parallel, resulting in >> total nuclear spin 1, while in para-water, the spins are antiparallel, >> resulting in total nuclear spin 0. The equilibrium ratio is all para >> at 0 K, and 3:1 ortho:para at high temperatures (> 50 K). The >> equilibration time is about 1 hour in liquid water and several months >> in ice. It appears possible to separate ortho and para-water, but I >> can't say more because I don't have access to the journal articles from >> here. [...] >> Similar considerations apply to the hydrogen molecule H2. When >> hydrogen gas is liquified, it retains the 3:1 ratio. As ortho-hydrogen >> slowly converts to para-hydrogen at cryogenic temperatures, the energy >> released causes a substantial evaporization of the liquid. For this >> reason, a catalyst is used to quickly convert the hydrogen to para >> form. This can double the storage lifetime of the liquid. Strangely enough, spin-polarized hydrogen has occasionally been mentioned as a very dense energy storage mechanism. Its instability makes it more of a bomb than a gas tank, alas. -- Steve Rowley <sgr(a)alum.mit.edu> http://alum.mit.edu/www/sgr/ ICQ: 52-377-390

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RE: Antwort: [math-fun] Floor function question
by dasimov＠earthlink.net 21 Sep '05

21 Sep '05

Hartmut writes: << The real numbers are characterised as being the unique totally ordered field, s.t. for every number x there exists a natural number n (= 1+...+1) with n>x. The set of all n with n>x has a unique minimum m =: ceiling(x). floor(x) is then defined in the usual way. A total ordering is necessary (complex numbers), that every number is eventually surpassed by the naturals is necessary as well (surreal numbers). >> Wouldn't *any* subfield F of the reals R also be a totally-ordered field satisfying the same condition? --Dan

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[math-fun] Floor function question
by David Wilson 21 Sep '05

21 Sep '05

Starting with a field-based axiomatization of the real numbers, how do we show that floor(x) exists for every x? -------------------------------- - David Wilson

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Re: [math-fun] 177 million tons of ice???
by dasimov＠earthlink.net 20 Sep '05

20 Sep '05

Tom writes: << The molecular size and charge distribution is almost identical, so the crystals form without distinguishing between H2O and DH2O. It would be rare to find any significant number of molecules of D2O. DH2O ice does not sink, in any case. >> I'm confused. Does DH2O mean DHO (one deuterium, one hydrogen, one oxygen)? (If not, please define. If so, why the 2 ?) --Dan

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Re: [math-fun] 177 million tons of ice???
by dasimov＠earthlink.net 20 Sep '05

20 Sep '05

Victor writes: << The Times has printed a correction: A front-page article on Saturday about mistakes by the Federal Emergency Management Agency after Hurricane Katrina referred incorrectly to the area being overseen by the agency's federal coordinating officer, William Lokey. It also misstated the amount of ice that had been delivered to the region by Friday. Mr. Lokey oversees Louisiana, not the three-state region. Relief workers delivered 177.6 million pounds, not tons. >> Yes, and not a second too soon. (3 1/3 days after I reported this error to both corrections(a)nytimes.com and to nytnews(a)nytimes.com around midnight Fri/Sat.) --Dan

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[math-fun] 177 million tons of ice???
by dasimov＠earthlink.net 20 Sep '05

20 Sep '05

A front-page Sat., Sept. 17 story on the NY Times website, found at http://snipurl.com/hqxi, claims that FEMA has delivered 177 million tons of ice to Louisiana, Mississippi and Alabama in the aftermath of hurricane Katrina. Is 177 million tons a credible amount? Seems very unlikely to me, but I'd like to know what others think. --Dan

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Re: [math-fun] 177 million tons of ice???
by dasimov＠earthlink.net 16 Sep '05

16 Sep '05

Thanks, Jud and Ed. Jud wrote: << Impossible, as stated. The limit a semi can carry is on the order of 30,000-50,000 pounds. Over 7 million truckloads? >> That's how I convinced myself, too -- est. 8.5 million truckfuls. (As I wrote in an e-mail to the Times, before deciding to ask math-fun in case I was missing something.) --Dan

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[math-fun] RE: A ternary notation
by franktaw＠netscape.net 16 Sep '05

16 Sep '05

"Eric Angelini" <keynews.tv(a)skynet.be> wrote: > Hello SeqFan and Math-Fun, > Consider this (hope this is not old-hat): > > Â 3^0 Â 3^1 Â 3^2 Â 3^3 Â 3^4 Â 3^5 Â 3^6 Â 3^7 ... >= Â 1 Â Â 3 Â Â 9 Â Â 27 Â 81 Â 243 Â 729 Â 2187 ... > >This seq. is known to be very efficient if you >want to weigh integer weights with a two-tray >balance, leaving no "holes" behind; use weights >of 1,3,9,27... units to measure all "natural" >quantities, from 1 to infinity: > >I had the idea, yesterday night, to represent >all natural numbers in the same way -- all I >had to do was to use three symbols : > Â 0 for a power of 3 I don't need > Â 1 for a power of 3 I need > Â 2 as a grahic symbol meaning "minus all the > Â Â rest" This is an old idea, but there is a conceptually much simpler approach to the third digit: it is the digit -1, representing putting just this weight in the other pan. The result is called "balanced ternary" - I first encountered it in Knuth's The Art of Computer Programming (I forget which volume). If you use 2 to represent the -1 digit, you will get: 0, 1, 12, 10, 11, 122, 120, 121, 102, 100, 101, ... This sequence is also not in the OEIS. -- Franklin T. Adams-Watters 16 W. Michigan Ave. Palatine, IL 60067 847-776-7645 __________________________________________________________________ Switch to Netscape Internet Service. As low as $9.95 a month -- Sign up today at http://isp.netscape.com/register Netscape. Just the Net You Need. New! Netscape Toolbar for Internet Explorer Search from anywhere on the Web and block those annoying pop-ups. Download now at http://channels.netscape.com/ns/search/install.jsp

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[math-fun] A ternary notation
by Eric Angelini 15 Sep '05

15 Sep '05

Hello SeqFan and Math-Fun, Consider this (hope this is not old-hat): 3^0 3^1 3^2 3^3 3^4 3^5 3^6 3^7 ... = 1 3 9 27 81 243 729 2187 ... This seq. is known to be very efficient if you want to weigh integer weights with a two-tray balance, leaving no "holes" behind; use weights of 1,3,9,27... units to measure all "natural" quantities, from 1 to infinity: 1 = 1 2 = 3-1 3 = 3 4 = 3+1 5 = 9-(3+1) 6 = 9-3 7 = 9-(3-1) 8 = 9-1 9 = 9 10 = 9+1 11 = 9+(3-1) 12 = 9+3 13 = 9+(3+1) 14 = 27-[9+(3+1)] etc. I had the idea, yesterday night, to represent all natural numbers in the same way -- all I had to do was to use three symbols : 0 for a power of 3 I don't need 1 for a power of 3 I need 2 as a grahic symbol meaning "minus all the rest" ... and use a regular addition rule (which will be explained below in a few seconds). Let's make first a parallel with the binary no- tation : 2^0 2^1 2^2 2^3 2^4 2^5 2^6 2^7 ... = 1 2 4 8 16 32 64 128 ... We use two symbols : 0 for a power of 2 we don't need 1 for a power of 2 we need ... and an easy addition rule. We start from the left, putting 0's and 1's where we want to, and then reverse the sequence of 0's and 1's. To represent 54 (my age) we look for the closest power < or = to 54 and mark it with a "1": 2^0 2^1 2^2 2^3 2^4 2^5 2^6 2^7 ... = 1 2 4 8 16 32 64 128 ... --> 1 54 - 32 = 22; we mark with another "1" the closest power < or = to 22: 2^0 2^1 2^2 2^3 2^4 2^5 2^6 2^7 ... = 1 2 4 8 16 32 64 128 ... --> 1 1 54 - (32+16) = 6, we mark accordingly 2^2: 2^0 2^1 2^2 2^3 2^4 2^5 2^6 2^7 ... = 1 2 4 8 16 32 64 128 ... --> 1 1 1 54 - (32+16+4) = 2, we mark 2^1: 2^0 2^1 2^2 2^3 2^4 2^5 2^6 2^7 ... = 1 2 4 8 16 32 64 128 ... --> 1 1 1 1 To complete the binary representation of 54, we put 0's under all unnecessary powers: 2^0 2^1 2^2 2^3 2^4 2^5 2^6 2^7 ... = 1 2 4 8 16 32 64 128 ... --> 0 1 1 0 1 1 ... last line is reversed to produce 110110, which is indeed the binary expression of 54. We can represent all natural integers with this method, of course: 2^0 2^1 2^2 2^3 2^4 2^5 2^6 2^7 ... = 1 2 4 8 16 32 64 128 ... 1 = 1 => 1 2 = 0 1 (to reverse)=> 10 3 = 1 1 => 11 4 = 0 0 1 => 100 5 = 1 0 1 => 101 6 = 0 1 1 => 110 7 = 1 1 1 => 111 8 = 0 0 0 1 => 1000 9 = 1 0 0 1 => 1001 etc. Let's do the same with the powers of 3; we will mark with a "1" the powers we need, with a "0" the ones we will discard and (novelty!) with a "2" the quantity we will substract (I will use the star symbol (*) to indicate hereunder an operation in base 10): 3^0 3^1 3^2 3^3 3^4 3^5 3^6 3^7 ... = 1 3 9 27 81 243 729 2187 ... 1 = 1 2 = *3-1* = 121 3 = 01 => 10 4 = *3+1* = 11 5 = *9-4* = 1211 6 = *9-3* = 1210 7 = *9-2* = 12121 8 = *9-1* = 1201 9 = 001 => 100 10 = *9+1* = 101 11 = *9+2* = 1121 12 = *9+3* = 110 13 = *9+4* = 111 14 = *27-13* = 12111 15 = *27-12* = 12110 16 = *27-11* = 121121 17 = *27-10* = 12101 18 = *27-9* = 12100 19 = *27-8* = 121201 20 = *27-7* = 1212121 21 = *27-6* = 121210 22 = *27-5* = 121211 23 = *27-4* = 12011 24 = *27-3* = 12010 25 = *27-2* = 120121 26 = *27-1* = 12001 27 = 0001 => 1000 28 = *27+1* = 1001 29 = *27+2* = 10121 30 = *27+3* = 10010 etc. Is this of interest? Is the introduction of a third symbol "economi- cally" efficient? (I think we will use less symbols to represent all first 1000 natural integers with this "ternary notation" than with the usual binary system -- the price might be to high, though)... Speaking of "notation efficiency" (how to re- present in the less "symbol-consuming" way all the natural numbers) -- is there a definitive answer? The "ternary notation" above can certainly be bettered as the "ternary artefacts" 20, 1220, 001 or 002 represent nothing. Best, É. (seq. 0,1,121,10,11,1211,1210,12121,1201,100,... is not in the OEIS)

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