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[math-fun] Multiplicative squares
by Christian Boyer 30 Sep '05

30 Sep '05

Bonjour, As said yesterday, 302400 is the smallest possible product for 5x5 multiplicative squares. And 332640 is another new possible product smaller than our previous example with 362880. Here is one of the numerous possible examples with 302400: 12 35 1 40 18 36 2 24 7 25 14 45 15 4 8 5 16 42 30 3 10 6 20 9 28 coming from the family: aab cd 1 aaac abb aabb a aaab d cc ad bbc bc aa aaa c aaaa abd abc b ac ab aac bb aad magic product a^6 * b^3 * c^2 * d (example with a=2, b=3, c=5, d=7 for 302400, but of course the same family can be used for 423360, 475200, ...) Other numerous families are possible with the same magic product a^6 * b^3 * c^2 * d. Some words on the method used for nxn multiplicative squares. For each "good" P (means with several little factors), the program computes all the magic series of n distinct integers having the product P. For example with n=5: P=302400 has 3527 series, 332640 has 3661 series, 362880 has 3734 series. Then it analyzes combinations of n series looking if it is possible to get n series using n² distinct integers. If n series are found, they are the n rows: arrange them trying to get n columns and 2 diagonals. No great secret in the method... A specific page on multiplicative squares will be added in the next update of www.multimagie.com/indexengl.htm, planned beginning of October. Christian. ------------------------------------------ The 10 smallest magic products for 5x5 are: Prod. = 2^ * 3^ * 5^ * 7^ * 11^ *13^ 302400 6 3 2 1 0 0 << The SMALLEST 332640 5 3 1 1 1 0 362880 7 4 1 1 0 0 << Previously known 393120 5 3 1 1 0 1 << id 332640 403200 8 2 2 1 0 0 415800 3 3 2 1 1 0 423360 6 3 1 2 0 0 << id 302400 443520 7 2 1 1 1 0 475200 6 3 2 0 1 0 << id 302400 491400 3 3 2 1 0 1 << id 415800 ------------------------------------------

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Re: [math-fun] NYTimes: melting ice cubes cause water torise ?
by dasimov＠earthlink.net 30 Sep '05

30 Sep '05

Dirk writes: << If you consider the spherical shape of the earth and the gravity not pointing in the same direction everywhere, I think it will be different (but I did no calculations). In the extreme case, when all the surface of a water planet is covered by ice, the ice will not sink at all because the gravity cancels out everywhere. When this ice block melts, the water level will rise... >> Surely there is more involved in this spherical ice shell than mere floating! (Imagine a spherical shell of radius ~1 meter, with thickness = 1/10 meter, made of iron, and now cut it into, say, 4 spherically-curved 120-degree triangular slabs, leaving them in position. Now imagine trying to push all 4 pieces towards the center at the same time. They will get in each others' way.) If instead we chop up the spherical shell of ice into pieces -- and get rid of some ice so that [the cones on the pieces from the earth's center] are mutually disjoint -- then starting from this situation instead of from the complete spherical shell of ice -- then when *this* ice all melts, the water level should not change perceptibly. I don't see any gravity cancellation here, or in the original situation. --Dan

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[math-fun] another kind of median (trying again)
by James Propp 30 Sep '05

30 Sep '05

The picture was supposed to show a rectangle of width 1 and height 2 whose bottom is centered at x=1, and to the right of it, a rectangle of width 1 and height 1 whose bottom is centered at x=2. The base of the first rectangle goes from x=1/2 to x=3/2, and the base of the second rectangle goes from x=3/2 to x=5/2. The total area under the histogram is (1)(2)+(1)(1) = 3. The area to the left of the line x=5/4 is (5/4-1/2)(2) = 3/2, which is half of the total area. So x=5/4 is the "median". Jim

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Antwort: Re: [math-fun] NYTimes: melting ice cubes cause water to rise ?
by dirk.lattermann＠bgs-ag.de 30 Sep '05

30 Sep '05

If you consider the spherical shape of the earth and the gravity not pointing in the same direction everywhere, I think it will be different (but I did no calculations). In the extreme case, when all the surface of a water planet is covered by ice, the ice will not sink at all because the gravity cancels out everywhere. When this ice block melts, the water level will rise... Dirk Henry Baker <hbaker1(a)pipeline.com>@mailman.xmission.com am 30.09.2005 05:52:51 Bitte antworten an math-fun <math-fun(a)mailman.xmission.com> Gesendet von: math-fun-bounces+dirk.lattermann=bgs-ag.de(a)mailman.xmission.com An: dasimov(a)earthlink.net, math-fun <math-fun(a)mailman.xmission.com> Kopie: Thema: Re: [math-fun] NYTimes: melting ice cubes cause water to rise ? That was the point of my original posting -- the NYTimes thought that melting floating ice would raise the water level -- they were obviously wrong. At 04:09 PM 9/29/2005, dasimov(a)earthlink.net, math-fun wrote: >(So what's the business about floating ice raising water levels when it melts?) > >--Dan _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] NYTimes: melting ice cubes cause water to rise ?
by Henry Baker 29 Sep '05

29 Sep '05

Let me see -- melting floating ice makes the water level rise? Isn't this a standard SAT question? I guess Mr Revkin's verbal SAT scores exceed his math SAT scores. ------------------ The New York Times September 29, 2005 In a Melting Trend, Less Arctic Ice to Go Around By ANDREW C. REVKIN The floating cap of sea ice on the Arctic Ocean shrank this summer to what is probably its smallest size in at least a century of record keeping, continuing a trend toward less summer ice, a team of climate experts reported yesterday. That shift is hard to explain without attributing it in part to human-caused global warming, the team's members and other experts on the region said. The change also appears to be headed toward becoming self-sustaining: the increased open water absorbs solar energy that would otherwise be reflected back into space by bright white ice, said Ted A. Scambos, a scientist at the National Snow and Ice Data Center in Boulder, Colo., which compiled the data along with the National Aeronautics and Space Administration. "Feedbacks in the system are starting to take hold," Dr. Scambos said. The data was released on the center's Web site, www.nsidc.org. The findings are consistent with recent computer simulations showing that a buildup of smokestack and tailpipe emissions of greenhouse gases could lead to a profoundly transformed Arctic later this century, when much of the once ice-locked ocean would routinely become open water in summers. Expanding areas of open water in the summer could be a boon to whales and cod stocks, and the ice retreat could create summertime shipping shortcuts between the Atlantic and the Pacific. But a host of troubles lie ahead as well. One of the most important consequences of Arctic warming will be increased flows of meltwater and icebergs from glaciers and ice sheets, and thus an accelerated >>rise in sea levels,<< threatening coastal areas. The loss of sea ice could also hurt both polar bears and Eskimo seal hunters. The Arctic ice cap always grows in the winter and shrinks in the summer. The average minimum area from 1979, when precise satellite mapping began, until 2000 was 2.69 million square miles, similar in size to the contiguous area of the United States. The new summer low, measured on Sept. 19, was 20 percent below that. Before 1979, scientists estimated the size of the ice cap based on reports from ships and airplanes. The difference between the average ice area and the area that persisted this summer was about 500,000 square miles, an area about twice the size of Texas, the scientists said. This summer was the fourth in a row with the ice cap areas sharply below the long-term average, said Mark C. Serreze, a senior scientist at the snow and ice center and a professor at the University of Colorado, Boulder. Dr. Scambos said the consecutive reductions in the ice cap "make it pretty certain a long-term decline is under way." A natural cycle in the polar atmosphere called the Arctic oscillation, which contributed to the reduction in Arctic ice in the past, did not appear to be a factor in the past several years, Dr. Serreze said. He said the role of accumulating greenhouse gas emissions had become increasingly apparent with rising air and sea temperatures. Still, many scientists say it is not yet possible to determine what portion of Arctic change is being caused by rising levels of carbon dioxide and other emissions from human sources and how much is just climate's usual wiggles. Dr. Serreze and other scientists said that more variability could lie ahead and that the area of sea ice could actually increase some years. But the scientists have found few hints that other factors, like more Arctic cloudiness in a warming world, will reverse the trend. "With all that dark open water, you start to see an increase in Arctic Ocean heat storage," Dr. Serreze said. "Come autumn and winter that makes it a lot harder to grow ice, and the next spring you're left with less and thinner ice. And it's easier to lose even more the next year." The result, he said, is that the Arctic is "becoming a profoundly different place than we grew up thinking about." Other experts on Arctic ice and climate disagreed on details. For example, Ignatius G. Rigor at the University of Washington said the change was probably linked to a mix of factors, including influences of the atmospheric cycle. But he agreed with Dr. Serreze that the influence from greenhouse gases had to be involved. "The global warming idea has to be a good part of the story," Dr. Rigor said. "I think we have a different climate state in the Arctic now. All of these feedbacks are starting to kick in and really snuffing the ice out by the end of summer." Other experts expressed some caution. Claire L. Parkinson, a sea ice expert at NASA's Goddard Space Flight Center in Greenbelt, Md., said a host of changes in the Arctic - including rising temperatures, melting permafrost and shrinking sea ice - were consistent with human-caused warming. But she emphasized that the complicated system was still far from completely understood. William L. Chapman, a sea ice researcher at the University of Illinois Urbana-Champaign, said it was important to keep in mind that the size of the ice cap could vary tremendously, in part because of changes in wind patterns, which can cause the ice to heap up against one Arctic shore or drift away from another.

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RE: [math-fun] Trimmed means and Multi-dimensional medians
by dasimov＠earthlink.net 29 Sep '05

29 Sep '05

Joshua writes: << . . . Here is another question about the median: is there a median that makes sense in two or more dimensions? Suppose (X,Y) ~ f(x,y) where f(x,y) is the continuous joint pdf of the random variables X and Y. Is there a reasonable quantity to call the median? >> For a *finite uniform* distribution, a reasonable way to generalize the 1D median (on R) to R^n is to use "convex peeling": do Take the convex hull of the data, then remove data on the boundary of its convex hull; while data remains. When eventually a removal leaves no remaining data, put these last points back and let their vector mean be the median. (For a continuous distribution, I suspect that by taking an arbitrarily large finite sample, the result of convex peeling should converge, with probability 1, to a point depending only on the original distribution. Better yet, there's probably some differential equation that implements this limit without resorting to samples. WPT ?) --Dan

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[math-fun] Re: Trimmed means and Multi-dimensional medians
by Thomas Colthurst 29 Sep '05

29 Sep '05

> Here is another question about the median: is there a median that makes > sense in two or more dimensions? Yes, the maximium likelihood point estimate of the center of the Lapace distribution I gave in my previous email makes sense in any dimension, and with suitable (positive definite) choice of S, will satisfy sum_i | x_i - mu | / ( x_i - mu ) = 0 > Here's an interesting question: suppose we have data X_1, ..., X_n drawn > from a Gaussian distribution with unknown mean > mu and known variance 1. We wish to estimate mu with a guess muhat. > Virtually everyone uses the sample mean of the dataset as an estimate of mu, > but note that mu is also the *median* of the distribution. Under what > circumstances would we be justified in prefering the sample median of the > data to estimate mu? If the data was drawn from a Laplace rather than a Gaussian distribution. > Since the sample average is a sufficient statistic, the > answer might be never, but I'm not sure. Might it be the case that that the > sample median is preferable if we are using L1 loss, i.e., seeking to > minimize E_mu |mu - muhat| ? As others have pointed out, the posterior distribution of mu is symmetric about the average, so any reasonable loss function (including L1 loss) is going to be minimized at the average. -Thomas C

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Re: [math-fun] NYTimes: melting ice cubes cause water to rise ?
by Henry Baker 29 Sep '05

29 Sep '05

The first line of the article says "The >>floating<< cap of sea ice on the Arctic Ocean shrank this summer..." At 07:48 AM 9/29/2005, ed pegg wrote: >Floating ice displaces an equal amount of water, true. > >However, the Arctic ice is partially anchored, and not >floating. > >Ed Pegg Jr > >--- Henry Baker <hbaker1(a)pipeline.com> wrote: > >> Let me see -- melting floating ice makes the water level rise? Isn't >> this a standard SAT question? I guess Mr Revkin's verbal SAT scores >> exceed his math SAT scores.

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[math-fun] Trimmed means and Multi-dimensional medians
by joshua sweetkind-singer 29 Sep '05

29 Sep '05

While we're at it, the "trimmed mean" is another way to combine the robustness of the median with the acuity of the mean. the k-trimmed-mean throws out the k most extreme data points (if I remember correctly, and probably k should be even) then takes the mean of the rest. The pure mean is then equivalent to the 0-trimmed mean, and the pure median is equivalent to the (n-1)-trimmed mean (where n is the number of data points in the set). Here's an interesting question: suppose we have data X_1, ..., X_n drawn from a Gaussian distribution with unknown mean mu and known variance 1. We wish to estimate mu with a guess muhat. Virtually everyone uses the sample mean of the dataset as an estimate of mu, but note that mu is also the *median* of the distribution. Under what circumstances would we be justified in prefering the sample median of the data to estimate mu? Since the sample average is a sufficient statistic, the answer might be never, but I'm not sure. Might it be the case that that the sample median is preferable if we are using L1 loss, i.e., seeking to minimize E_mu |mu - muhat| ? Here is another question about the median: is there a median that makes sense in two or more dimensions? Suppose (X,Y) ~ f(x,y) where f(x,y) is the continuous joint pdf of the random variables X and Y. Is there a reasonable quantity to call the median? -Joshua On 9/29/05, Mike Speciner <speciner(a)ll.mit.edu> wrote: > > So, if y(x) is the histogram, the median is the m such that > > integral(x<m) y(x) = integral(x>m) y(x) > > while the mean is the m such that > > integral(x<m) |x-m|*y(x) = integral(x>m) |x-m|*y(x) > > This suggests a whole family of averages (using various functions of > (x-m) for the weighting), though what use they might have escapes me. > > --ms > > > David Gale wrote: > > > > > Jim, what is the Propp median if there are m zeros and m fives (and zero > > everything else)? > > Dan, if you're going to bring in averages at all then why not go all the > > way and use THE average? But maybe the CDC was using some sort of hybrid > > like the one you suggest. > > > > D > > > > At 09:14 PM 9/28/2005, you wrote: > > > >> The picture was supposed to show a rectangle of width 1 and height 2 > >> whose bottom is centered at x=1, and to the right of it, a rectangle > >> of width 1 and height 1 whose bottom is centered at x=2. > >> > >> The base of the first rectangle goes from x=1/2 to x=3/2, and the base > >> of the second rectangle goes from x=3/2 to x=5/2. > >> > >> The total area under the histogram is (1)(2)+(1)(1) = 3. > >> > >> The area to the left of the line x=5/4 is (5/4-1/2)(2) = 3/2, which > >> is half of the total area. So x=5/4 is the "median". > >> > >> Jim > >> > >> _______________________________________________ > >> 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 >

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Re: [math-fun] Sex, Lies and Government Conducted Surveys
by Richard Schroeppel 29 Sep '05

29 Sep '05

(forwarded, with headers eviscerated -- Rich) From: Dylan Thurston <dthurston(a)barnard.edu> Date: Thu, 29 Sep 2005 11:37:07 -0400 On Wed, Sep 28, 2005 at 02:50:45PM -0400, Michael B Greenwald wrote: > .... For the purpose of a mathematical example, and not to cater to > stereotypical prejudices, let's assume that the men want to, and > willingly do, pair with every possible partner (the 20 women for the > lone heterosexual man, and the other 18 homosexual men for the 19 > homosexual men). The 20 monogamous women, each partner with the one > heterosexual man (presumably unaware that he's not monogamous). The > homosexual men each pair up with every possible homosexual partner. > Then we'd have a mean of 1 partner for the women, but a mean of > about 18.1 partners for the men. David quote the report imprecisely: if you look at the report, it asks for the number of female partners of males, and the number of male partners of females. The point about an unequal total population might still work, but wouldn't put the averages off by a factor of 2. Of course they're reporting median, not average. Peace, Dylan

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