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Re: [math-fun] Probabilistic harmonic series
by asimovd＠aol.com 09 May '03

09 May '03

Hi, Victor. Good to hear from you! And I'm glad at least one person's interested in the proof of analyticity. The gist of the proof is to show that the characteristic function (as explicitly described in the paper of Schmuland) of the distibution in question has sufficiently bounded tails. According to a theorem in functional analysis (which I had to dig up, being no functional analyst), this guarantees analyticity of the original density. How am I doing? I guess I could be doing better. After working for 5 years at NASA Ames Research center on visualization research, followed by a 1 1/2 year visit to AT&T Labs just as they opened in Florham Park, I've had trouble finding a job lately. So, recently I began tutoring H.S. students near Berkeley, where I live. Does your e-address mean you're working for IDA ? Regards, Dan

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Re: [math-fun] Probabilistic harmonic series
by asimovd＠aol.com 09 May '03

09 May '03

The question was, What is the distribution of the sum of the random harmonic series ±1 ± 1/2 ± 1/3 ± ... ? By amazing synchronicity, an article on this very subject, "Random harmonic series" by Byron Schmuland, appears in the May, 2003 issue of the American Mathematical Monthly and is available from the author's website. (Thanks, Neil, for pointing this out.) As E. Clark mentioned, the article's available for downloading at <http://www.stat.ualberta.ca/people/schmu/publications.html>. I haven't studied every detail of this paper, but it's very interesting, with a number of graphs of the distribution function and related functions (like the distribution function if the same question is asked about placing random signs in front of the series of *squares* of reciprocal integers). This paper rewards whatever level of perusal you feel like giving it. It uses a clever series-rearrangement trick to nail down the distribution function in question. The author proves that this distribution function is infinitely differentiable, though he wrote me in private e-mail that for 15 years before solving the question, he believed it would not even be differentiable at all. He wrote that he did not know whether it is real analytic. In case anyone is interested, I'll mention that I've proved it's in fact real analytic. --Dan

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[math-fun] Ferguson's sculptures
by David H Bailey 07 May '03

07 May '03

Our friend Ferguson got a nice plug in the Washington City Paper, current issue: DHB ---------------------- WASHINGTON CITY PAPER Washington's Free Weekly Vol. 23, No. 17, May 2-8, 2003 Washington, D. C. The Beauty of Phenomena: Art in the Communication of Science Over the years, the National Academy of Sciences has developed a knack for mounting eye-popping exhibitions of works that either embody or explain scientific principles. "The Beauty of Phenomena: Art in the Com- unication of Science" follows this pattern, featuring eight scientifically minded artists. Three, however, are particularly noteworthy: David S. Good- sell, author of several illustrated books, offers a series of eight watercolor panels that delve into the microscopic universe of the cell in a color palette so closely resembling that of a Tintin comic book that you half expect Snowy to poke his head out of the thicket of intracellular struc- tures. Physicist Sidney Nagel offers several memorable photographs (Emul- sion is pictured) that capture the graceful geometries of humble physical processes--drips falling from a faucet; a glass straw sucking up liquids. The most impressive oeuvre in the exhibit, however, may be that of sculp- tor Helaman Ferguson, who uses mathematical precepts to rigorously undergird his works in bronze and granite. While the underlying math will likely exceed the knowledge of many visitors, the resulting three-dimen- sional forms--some bulbous, some honeycombed, some elegantly stream- lined--dazzle the mind in their own right. On view from 9 a.m.to 5 p.m. Monday to Friday, to Sunday, September 14, at the National Academy of Sciences, 2100 C St. NW. Free. (202) 334-2436. (Louis Jacobson)

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[math-fun] problem with prime gap proof
by Thane Plambeck 07 May '03

07 May '03

http://www.siliconvalley.com/mld/siliconvalley/5804885.htm Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm

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Re: [math-fun] m & m's
by Henry Baker 06 May '03

06 May '03

Hersey's is obviously trying to rediscover Zipf's Law: BRN YEL RED ORG BLU GRN 30% 20% 20% 10% 10% 10% 31% 21% 16% 13% 10% 9% True Zipf's Law %'s Zipf's Law is "satisfying" to humans, because it has just the right combination of comfort and surprise -- the words of languages occur in an approximation to Zipf's Law. The popularity of TV channels is an approximation to Zipf's Law. The popularity of web pages is an approximation to Zipf's Law. At 04:19 PM 5/6/03 -0400, Thomas Colthurst wrote: >According to >http://global.mms.com/us/about/products/milkchocolate.jsp, >30% of M&M's are brown, >20% are yellow, >20% are red, >and orange, blue, and green all account for 10% each.

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Re: [math-fun] A proposed new chess rule
by Dan Hoey 06 May '03

06 May '03

Of course pinned pieces can check just like any others; I can't imagine players of real chess wanting it different. Still, a variant chess without check-pins might be fun. But then can a pinned piece pin another piece? If A pins B pins C pins D pins A, then I guess any one piece could move, but its ally would then be "really" pinned. I seem to recall a problem where there is a mate in N moves after 2N+2 checks, all by pinned pieces. Or maybe it was 2N+3 checks. Dan

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[math-fun] A proposed new chess rule
by asimovd＠aol.com 06 May '03

06 May '03

I don't think this comes up often in practice, but unless I'm mistaken one's king may not be on a square that is attacked by any of the opponent's pieces. AFAIK, this even includes the situation when the said opponent's piece is immobilized due to its being pinned against its own king. Thus a checkmate could occur even when the checkmate hinges on a king's being forbidden to escape to a square "attacked" by an opponent's piece that is at the time immobilized by a king pin. Is this correct? It won't change the course of chess very much, but if this is possible I think the rule should be changed so that the king is allowed to move to any unoccupied square unless the opponent could conceivably "take" the king on that square on the very next move. Comments? --Dan

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Re: [math-fun] m & m's
by Mike Stay 06 May '03

06 May '03

>From the FAQ on m-ms.com: Question What is the percent of each colour in M&M's® packs? Answer Hello, M&M's® Milk Chocolates are made up as follows: 31% Brown, 19% Red and 12.5% each of yellow, orange, green and blue. M&M's® Peanut and Crispy are made up as follows: 16.6% each of brown, yellow, red, blue, green or orange. Because we make so many at any given time, it is not likely that you will open a pack and get this exact percentage split of colours. To test if this split actually works you'd need about 5,000 M&M's® and that's a lot of M&M's® to eat ! Regards M&M's® -- Mike Stay staym(a)clear.net.nz http://www.xaim.com/staym

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Re: [math-fun] m & m's
by Thomas Colthurst 06 May '03

06 May '03

According to http://global.mms.com/us/about/products/milkchocolate.jsp, 30% of M&M's are brown, 20% are yellow, 20% are red, and orange, blue, and green all account for 10% each. I've seen more than one elementary school teacher assign the problem of estimating these proportions from empirical data (such as thane(a)best.com presents), but not of course the problem of whether the differences are statistically significant. I would guess that the color proportions are primarily determined by aesthetics (single blue or green splotches on a brown background look more appealing to me than brown splotches on a blue or green background). Also, history plays a role: the original M&Ms were apparently all brown, with red, green and yellow all added in 1960. [Orange was introduced in 1976, blue in 1995. Red was yanked in 1976 and reinstated in 1987. Purple is scheduled to be added to the standard mix sometime soon.] -Thomas C

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[math-fun] m & m's
by Thane Plambeck 06 May '03

06 May '03

My son complained that his bag of m&m's contained "all browns, and no blues." His evidence being unavailable (he ate them all), and suspecting exaggeration, I bought three bags at two locations with these results Milk Chocolate M&M's 1.69oz (47.9g) bag, 240 calories Package shows all colors but brown. BAG 1 6 May 2003 10:30am - Safeway, Middlefield Road, Palo Alto 20 red 16 brown 5 orange 7 yellow 6 green 4 blue 58 Total BAG 2A 6 May 2003, 11am -- Long's Drugs, downtown Palo Alto 21 brown 12 green 11 orange 6 yellow 4 blue 3 red 57 Total BAG 2B 6 May 2003, 11am - Long's Drugs, downtown Palo Alto 20 brown 9 yellow 9 blue 9 red 7 orange 3 green 57 Total Brown does seem to be overrepresented. Statistically significant, or merely tasty? Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm

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