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[math-fun] Continuous derivations on the reals
by Dan Asimov 19 Jun '07

19 Jun '07

Marc wrote: << Is there term for when f(a b) = a f(b) + b f(a)? And the answer, supplied by Neil, is "derivation". Okay, sure, if say C^oo(R) is the set of infinitely differentiable functions from R to R, then D: C^oo(R) -> C^oo(R) defined by D(f)(x) = f'(x) is perhaps the epitome of a derivation. -------------------------------------------------------------------- But what are the continuous derivations d: R -> R ? I.e., for all real u,v, we must have d(uv) = d(u)*v + u*d(v) The 0 function works. What is the general solution? --Dan

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Re: [math-fun] Continuous derivations on the reals
by Dan Asimov 19 Jun '07

19 Jun '07

Ki Song wrote: << I thought 'derivation' was restricted to linear operators. Perhaps these operators should be called something like quasi-derivation, or pseudo-derivation? >> Derivations are defined for rings, fields, and algebras. Cf. http://en.wikipedia.org/wiki/Derivation_%28abstract_algebra%29 . --Dan

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Re: [math-fun] term for f(a b) = a f(b) + b f(a)?
by N. J. A. Sloane 16 Jun '07

16 Jun '07

Marc asked: Is there term for when f(a b) = a f(b) + b f(a)? If not, what would you suggest? This is called a "derivation" and is better written f(a b) = a f(b) + f(a) b See e.g. Bourbaki, Lie Groups, Vol 1 , Chap 1, section 1, para 1, page 2 ! Neil

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Re: [math-fun] Triangle 1/7th of generating ABC triangle
by Dan Asimov 16 Jun '07

16 Jun '07

ERic wroter: << Draw a triangle ABC (clockwise labels) with each side devided into 3 equal parts (labels A,A1,A2,B,B1,B2,C,C1,C2): AA1=A1A2=A2B, BB1=B1B2=B2C, CC1=C1C2=C2A. Now draw AB1, BC1, CA1 -- those lines shape an "inside" triangle "I" having no common point with ABC. Question: Does an ABC triangle exist such that the surface of the inner "I" triangle is exactly 1/7th of ABC's surface? >> This looks to me like putting the answer before the question! A slightly more general problem, and fun to solve, is: Suppose 0 <= p <= 1. Given a triangle ABC, mark the 3 points that are the fraction p of the way from A to B; from B to C; and from C to A. Now draw the segment from each of A,B,C to the marked point on the opposite side. Express the area of the interior triangle thus created, as a fraction F(p) of the area of ABC ? (As a check, F(1/3) = 1/7.) --Dan

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[math-fun] term for f(a b) = a f(b) + b f(a)?
by Marc LeBrun 16 Jun '07

16 Jun '07

A function f when satisfying f(a b) = f(a) f(b) is described as "multiplicative". Is there term for when f(a b) = a f(b) + b f(a)? If not, what would you suggest? (I was thinking of saying "derivative", but maybe that's too confusing?) Thanks!

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[math-fun] New smallest known multiplicative magic cube
by Christian Boyer 14 Jun '07

14 Jun '07

The smallest possible multiplicative magic cube (here smallest = using the smallest possible integers) is an unsolved problem. It is also a good factorization problem. Until now, the best known cube used integers <= 416, in a 4x4x4 cube. A 3x3x3 cube can't be "smaller", because the best possible 3x3x3 use 900. Here is the NEW best known cube: it uses integers <= 364, again in a 4x4x4 cube. You can check that its integers are distinct, that its rows, columns, pillars and 4 main diagonals give the same product P = 17 297 280, each line being a different factorization of this same P. It was not asked (it is not a "perfect" magic cube), but a supplemental property: some small diagonals give again the same product. 52 168 15 132 36 55 273 32 231 12 96 65 40 156 44 63 42 11 234 160 260 144 3 154 8 182 220 54 198 60 112 13 165 72 16 91 56 26 264 45 312 120 21 22 6 77 195 192 48 130 308 9 33 84 80 78 30 66 39 224 364 24 18 110 Is it possible to construct a multiplicative magic cube with smaller integers? Christian. www.multimagie.com/indexengl.htm

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[math-fun] LATimes: Mr. Wizard dead at 89
by Henry Baker 12 Jun '07

12 Jun '07

FYI -- For those of us in the U.S. of a certain age, Mr. Herbert was "the" wizard. http://www.latimes.com/news/obituaries/la-ex-herbert12jun13,0,4867128.story >From the Los Angeles Times Don Herbert, 89; TV's 'Mr. Wizard' taught science to young baby boomers By Dennis McLellan Times Staff Writer 7:03 PM PDT, June 12, 2007 Don Herbert, who explained the wonderful world of science to millions of young baby boomers on television in the 1950s and '60s as "Mr. Wizard" and did the same for another generation of youngsters on the Nickelodeon cable TV channel in the 1980s, died Tuesday. He was 89. Herbert died at his home in Bell Canyon after a long battle with multiple myeloma, said Tom Nikosey, Herbert's son-in-law. A low-key, avuncular presence who wore a white dress shirt with the sleeves rolled up and a tie, Herbert launched his weekly half-hour science show for children on NBC in 1951. Broadcast live from Chicago on Saturdays the first few years and then from New York City, "Watch Mr. Wizard" ran for 14 years. Herbert used basic experiments to teach scientific principles to his TV audience via an in-studio guest boy or girl who assisted in the experiments. "I was a grade school kid in the '50s and watched 'Mr. Wizard' Saturday mornings and was just glued to the television," said Nikosey, president of Mr. Wizard Studios, which sells Herbert's science books and TV shows on DVD. "The show just heightened my curiosity about science and the way things worked," Nikosey said. "I learned an awful lot from him, as did millions of other kids." By 1955, there were about 5,000 Mr. Wizard Science Clubs nationwide, with more than 100,000 members. And as "Mr. Wizard," Herbert was a true TV star, who was featured in an array of magazines, including TV Guide, Life, Time, Newsweek, Science Digest, Boy's Life and even Glamour. Herbert was taken aback by the show's success. "What really did it for us was the inclusion of a child," he told the St. Louis Post-Dispatch in 2004. "When we started out, it was just me up there alone. That was too much like having a professor give a lecture. We cast a boy and girl to come in and talk with me about science. That's when it took off. "The children watching could identify with someone like them." In explaining how he brought a sense of wonder to elementary scientific experiments, Herbert told the New York Times in 2004 that he "would perform the trick, as it were, to hook the kids, and then explain the science later. "We thought we needed it to seem like magic to hook the audience, but then we realized that viewers would be engaged with just a simple scientific question, like, why do birds fly and not humans? A lot of scientists criticized us for using the words 'magic' and 'mystery' in the show's subtitle, but they came around eventually." "Watch Mr. Wizard" garnered numerous honors, including a Peabody award, four Ohio State awards and the Thomas Alva Edison Foundation award for "Best Science TV Program for Youth." And Herbert had a lasting effect. "Over the years, Don has been personally responsible for more people going into the sciences than any other single person in this country," George Tressel, a National Science Foundation official, said in 1989. "I fully realize the number is virtually endless when I talk to scientists," he said. "They all say that Mr. Wizard taught them to think." Herbert's experiments on the show typically used household items. As a 1951 Time magazine story noted: "Herbert's object is to show his audience what goes on in the world Â why the wind blows, what makes a cake rise, how water comes out of a kitchen tap. "To explain rain, he boils water in a coffee pot, compares the steam to clouds, and shows how 'rain' will condense on the sides of a glass held over the spout." Not every Mr. Wizard experiment went according to plan. In "Saturday Morning TV," a 1981 book by Gary H. Grossman, Herbert recalled pouring two colorless solutions into one glass and then announced that the solution would turn black before he counted to nine. "I got up to 20 and decided I'd better stop," he recalled. "I explained that apparently other factors like temperature and acidity had interfered with the experiment." But as he finished his explanation, the liquid changed color. "It was embarrassing, certainly, but I discovered the answer," he said. "We hadn't used a fresh solution, so the reaction was slower than expected." After "Watch Mr. Wizard" ended its 14-year-run in 1965, Herbert showed up frequently on talk shows, including "The Tonight Show" and "Late Night With David Letterman." "Watch Mr. Wizard" was revived on NBC in 1971 for a season, and "Mr. Wizard's World" ran on Nickelodeon from 1983 to 1990. Born July 10, 1917, in Waconia, Minn., Herbert grew up in La Crosse, Wis. He graduated from LaCrosse State Teachers College in 1940 and could have taught English or general science Â his majors Â but he recalled later that he was more interested in the theater. He worked as an actor and stagehand in the Minnesota Stock Company before moving to New York City in 1941. A year later, he volunteered for the Army Air Forces. As a B-24 bomber pilot, he flew 56 missions over Italy, Germany and Yugoslavia and received the Distinguished Flying Cross and the Air Medal with three oak-leaf clusters. Herbert wrote several books, including "Mr. Wizard's Supermarket Science" and "Mr. Wizard's Experiments for Young Scientists." In recent years, he helped set up his website, www.mrwizardstudios.com. He is survived by his wife of 34 years, Norma; his two sons and a daughter from his first marriage, Jay and Jeffrey and Jill Rogers; his stepdaughters, Kendra Jeffcoat and Kris Nikosey; his stepson, Kim Kasell; and 13 grandchildren. The family plans to hold a private memorial service. dennis.mclellan(a)latimes.com

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[math-fun] Primogeniture question
by Dan Asimov 12 Jun '07

12 Jun '07

Let f(X) be a nonconstant polynomial in Z[X] not of the form f(X) = g(X) h(X) where g(X), h(X) are in Z[X] - {1,-1}. Does there necessarily exist an integer N such that f(N) is a (positive or negative) prime number ? If so, is there known to be a minimum number m(d) of such N, where d = deg(f) ? (What if f(X) is further assumed to be monic?) --Dan

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Re: [math-fun] Primogeniture question
by Dan Asimov 11 Jun '07

11 Jun '07

Ed wrote: << See http://demonstrations.wolfram.com/TheBouniakowskyConjecture/ In 1857, Bouniakowsky conjectured that if f(x) is an irreducible polynomial, f(x)/gcd(f(0),f(1)) generates an infinite number of primes. It's an unsolved question. >> Wow, that's fascinating. Thanks, Ed and everyone for your answers. (Btw, if on the other hand an integer polynomial of degree d take prime values at 2d+1 integers, then it must be irreducible, I think.) --Dan

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Re: [math-fun] Primogeniture question
by Michael Reid 11 Jun '07

11 Jun '07

X^2 + X + 4 will never produce prime values. mike > Let f(X) be a nonconstant polynomial in Z[X] > not of the form > > f(X) = g(X) h(X) > > where g(X), h(X) are in Z[X] - {1,-1}. > > Does there necessarily exist an integer N such that > f(N) is a (positive or negative) prime number ? > > If so, is there known to be a minimum number m(d) > of such N, where d = deg(f) ? > > (What if f(X) is further assumed to be monic?) > > --Dan >

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