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[math-fun] More on my game
by Jon Perry 29 Aug '03

29 Aug '03

A small mathematical aside - split a pack of cards into 2*26. In each pack, count the number of numbers where all 4 of the suits are present, e.g. in pack A, increase the count if say 3C, 3D, 3H and 3S are all present. What is the expected final count when both packs have been fully examined? Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] inversion in finite field ~= fourier transform?
by Mike Stay 29 Aug '03

29 Aug '03

I noticed this a few years ago and recently came across it again. In GF(2^n), inversion behaves very much like a change between conjugate observables; when you add a "delta function" h, inversion spreads it out over the whole spectrum: (x+h)^(-1) = (x+h)^(N) = sum (N choose k) x^(N-k) h^k. Is there some kind of equivalent of a gaussian that preserves form under addition and inversion? -- Mike Stay staym(a)clear.net.nz http://www.xaim.com/staym

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[math-fun]
by Jon Perry 28 Aug '03

28 Aug '03

You are invited to join the MathsNotMaths group, a group for Mathematicians to not talk about Maths, make announcements, etc... http://groups.yahoo.com/group/mathsnotmaths Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] New book: Mathematical Constants
by Steven Finch 28 Aug '03

28 Aug '03

Hello! I am pleased to announce the publication of my book, "Mathematical Constants", by Cambridge University Press (2003), ISBN 0-521-81805-2. It is an encyclopedia bursting with information and over 600 pages long. A table of contents and two sample essays are available (in PDF) at: http://us.cambridge.org/titles/catalogue.asp?isbn=0521818052 where one may also order a copy for $95. My website: http://pauillac.inria.fr/algo/bsolve/ has three additional sample essays from the book, as well as supplementary materials (which will continue to expand as time passes). Cambridge has done a beautiful job in producing my book. It is far more encompassing and detailed than my website ever was. By the way, please note my new e-mail address. My old e-mail address (at MathSoft) should no longer be used. As far as I know, there is no book like this! The breadth and depth of the coverage, as well as many exotic topics (e.g., from statistical mechanics and theoretical computer science) and unsolved problems, should make "Mathematical Constants" a joy to read. Thank you! Steve Finch _________________________________________________________________ Get MSN 8 and enjoy automatic e-mail virus protection. http://join.msn.com/?page=features/virus

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Re: [math-fun] Planet brightness problem
by asimovd＠aol.com 27 Aug '03

27 Aug '03

Jud wrote: << ... When is [a planet whose orbit is inside the Earth’s] the brightest? What function describes the brightness in terms of the angle between the Earth and the planet viewed from the Sun? (Assume that the orbits are circular and co-planar.) There could be two versions of this problem: (1) assume that all sunlight portions of the planet are equally bright as seen from the Sun (2) take into account the differences in brightness of parts of the planet (but still assume that all parts of the planet inherently reflect the same amount of sunlight). >> Some thoughts: Physically, maybe a fair way to answer this question is to say which position of the planet in its orbit reflects the maximum amount of the sun's energy to Earth. (Or more to the point, we could take a single point of earth at a given time on a given day of the year, and calculate the energy density of the sun's energy that reaches that point after reflecting from the planet.) But if we use the most basic model of reflectivity used in computer graphics, we assume a given surface's reflectivity characteristics is a convex combination of "diffuse" and "specular" reflection. Pure specular reflection is as though the surface were a perfect mirror (and is assumed to be independent of which colors comprise the incoming light ray). Pure diffuse reflection is assumed to reflect equally in (the hemisphere of) all directions leaving any point of the surface. (It's also assumed to have a reflectance spectrum given by assigning a number in [0,1] to each of R, G, and B -- since that's what CRT monitors are based on.) Light sources are usually modeled as point sources, which works fairly well. Another reasonable simplification that's used is the light source is assumed to be "at infinity" so that its rays are emitted all parallel to each other, making angle calculations easier. (The moon is often given as an example of a pure diffuse reflection -- as you look at the lunar disk of a full moon, the reflected light is virtually independent of whether you look, from the the center of the disk to right near its edge.) If we make the simplest assumption -- that the planet in question has pure diffuse reflectivity -- then a good approximation to how bright any point of it will look is proportional to 1/(d(S,P)^2 * d(P,E)^2), where d = distance, S = the sun, P = planet, E = Earth. But of course, only the side of the planet facing the sun will be seen from Earth, so the above brightness should be multiplied by the area of the lune of the planet's disk that is sunlit and can be seen from earth. For the other extreme -- pure specular reflectivity -- and continue assuming the sun is a point light source, then only the points of the planet whose line to the sun makes an equal & opposite angle with a line that meets the earth will appear to be illuminated at all. In this case only a typically small portion of the planet will reflect light that reaches the earth. (This will be approximately the result of taking the spherical disk of the planet's surface that is centrally projected to Earth, and moving each of its points halfway to the point Q on the planet that's closest to the sun.) It will look like an oval from the earth, possibly a bit kidney-shaped. Is there anyone out there who's willing to take on the task of filling in the details and programming this up to get a brightness function determined by the location of the center of a planet with respect to Earth and the sun? Then this function could be maximized when restricted to the planet's orbit. --Dan A.

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[math-fun] Thought for food
by Jon Perry 27 Aug '03

27 Aug '03

"What is blasphemy when there is no God?" "I always doubt myself" "The Ministry of Mis-Information does not exist" "I am therefore I think" "This sentence will explode in 10 seconds" "Logic is nonsense, and nonsense is logical" Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] Game with no strategy
by Jon Perry 27 Aug '03

27 Aug '03

I invented this card game while on holiday in France. It's a game for 3 players and uses a standard pack of cards. Deal 17 cards to each player and turn the last card face up. In this game, suits mean nothing. Each player lays down all of their cards that are the card in question, one higher and one lower. e.g. if the upturned card is a 3, then all 2's, 3's and 4's must be laid. Ace's wrap round, so if a king or ace is played, then KA2 and QKA are the valid cards to be laid down. Play begins with the player to the left of the dealer, and they choose 1 (exactly one) card from their hand. The other 2 players can then discard as before, but note that the person who laid the original card CANNOT discard any cards they may have rendered valid in this go. Play proceed anti-clockwise, and the first person to get rid of all their cards wins. Is there a winning strategy? The game can be played with more players, each dealt less cards, turning a card after dealing is optional, and a 'dead pack' can be created so that the cards played have (slightly) less significance. Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] Update on Math Exam flaps
by Henry Baker 27 Aug '03

27 Aug '03

FYI -- Testing has become a political football. If tests show _any_ discrepancy between different groups (ethnic, gender, etc.), then the politicians & teachers want to change the test until no discrepancies remain. Some test! Some math questions to add to the test: 1. What is the probability of 3 or more "hard questions" appearing sequentially in a test of 35 randomly chosen questions? 2. What is the probability of test averages showing changes _even when the underlying probabilities haven't changed_ ? 3. What is the probability that, after millions of tests, under every conceivable circumstance, by hundreds of different types of agencies, for a wide variety of purposes, that gender differences in math performance could be the result of "test bias" ? ----- http://www.nytimes.com/2003/08/27/nyregion/27REGE.html?pagewanted=print&pos… August 27, 2003 Trials of Regents Test Foresaw Failure at a High Rate By KAREN W. ARENSON .. ¶A string of especially tough problems partway through the test appeared to frustrate some students, who simply gave up on the exam. .. ¶Trigonometry, which was part of the state's Math A syllabus and of previous tests, was not included among the 35 questions on the test. But there were three items that used the Pythagorean theorem, leaving any students who were weak in that area at a disadvantage. .. Asked why the Education Department gave a test when it expected the average score to be so low, Alan Ray, a department spokesman, said, "The purpose of the test is to measure whether students meet the standards, not to make sure that a certain number of students pass or fail." .. ---- http://www.nytimes.com/2003/08/27/education/27SAT.html?pagewanted=print&pos… August 27, 2003 High School Seniors Get Highest SAT Math Scores in 35 Years By DIANA JEAN SCHEMO WASHINGTON, Aug. 26 The scores of high school seniors on the SAT math exam have climbed to their highest level in 35 years. Experts tied the gain to the growing number of students learning algebra in the eighth grade and taking more advanced courses in math after reaching high school. The average math score was 519, out of a possible 800. Educators around the country hailed the results of the 2003 SAT tests, released here today, as good news, with some reservations: the scores of students from rural areas and large cities continued to lag, and women continued to lag behind men, particularly in math. .. Today's results appeared to contrast sharply with the results of the ACT, another college entrance exam, released last week. Those results showed student scores holding steady, but at a level indicating that a majority of students were unprepared for college-level work in math or science. .. In addition, said Cyndie Schmeiser, the vice president for development at ACT, the two exams measure different kinds of learning. The ACT tests knowledge of the curriculum for a given subject, while the SAT more generally measures aptitude and the ability to reason. Unlike the SAT, the ACT results showed no gap between the performance of men and women. The average math SAT score of 519 represented a 16-point gain over the last decade, while the average verbal score of 507 represented a 7-point gain. ..

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[math-fun] photos of mathematicians at work
by Thane Plambeck 22 Aug '03

22 Aug '03

http://www.msri.org/media/photography/ed_alcock_spring_2003/index_html Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm

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[math-fun] ocean matrix
by Meeussen Wouter (bkarnd) 22 Aug '03

22 Aug '03

if aesthetics is important, then this is an important binary matrix : it=Nest[BlockMatrix[{{1-#, #},{ #, Reverse[Reverse/@ #] }}]&,{{1}}, n=4 ] g[{1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1}] g[{0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1}] g[{0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1}] g[{1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0}] g[{0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1}] g[{1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0}] g[{1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0}] g[{0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1}] g[{0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1}] g[{1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0}] g[{1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0}] g[{0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1}] g[{1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0}] g[{0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1}] g[{0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1}] g[{1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0}] look at it for n=4 .. 7 using Show[ListDensityPlot[Re@it],AspectRatio->1,Mesh->False] looks like ocean waves. it has Determinant = Ceiling[ 2^n /3] = EIS A005578 W. =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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