August 2006 - mathfun@mailman.xmission.com

RE: [math-fun] Possibly naive question about algebraic numbers
by Andy Latto 29 Aug '06

29 Aug '06

> -----Original Message----- > From: math-fun-bounces+andy.latto=pobox.com(a)mailman.xmission.com > [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com]On > Behalf Of Daniel Asimov > Sent: Monday, August 28, 2006 11:48 PM > To: math-fun > Subject: Re: [math-fun] Possibly naive question about > algebraic numbers > > Also, given such an integer-polynomial function > P: C^n -> C^n such that its Jacobian at each of its > roots Z in C^n is non-singular: > > QUESTION: What can be said about the size of the > (necessarily) 0-dimensional set of such roots? If you ask the question in projective space, rather than in affine space, you get the answer you want, which is that provided all the roots are simple (that is, the Jacobian is non-singular), then the number of roots is exactly equal to the product of the degrees of the polynomials. If you ask the question in affine space, the best you can say is that the number of roots is <= this product, because you have left out the "roots at infinity". In two dimensions, this goes by the name of Bezout's theorem. Andy Latto andy.latto(a)pobox.com > > (Where of course a root of P means a Z in C^n where > P(Z) is the origin in C^n.) > > --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] Possibly naive question about algebraic numbers
by Daniel Asimov 29 Aug '06

29 Aug '06

Suppose we have not one integer polynomial in one complex variable, but n > 1 integer polynomials P_1, . . . P_n in n complex variables z_1, . . . , z_n. Set Z = (z_1, . . ., z_n) and P(Z) = (P_1(Z), . . ., P_n(Z)) in C^n, for any z_1, . . ., z_n in C. Suppose (*) Z_0 in C^n satisfies P(Z_0) = 0. ------------------------------------------------------------------- QUESTIONS: 1. Does (*) imply that the complex coordinates of Z_0 are all algebraic numbers ??? 2, If the answer to this is "not necessarily", what if we add the condition (**) det(JP(Z_0)) is non-zero, where JP is the n x n Jacobian matrix (df_j/dz_k (Z_0)), 1 <= i, j <= n ??? 3. If still no, then what kind of numbers do we get, anyway, esp. if we assume both (*) and (**) -- supposing we toss in all complex coordinates of all such "algebraic vectors" Z_0 for all n >= 1 ??? Or for a fixed n ??? Or for all n <= N, for some fixed N ??? Just curious. --Dan

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[math-fun] Postmarks
by Robert Baillie 25 Aug '06

25 Aug '06

You have probably received mail that has postmarks in the form of a square consisting of pixels, some of which are filled in. Nearest I can can count, the square is about 42 by 42 = 1764 pixels. If each pixel could be on or off, a lot of data could be encoded there! Does anybody know what sort of data is encoded there, and how it's stored? Bob Baillie

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[math-fun] Chess problem
by David Wilson 25 Aug '06

25 Aug '06

What is the largest possible number of plays by White from a legal chess position?

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RE: [math-fun] Chess problem
by Andy Latto 24 Aug '06

24 Aug '06

> -----Original Message----- > From: math-fun-bounces+andy.latto=pobox.com(a)mailman.xmission.com > [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com]On > Behalf Of Jud McCranie > Sent: Wednesday, August 23, 2006 8:30 PM > To: math-fun > > I get 201 with this position, but it isn't optimal > 201 > _ Q _ _ _ _ _ _ > _ _ _ Q _ _ _ Q > _ _ _ _ _ Q _ _ > Q _ _ _ _ _ _ _ > _ _ Q _ _ _ _ _ > K _ _ _ Q _ _ _ > _ R _ _ _ _ Q _ > k B _ Q _ _ _ _ Also, if we want to restrict ourselves to positions that can occur in actual play, this is not a possible position (what was black's last move?) Andy Latto andy.latto(a)pobox.com

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[math-fun] Egregious inaccuracy
by Fred lunnon 24 Aug '06

24 Aug '06

The (sometime Manchester) Guardian newspaper for 26th Aug 2006, on page 32 of its G2 section, reports that (sic) "A rock needs to have a mass of about 5,1020kg for gravity to give it a nice round planet-y sort of shape ..." Even for a newspaper with a noble tradition of embarrassing misprints, an error of 16 (presumed) orders of magnitude must surely be close to a record! WFL

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RE: [math-fun] Chess problem
by Andy Latto 24 Aug '06

24 Aug '06

> -----Original Message----- > From: math-fun-bounces+andy.latto=pobox.com(a)mailman.xmission.com > [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com]On > Behalf Of David Wilson > Sent: Wednesday, August 23, 2006 4:33 PM > To: math-fun > Subject: Re: [math-fun] Chess problem > > > The problem is to find a legal chess position with the > maximal number of > possible plays for White. > > The problem arose in trying to determine whether a two-computer chess > program could be played where the computers transmitted one > 8-bit byte per > move. The best encoding for moves seems to be to order all > possible moves > canonically (say lexically on start and destination square) > and send the > index of the move. If some position admits more than 256 > possible moves, > this scheme, and presumably the whole idea, fails. The idea can be patched in any number of ways, though. Reserve the byte value 255 to mean "I want to make a move that isn't in the first 255 possible moves from this position; I will transmit a second byte (and a third, in the almost certainly impossible situation that there are more than 511 moves from a single position) to further detail my move". If the protocol requires that the two computers alternate sending bytes for some reason, the special 255 byte can be conventionally replied to with a 0 byte, meaning "No move yet: continue specifying your move".

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[math-fun] A request to avoid receiving multiple identical posts
by Daniel Asimov 24 Aug '06

24 Aug '06

Dear Fellow Funsters, Normally, if possible, my e-address doesn't need to be on the recipients' e-address list if math-fun already is -- as has happened innumerable times in response to something I've posted. If you can spare the split-second needed to help me avoid receiving duplicate postings, I'd greatly appreciate it. Very recently, however, math-fun has been listed twice as an addressee, with the expected duplicate results. E.g., a very recent post has this addressee header: << To: math-fun <math-fun(a)mailman.xmission.com> Cc: math-fun <math-fun(a)mailman.xmission.com> >> Surely we can avoid this kind of thing if we just take a peek at who the addressees are before hitting Send. Thanks a lot, Dan

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[math-fun] Multiple posts
by Daniel Asimov 24 Aug '06

24 Aug '06

Actually, I meant to display *this* addressee header, which came very recently: << To: math-fun <math-fun(a)mailman.xmission.com> Cc: math-fun <math-fun(a)mailman.xmission.com>, math-fun <math-fun(a)mailman.xmission.com> >> --Dan

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Re: [math-fun] Chess problem
by Jud McCranie 24 Aug '06

24 Aug '06

At 08:05 PM 8/23/2006, Joshua Zucker wrote: >But in May there was a 517 move endgame win discovered! >One announcement of that is at >http://216.25.93.108/forum/viewtopic.php?t=2860 and it's also >mentioned in the wikipedia article. That's the one I was thinking about. They were assuming a max of 511 or 512, then that one was found.

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