On 8/13/12, math-fun-request@mailman.xmission.com <math-fun-request@mailman.xmission.com> wrote:
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Today's Topics:
1. they have found 5 boxes of notes of Alexandre Grothendieck (Simon Plouffe) 2. Re: they have found 5 boxes of notes of Alexandre Grothendieck (Dan Asimov) 3. Re: peculiar 2x2 products (Bill Gosper) 4. MathJax: Beautiful math in all browsers (Henry Baker) 5. Re: MathJax: Beautiful math in all browsers (Robert Smith) 6. Re: MathJax: Beautiful math in all browsers (Joerg Arndt)
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Message: 1 Date: Sun, 12 Aug 2012 13:12:00 +0200 From: Simon Plouffe <simon.plouffe@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] they have found 5 boxes of notes of Alexandre Grothendieck Message-ID: <50278F80.1070607@gmail.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Hello,
there is one big news here circulating in France about Alexandre Grothendieck, they have found 5 boxes, 20000 pages of documents from him, written pages about Algebraic Geometry,
Alexandre Grothendieck is retired (sort of) in west-south france in the Pyren?es, for about 20 years, he won fields medal in the 60's and then retired completely in 68-69, refusing honnors and money,
There is a very good account of the life and mathematics of him here : http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-m... but this article is in french, I hope that google translate will do a decent job for those speaking english only,,
Bonne lecture,
Simon Plouffe
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Message: 2 Date: Sun, 12 Aug 2012 07:44:43 -0700 From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] they have found 5 boxes of notes of Alexandre Grothendieck Message-ID: <70615F75-6EC3-41D1-A548-7358E3796766@earthlink.net> Content-Type: text/plain; charset=iso-8859-1
Also, a two-part article about his life & work is at:
http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf and http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf
--Dan
On 2012-08-12, at 4:12 AM, Simon Plouffe wrote:
<< Hello,
there is one big news here circulating in France about Alexandre Grothendieck, they have found 5 boxes, 20000 pages of documents from him, written pages about Algebraic Geometry,
Alexandre Grothendieck is retired (sort of) in west-south france in the Pyren?es, for about 20 years, he won fields medal in the 60's and then retired completely in 68-69, refusing honnors and money,
There is a very good account of the life and mathematics of him here : http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-m... but this article is in french, I hope that google translate will do a decent job for those speaking english only,,
Bonne lecture,
Simon Plouffe
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Message: 3 Date: Sun, 12 Aug 2012 18:40:33 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] peculiar 2x2 products Message-ID: <CAA-4O0GcziOwe82GSYPJw-LkqjGAOK48X75HT-YH=yQ9kPuvbQ@mail.gmail.com> Content-Type: text/plain; charset=UTF-8
On Fri, Aug 10, 2012 at 2:39 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Mon, Jul 23, 2012 at 12:01 AM, Bill Gosper <billgosper@gmail.com>wrote:
We're used to evaluating an infinite continued fraction as a limit of ratios of elements of a running product of 2x2 matrices, while the actual elements of the product blow up. E.g., for 4 arctan(1)
In[1153]:= FoldList[Dot, {{0, 4}, {1, 0}}, Table[{{2*k - 1, k^2}, {1, 0}}, {k, 9}]]
Out[1153]= {{{0, 4}, {1, 0}}, {{4, 0}, {1, 1}}, {{12, 16}, {4, 4}}, {{76, 108}, {24, 36}}, {{640, 1216}, {204, 384}}, {{6976, 16000}, {2220, 5100}}, {{92736, 251136}, {29520, 79920}}, {{1456704, 4544064}, {463680, 1446480}}, {{26394624, 93229056}, {8401680, 29675520}}, {{541937664, 2137964544}, {172504080, 680536080}}}
In[1154]:= N[Divide @@ # & /@ Rest[%]]
Out[1154]= {{4., 0.}, {3., 4.}, {3.16667, 3.}, {3.13725, 3.16667}, {3.14234, 3.13725}, {3.14146, 3.14234}, {3.14161, 3.14146}, {3.14159, 3.14161}, {3.14159, 3.14159}}
But here's an infinite matrix product where the elements converge individually:
Product[MatrixForm[{{(3*a^2 - 3*a*j + j^2)/(j*(-1 - 3*a + b + j)), a^3/(j*(-1 - 3*a + b + j))}, {1/j, 1}}], {j, b, ?}] == MatrixForm[{{"?", (a^3*HypergeometricPFQ[{1 + a, 1 + a, 1 + a}, {1 + b, 1 + b}, 1])/b^2}, {"?", HypergeometricPFQ[{a, a, a}, {b, b}, 1]}}]
(Product means matrix product, not Mathematica's product.) The ?s converge, too, but I don't know to what. Note that the determinant ? 1, but approaches 1 in a very particular way. It's not obvious to me how to scale a matrix to make its running product converge elementwise.
I actually have an elementwise convergent 2x2 product for the general 3F2[1], but idiot Mathematica 8.04 takes (literally?) forever to convert a simple sum of Pochhammers to a 3F2. --rwg
I gave up waiting and simply guessed the general case, and while numerically testing, guessed the "?"s:
I.e., the right column can be proven by path invariance, but the left is conjectural.
MProd[MatrixForm[{{((-a)*(b + c - j) + (b - j)*(-c + j))/ ((1 + a + b + c - d - j)*j), -((a*b*c)/((1 + a + b + c - d - j)*j))}, {1/j, 1}}], {j, e, ?}] == MatrixForm[ {{(a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, {1 + d, 1 + e}, 1])/((-1 + d)*d*e), (a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, {1 + d, 1 + e}, 1])/(d*e)}, {HypergeometricPFQ[{a, b, c}, {d, e}, 1]/(-1 + d), HypergeometricPFQ[{a, b, c}, {d, e}, 1]}}]
I.e., the "?"s were just the right column divided by d-1. So we "only" get two useful results instead of four. But from a 2x2! Except the convergence is lousy. I'm working on it.
Fail. It appears impossible to rapidly compute two pFqs with a 2x2. They seem always to converge like 1/j. Even 2F1[z]!
MProd[MatrixForm[{{-(((a + b - j)*z)/(j*(-1 + z))), (a*b*z)/(j - j*z)}, {1/j, 1}}], {j, e, ?}] == MatrixForm[{{0, (a*b*z*Hypergeometric2F1[1 + a, 1 + b, 1 + e, z])/e}, {0, HypergeometricPFQ[{a, b}, {e}, z]}}]
Other attempts to accelerate convergence were messy, producing, e.g. a slow octic/cubic CF.
Here are the two Kochflake ring<http://gosper.org/kocharmor-2-8-colored.png> pictures <http://gosper.org/kocharmor-2-8-coloredmore.png> Neil withheld because he was embarrassed about the colors, which were hastily grabbed from a Paint palate for the sole purpose of visualizing the recurrence relations.
No, he didn't mouthpaint. Duh, I meant palette.
It looks to me like the boundary dimension is reducing
(confirmed by Julian based on
In[125]:= N[Max[Abs[Eigenvalues[{{0, 0, 0, -2}, {1, 0, 0, 2}, {0, 1, 0, 3}, {0, 0, 1, 0}}]]], 30]
Out[125]= 1.87018472683652509239367973786 )
dramatically--possibly to 1?! --rwg
Also, Julian has discovered how pentaflakes reproduce<http://gosper.org/pentahouse.png>! Tsk. --rwg
I think during an earlier discussion here of the necessity of algebra instruction I mentioned a childhood memory of my great aunt recounting her struggles filling out a New York State tax form after her carpenter husband neglected to charge a customer some prescribed sales tax. She was trying to compute by successive approximations what pre-tax principal he would have to charge so that the added tax totaled what he actually charged. Ironically, I had already read her sister's (my grandmother's) highschool algebra text (which I found in a carton), and could have easily solved her problem. But I said nothing, because I could tell from her tone that she knew I knew.
The book was a plain green hardcover containing zero illustrations. I doubt I would have learned anything at all from one of those ridiculous damned picture books foisted on the current generation.
My eavesdropping teacher-cousin calls them hardcover comic books.
It must come as a shock when today's college freshmen first open their texts and find only text.
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Message: 4 Date: Mon, 13 Aug 2012 06:51:20 -0700 From: Henry Baker <hbaker1@pipeline.com> To: math-fun@mailman.xmission.com Subject: [math-fun] MathJax: Beautiful math in all browsers Message-ID: <E1T0v4c-0006qV-Iv@elasmtp-mealy.atl.sa.earthlink.net> Content-Type: text/plain; charset="us-ascii"
I just came across this package, which produces very high quality math on web pages. It's a bit slow on my cheap laptop w/Firefox, but it is pretty.
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Message: 5 Date: Mon, 13 Aug 2012 10:22:05 -0500 From: Robert Smith <quadricode@gmail.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] MathJax: Beautiful math in all browsers Message-ID: <50291B9D.1070802@gmail.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Hey:
This is actually what I use on my website (http://www.symbo1ics.com/blog).
Be sure to install the STIX font set for high quality renderings using native fonts as opposed to images! (MathJax might have moved on to using webfonts, so even that might not be necessary.)
The fonts are located here:
http://sourceforge.net/projects/stixfonts/files/Current%20Release/
Specifically, the file: STIXv1.1.0.zip
Cheers!
-Robert
On 8/13/2012 8:51 AM, Henry Baker wrote:
I just came across this package, which produces very high quality math on web pages. It's a bit slow on my cheap laptop w/Firefox, but it is pretty.
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Message: 6 Date: Mon, 13 Aug 2012 19:14:43 +0200 From: Joerg Arndt <arndt@jjj.de> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] MathJax: Beautiful math in all browsers Message-ID: <20120813171443.GA8132@jjj.de> Content-Type: text/plain; charset=us-ascii
* Henry Baker <hbaker1@pipeline.com> [Aug 13. 2012 17:47]:
I just came across this package, which produces very high quality math on web pages. It's a bit slow on my cheap laptop w/Firefox, but it is pretty.
Mathoverflow ( http://mathoverflow.net/ ) is one example, e.g. (rather random example)
http://mathoverflow.net/questions/17997/deriving-symmetries-of-a-gauge-theor...
[...]
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