April 2003 - mathfun@mailman.xmission.com

Re: [math-fun] Probabilistic harmonic series
by asimovd＠aol.com 29 Apr '03

29 Apr '03

Bill, Thanks for the insight about this question. Did you get my recent q. asking when you first formulated the geometrization conjecture? I haven't heard a response, so I hope all is OK. --Dan

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[math-fun] Re: (x^x)^(x^x), x^(x^(x^x)), etc...
by N. J. A. Sloane 29 Apr '03

29 Apr '03

Isn't that the subject of the following well-known article? %D A003018 R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis. Amer. Math. Monthly 80 (1973), 868-876. (there are numerous references to it in the OEIS) NJAS

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[math-fun] RE: (x^x)^(x^x), x^(x^(x^x)), etc...
by Meeussen Wouter (bkarnd) 29 Apr '03

29 Apr '03

specialising x to Sqrt[2], only these remain: 1,2,4,8,17,38,88,206,498 ... Wouter. -----Original Message----- From: Edwin Clark [mailto:eclark@math.usf.edu] Sent: maandag 28 april 2003 17:30 To: seqfan Cc: math-fun(a)mailman.xmission.com Subject: (x^x)^(x^x), x^(x^(x^x)), etc... Consider the following five functions defined on positive reals: f1(x) = ((x^x)^x)^x; f2(x) = (x^(x^x))^x; f3(x) = x^((x^x)^x); f4(x) = x^(x^(x^x )); f5(x) = (x^x)^(x^x); Note that f2 = f5. There are really only 4 different functions of this type. We know the Catalan number C(n-1) gives the number of ways to parenthesize a product of n variables. If we use x^x for the binary operation and identify products that give equal functions when x is positive, how many products do we get? If the number of x's is n let e(n) be the number of distinct functions so obtained. I used Maple to simplify the set of C(n-1) expressions and got the following cardinalities for the simplified sets for n from 1 to 11: (*) 1,1,2,4,9,20,48,115,286,719,1842 Except for some of the smaller values, I cannot be sure that Maple has not failed to identify some equal functions. So this may not be the sequence e(n). The sequence (*) coincides with A000081: rooted trees with n nodes --- at least as far as I have computed. Is this known? If not can anyone prove that the sequences coincide? Maple undoubtedly uses standard laws of exponents to simplify these expressions. This in itself raises a number of questions. Edwin Clark =============================== 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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[math-fun] (x^x)^(x^x), x^(x^(x^x)), etc...
by Edwin Clark 28 Apr '03

28 Apr '03

Consider the following five functions defined on positive reals: f1(x) = ((x^x)^x)^x; f2(x) = (x^(x^x))^x; f3(x) = x^((x^x)^x); f4(x) = x^(x^(x^x )); f5(x) = (x^x)^(x^x); Note that f2 = f5. There are really only 4 different functions of this type. We know the Catalan number C(n-1) gives the number of ways to parenthesize a product of n variables. If we use x^x for the binary operation and identify products that give equal functions when x is positive, how many products do we get? If the number of x's is n let e(n) be the number of distinct functions so obtained. I used Maple to simplify the set of C(n-1) expressions and got the following cardinalities for the simplified sets for n from 1 to 11: (*) 1,1,2,4,9,20,48,115,286,719,1842 Except for some of the smaller values, I cannot be sure that Maple has not failed to identify some equal functions. So this may not be the sequence e(n). The sequence (*) coincides with A000081: rooted trees with n nodes --- at least as far as I have computed. Is this known? If not can anyone prove that the sequences coincide? Maple undoubtedly uses standard laws of exponents to simplify these expressions. This in itself raises a number of questions. Edwin Clark

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Re: [math-fun] a(n old?) riddle ***SPOILER***
by asimovd＠aol.com 27 Apr '03

27 Apr '03

He who Gosps asks: << Q: How is Margaret not a gem? A: Backwards, more or less? Reason: 10^12 g = 10^9 kg ~ 2.2 x 10^9 lb ~ 10^6 x (2000 lb) On the other hand: Q: Why *is* Margaret a gem ? --Dan P.S. Mnemonic for pi: "Pi seconds is a nanocentury."

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[math-fun] Two quartically convergent sums and a(n old?) riddle
by R. William Gosper 27 Apr '03

27 Apr '03

Old business: I asked, >So, can you construct any circular arc which divides a given disc into >areas 1:2 ?-) The only ones I know are full circles of radius sqrt(1/3) or sqrt(2/3), internally tangent to the propositus. Unlikely new, but interesting: inf ==== k \ k 4 z 1/4 > theta (2 z, q ) = theta (-, q ) - 1 / 2 3 2 ==== k = 0 1 -- k z 4 2 inf theta (--, q ) z ==== 4 k ------ \ 2 4 %pi log(q) > --------------- = 2 theta (2 z, q ) - sqrt(- ------) %e / k 3 log(q) ==== 2 k = 0 Of equally dubitable novelty: Q: How is Margaret not a gem?

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Re: [math-fun] math writing
by asimovd＠aol.com 26 Apr '03

26 Apr '03

<< 1. An n-dimensional manifold is a topological space whose points have neighborhoods homeomorphic to an n-dimensional ball. That's too technical, so I'll say the same thing without the jargon. 2. An n-dimensional manifold is such that every point has a region around it like an n-dimensional ball. Dropping the jargon makes it possible to equivocate about what "region around it" and "like ball" mean. >> The typical layperson still hasn't the foggiest notion of an n-dimensional ball for n > 3. --Dan

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[math-fun] 20000 math problems online: AMM,Math.Mag,Crux,J.Rec.M,Ed.Times,...
by Bill Dubuque 25 Apr '03

25 Apr '03

The message below has a link to a 50M zip file of gif's of almost 20000 math problems. Now you've really got problems. Enjoy From: "Julien Santini" <santini.julien(a)wanadoo.fr> Subject: That may interest you (for profs and students) Date: Fri, 25 Apr 2003 09:56:46 +0200 Message-ID: <b8apo1$5f8$1(a)news-reader11.wanadoo.fr> Hi, I wanted to inform all of you that I compiled a zip file containing: *6000 AMM problems (from 1894 to 1997) *13656 other problems from Crux, Math Gazette, Math Magazine, and other magazines and olympiads. It took me 20 hours of work to compile such a file, so I want it to be shared a MAX. Each problem is a *.gif file (pic file) and the zipped problems are contained in directories whose names indicate which magazine/olympiad the problem appeared in. You should unzip the file "19656_problems.zip" using Winzip, Winrar, or PowerArchiver (*excellent* freeware) but NOT the program given by Windows XP (i don't know its name in english, sorry) because of the HUGE number of files manipulated. I hope you will enjoy this resource (the problems were all provided by a uni server on the web for personal use). In order to download the zip file: http://perso.wanadoo.fr/julien.santini/19656_problems.zip Additionally, you could download only the 6000 AMM problems at: http://perso.wanadoo.fr/julien.santini/AMM.zip if you have a slow connection [only some 15 Mo]. All the problems are in english, but the rest of my site is in french.

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[math-fun] FYI: WBUR show with Wolfram on ANKoS
by Beeler, Michael 24 Apr '03

24 Apr '03

A friend pointed this out - a brief web page with a clickable icon to listen to the recorded show. I haven't listened to it. -- Mike Subject: Boston University World of Ideas with Ted O'Brien: Sunday, April 06, 2003 http://www.buwi.org/shows/2003/04/20030406.asp

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Re: [math-fun] brick tilings
by asimovd＠aol.com 24 Apr '03

24 Apr '03

Using only 1xm and nx1 tiles (m,n > 1) there are of course tilings of the *entire* plane having no rectangles at all other than the tiles themselves. --Dan

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