September 2006 - mathfun@mailman.xmission.com

[math-fun] Re: mis-defining limits
by James Propp 10 Sep '06

10 Sep '06

I'm also interested in defective definitions that may not have been proposed in writing, but are the sort that someone might have come up with en route to the correct definition. E.g., "We say that $\lim_{x \rightarrow a} f(x) = L$ if for every number $\epsilon > 0$ there is a corresponding number $\delta$ such that if $0 < |x-a| < \delta$ and $f(x)$ is defined, then $|f(x)-L| < \epsilon$." (Under this definition, the limit of sqrt(x) as x approaches -1 is anything you please!) As I explained to my students on the first day of class: In a non-honors course, one treats calculus as a vehicle for getting from point A to point B. In an honors course, one wants the students to poke around under the hood and find out what the different parts do by removing them from the vehicle and then driving the vehicle off a cliff to see what happens. (Nearly all of the students in the class are declared engineering majors, and I think they find this motivation for rigor more appealing than fussy appeals to intellectual hygiene.) Jim Propp

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[math-fun] Re: mis-defining limits
by Chris Landauer 10 Sep '06

10 Sep '06

hihi, all - in 1974 or so, i taught a calculus for non-physical science majors at U. Wisconsin Madison after a couple of months of doing derivatives and integrals, i was talking about some problem or other, and said ``the limit of this expression is ...'' one of the students piped up and asked ``how do you know?'' i gave them an intentionally incomplete answer about getting close, and another student pushed me on that, and over the course of an entire hour, (well maybe half an hour), the class collectively pinned me to the wall until i wrote out the entire epsilon delta definition of limits, and they said ``why is that so hard?'' - it was one of the most remarkable class sessions i have ever been in getting close how close always close or just sometimes eventually always close than this as a second suggestion, i also taught an honors calculus class there for two semesters - in my class, every homework and every quiz and every test had a mathematical induction problem, initially as extra credit in the second semester, i got the runoff from two parallel first semester classes, one of them mine - so i put an induction problem on the first test, and every one from my class but one got it, and everyone from the other class but one did not more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org

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[math-fun] mis-defining limits
by James Propp 10 Sep '06

10 Sep '06

I'm teaching an honors calculus class this semester, and to motivate the formal epsilon-delta definition of limits, I'd like to show the students some appealing, informal definitions of limits and the hidden errors and ambiguities of those definitions. Do any of you have any favorite examples of this? E.g.,, I seem to recall that some definitions of limits say that f(x) should approach but never equal b as x approaches a (which would make lim_{x -> 0} x sin x problematical). Jim Propp

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Final Hint Re: [math-fun] 1,2,3,5,18 puzzle
by Daniel Asimov 09 Sep '06

09 Sep '06

<< Puzzle: What do the positive integers 1, 2, 3, 5, 18 have in common with each other, but with no other positive integer (conjecturally) ? >> Hint: These number are cycle lengths. --Dan

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Re: [math-fun] Choreography of large complete graphs
by Edwin Clark 08 Sep '06

08 Sep '06

On Wed, 30 Aug 2006, Ed Pegg Jr wrote: > At a convention, 128 people all need to meet with each > other, 1 on 1, for a paper exchange. > > Suppose a room is divided into a 8x8 square, and a > clock is set up to ding 127 times, once every 30 seconds. > Each person is given a map of their starting square, > and a map of their route over the next hour and 4 minutes. > Only two people can be in each cell. > Here is a solution which requires 64*134 unit person moves. (Instead of cells I'm using 64 tables. Each with two chairs-- one on the left and one on the right.) One person moving from one table to an adjacent table is what I'm calling a "unit person move". While explaining it to Ricardo Restrepo he noticed that a lower bound would be 64*127 (at each of the 127 bell ringings at least one person must move from each of the 64 tables since in all there must be binomial(128,2) = 64*127 meetings. So the 64*134 is pretty close to the minimum. The idea is the following: Stage #1: Start with the 8x8 array of tables. There is an obvious Hamiltonian cycle through the array (think of it as a grid graph). Say that each table has a left side and a right side. Each person on the right of a table moves to the next table along the Hamiltonian cycle. Those on the left of each table remain seated. After 63 moves everybody on the left has been paired with everybody on the right of the tables. This gives a total of 64*63 unit person moves. Stage #2: Now the lefts must be paired with the lefts and the rights with the rights. To do this we do some major shifting: Each person sitting on the right side of a table in the first four columns interchanges places with the person on the left side of the table 4 places to his right. This gives us two 8 x 4 grids of previously unpaired people. This shifting requires 64*4 unit person moves. (Each interchange is 2*4 unit person moves.) Stage #3: Now as before each of the 8x4 grids of unpaired people has a Hamiltonian cycle. Again those on the left of a table stay fixed and those on the right move to the next table along the Hamiltonian cycle. After 31 moves everybody on the left has been paired with everybody on the right in each of the two 8x4 grids. This adds another 64*31 unit person moves. Stage #4: Now we shift people again in each of the 8x4 girds so that those on the left are moved to the top 4 rows and the rights are moved to the bottom 4 rows. Doing this for each of the 2 8x4 grids gives us 4 4x4 grids of tables of previously unmatched people. To get to this stage requires 64*4 unit person steps. Stage #5: Then using a Hamiltonian shift as above for each of the 4x4 grids we pair up those on the left with those on the right. In all this requires 64*15 unit person moves. Stage #6: Next the 4 4x4 grids are shifted to 8 4x2 grids in 64*2 unit person moves. .....And so forth till one reaches 64 1x1 grids. When all is completed, the total is 64*134 unit person moves. Clearly this method generalizes from 128 to any number of the form 2^(2n+1). In which case the number of unit person moves taken by this method is 2^(2n)(2^(2n+1)-4+2^(n+1)-2n) Could this be the true minimum? --Edwin

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P.S. Re: [math-fun] Chinese Mathematicians Prove Poincare Conjecture
by dasimov＠earthlink.net 08 Sep '06

08 Sep '06

P.S. I did not see this in the Ximhua press release, but the Asian Journal of Mathematics says (in the link provided by Ray) that the article in question finishes the proof of the Thurston Geometrization Conjecture (of which the Poincare conjecture is just one of many important consequences). Up to now, a few media (Scientific American in 1994, New York Times in the past year) have claimed the Poincare conjecture has been proved, although no respectable professional journal has reported any such thing (only comments like "time will tell.") And not even media outlets have reported that the Geometrization conjecture was proved. Hmm. I guess time will tell. --Dan

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[math-fun] 1,2,3,5,18 puzzle
by Daniel Asimov 08 Sep '06

08 Sep '06

Puzzle: What do the positive integers 1, 2, 3, 5, 18 have in common with each other, but with no other positive integer (conjecturally) ? --Dan

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[math-fun] Re: Melbourne, city of Math
by James Propp 05 Sep '06

05 Sep '06

>A nice cube is the Atomium. >http://mathworld.wolfram.com/Cube.html I propose that someone build a tetrahedron with vertices at 33.87 S, 151.20 E Sydney, Australia 24.80 S, 70.00 W northern Chile, S of Antofagasta 66.69 N, 151.70 W central Alaska, near Bettles Field 3.35 N, 39.92 E northeast Kenya (Thanks to Fred Helenius for identifying this "land tetrahedron": see http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/geographicals .) I picture something like the Atomium, with struts that disappear into the earth. Presumably the struts would terminate a few feet below the ground, although the viewer would be invited to imagine them as uninterrupted, several-thousand-mile-long beams. Who would pay for this? Well, if the struts are scaled-up version of the Zome construction system, the Zome guys might view the project as free advertising. Alternatively, the tetrahedron's latent It's-a-small-world-after-all, Building-bridges-between-nations political message might attract some potential funders of this conceptual art project. (To underscore the one-world theme, one could add video cams that let the viewers at each site watch what's happening at the other sites, along with a globe that shows the four vertices and the signal-paths that join them.) Or, maybe Sydney would fund the whole project, as a way of besting Melbourne? :-) Jim Propp

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RE: RE: [math-fun] Lost in fraction
by Eric Angelini 05 Sep '06

05 Sep '06

Yes, thanks, Michael -- I had quoted yr paper on my website: see october first, 2003 (in french) ! http://www.cetteadressecomportecinquantesignes.com/Klei03.htm Best and thanks to all of you ! É. -----Message d'origine----- De : math-fun-bounces+eric.angelini=kntv.be(a)mailman.xmission.com [mailto:math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com] De la part de Michael Kleber Envoyé : mardi 5 septembre 2006 18:45 À : math-fun Objet : Re: RE: [math-fun] Lost in fraction Eric Angelini wrote: > ... take any digit of (a); when written > in french, this digit has as much letters > as described by the next digit : > > UN (2 letters) > DEUX (4 letters) > QUATRE (6 letters) > SIX (3 letters) > TROIS (5 letters) > CINQ (4 letters) > QUATRE (6 letters) > ... In English, the map (n -> # letters in the name of n) always ends up at its unique fixed point: FOUR has 4 letters. One of my early Intelligencer Entertainments columns riffed on the slighly more interesting point of view where you say that the map is, instead, (n -> product of lengths of words in the name of n). So 24 -> "twenty four" -> 6*4 = 24. If you find this amusing, the column is here: http://people.brandeis.edu/~kleber/Papers/twenty-four.pdf --Michael -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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RE: [math-fun] Lost in fraction
by Eric Angelini 05 Sep '06

05 Sep '06

-----Message d'origine----- De : math-fun-bounces+eric.angelini=kntv.be(a)mailman.xmission.com [mailto:math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com] De la part de Christian Boyer Envoyé : mardi 5 septembre 2006 18:16 À : 'math-fun' Objet : RE: [math-fun] Lost in fraction x = 0.4635 4635 4635... 10000*x = 4635 + x x = 4635/9999 12.4635 4635 4635 = 12 + 4635/9999 = 124623/9999 = 13847/1111 Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr(a)mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Eric Angelini Envoyé : mardi 5 septembre 2006 17:31 À : math-fun Objet : [math-fun] Lost in fraction Hello, is there a (simple) fraction whose result is : 12.4635 4635 4635 4635 ... ? Best, É. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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