February 2012 - mathfun@mailman.xmission.com

[math-fun] What is the meaning of diff(express1,express2) ?
by Henry Baker 16 Feb '12

16 Feb '12

There's been a little discussion on the Maxima email list about diff(expression1,expression2), where expression2 isn't an atomic symbol, and what this might mean. For example, we know that diff(x^2,x)=2*x. But what is the meaning of diff(x^2,x*y) ? In Maxima, diff(expression1) (without a 2nd argument), does a "total differential", and leaves del(x), del(y), etc., in the expression. Thus, diff(x*y) = x*del(y)+y*del(x). One possibility for defining diff(expression1,expression2) is diff(expression1,expression2) = diff(expression1)/diff(expression2),factor (we hope that factor will cancel out any del(x), del(y), etc.) Thus, diff(x^2*y^2,x*y) = 2*x*y*(x*del(y)+y*del(x))/(x*del(y)+y*del(x)) = 2*x*y Problem: There is no guarantee that we can get all the "del(variable)" items to cancel, is there ?? Now we have an analogous problem with integrate(expression1,expression2). integrate(x^2,x) = x^3/3 But what is integrate(x*y,x*y)? If we let a new variable z=x*y, then we can do the following: integrate(x*y,x*y) = integrate(z,z) = z^2/2 = (x*y)^2/2 What about integrate(sin(x),x*y) ? integrate(sin(x),x*y) = integrate(sin(x),z) = integrate(sin(z/y),z) = -y*cos(z/y) = -y*cos(x) Is there any standard way to do this? Can we extend this to definite integration: integrate(expression1,expression2,low,high) ?? What is the relationship between diff(expression1,expression2) and integrate(expression1,expression2) ?? Normally, we expect diff(integrate(expression,x),x)=expression, but what if x is no longer a variable, but an expression??

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[math-fun] sideways math fun
by Dave Dyer 16 Feb '12

16 Feb '12

Check out http://colorku.com/ ColorKu is exactly Sudoku, but with 9 colors instead of 9 digits. It's an interesting idea from several directions. I immediately wondered if the perception of the board would affect the ability to solve - for better or worse. Perceiving sets of colors is bound to involve different brain areas than sets of numbers. I can definitely imagine being able to "see" solutions based on perception alone, in a way that is hard to imagine when the symbols are numbers. And in the reverse direction, the kind of explicit reasoning needed to solve sudoku, if translated into color space instead of number space, would surely be harder to do.

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Re: [math-fun] to break RSA...
by Stuart Anderson 16 Feb '12

16 Feb '12

Another cat, another bag .... http://www.smh.com.au/it-pro/security-it/researchers-find-flaw-in-online-en…

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Re: [math-fun] We don't need no stinking factorizer to break RSA... duh!!!!!!!!!
by Dan Asimov 16 Feb '12

16 Feb '12

This gives me an idea to put up a website where you can register your address, phone number, birthdate and social security number, for a very small fee (paid for the sense of security this will confer). --Dan Mike Speciner wrote: << So, given that there's no limit on human stupidity, has someone put up a website (commercial, or public service) where you attempt to register your public RSA key and it tells you if it matches someone else's, or if it shares a factor with someone else's (in which case it informs the someone else[s] if they've registered), or if it is a weak key for some other reason (e.g., easily factored, easily guessed private exponent), or it says it's OK and registers it? . . . >> ________________________________________________________________________________________ It goes without saying that .

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Re: [math-fun] yet another triangle dissection puzzle
by Michael Reid 16 Feb '12

16 Feb '12

> Any triangle can be dissected into three mirror-symmetric pieces. If we are limited to > just two pieces, is the number of (non-congruent) triangles that admit a dissection (into > mirror-symmetric pieces) finite or infinite? By chance, recently I was looking through some back issues of Journal of Recreational Mathematics, and saw this related problem: >From Journal of Recreational Mathematics, v. 27 (1995) no. 4, p. 302. Problem 2283. Dissection for Coincidence by Heinrich Hemme, Aachen, Germany On page 56 of Charles W. Trigg's Mathematical Quickies (McGraw-Hill Book Co., New York, 1967), there is the following problem: Q 204: Consider two congruent triangles that can be brought into coincidence only by rotation of one of them through a third dimension. How, by cutting the triangles, could coincidence be effected by motion in a plane only? The solution (by Louis R. Chase) to the problem is given on page 172 as follows (Figure 16): The result can be achieved by dissecting one of the triangles into isosceles triangles. A right triangle will be divided by the median upon the hypotenuse. An obtuse triangle can first be divided into right triangles by the altitude upon the longest side, An acute triangle will be divided into three isosceles triangles by the circumradii to the vertices. It can be shown that this solution may be optimized even further. To that end, the problem might be posed as follows: What types of triangles can be transformed into their respective mirror images by dissection into (a) one piece, (b) two pieces, (c) three pieces? OK, that quote is a mouthful, er, a keyboardful. The connection to Veit's problem is that the dissections for (b), the interesting part, come from two-piece dissections into pieces with reflective symmetry. Is it obvious ... provable ... true(?!) that a two piece dissection must be into pieces with reflective symmetry. Hmmm ... not true for isosceles triangles, when a one-piece "dissection" suffices. For scalene triangles, the question still stands. Solutions to the JRM problem by the Proposer (Hemme), Markus Goetz and Helmut Postl, were published in JRM volume 28, (1996-97) no. 4, on pages 313-315. There is no claim that they have found all triangles which admit such a two-piece dissection. I have not located Trigg's book, so I do not know if it has any information about two-piece dissections. I do recall, many years ago, being given the problem of finding such a two-piece dissection for a particular triangle, given by its angles. I'm not certain any more what those angles were, perhaps 50, 55 and 75 degrees. In any event, those angles work for the particular dissection I eventually found, which is one of several the JRM solvers found. Veit, or others, do you have a reference for the two-piece dissection problem? Perhaps Trigg's book has one, or is the origin of it, but I haven't seen the book. Michael Reid > Any triangle can be dissected into three mirror-symmetric pieces. If we are limited to > just two pieces, is the number of (non-congruent) triangles that admit a dissection (into > mirror-symmetric pieces) finite or infinite? > > Veit > > On Feb 13, 2012, at 5:42 PM, Veit Elser wrote: > > > Okay, I'll try again: > > > > Which triangles can be dissected into two pieces, both of which are mirror-symmetric? > > > > On Feb 13, 2012, at 4:09 PM, Dan Asimov wrote: > > > >> Veit -- Can the word "just" be omitted without changing the meaning here, > >> or is it in some way limiting the number of mirror-symmetric pieces > >> that such a triangle can be dissected into? > >> > >> Thanks, > >> > >> Dan > >> > >> Veit asked: > >> > >> << Which triangles can be dissected into just two mirror-symmetric pieces? > >>

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[math-fun] Heroic upload
by Fred lunnon 15 Feb '12

15 Feb '12

The Heronian tetrahedron lattice embedding proof has been uploaded at http://arxiv.org/abs/1202.3198 I should be most grateful if someone would make a determined attempt to shoot it down (though probably not quite so pleased should they manage to succeed). [I have suppressed some previous tangential material, particularly concerning the volume lying in 168 |Z, after shooting that down all by myself --- maybe more of that saga later ...] I finally grasped the nettle and took on the arXiv automated LaTeX-muncher. I read the instructions (large type, unambiguous, comprehensive) for signing up and submitting; I followed them without being obliged to utter a single vulgar expletive or punch my desk even once; my source compiled first shot, and 24 hours later it's been moderated and mounted. Why am I telling everybody this? Because it's a *&^%$£@?! miracle, that's why! In the dark days ahead, as I wrestle with the stunningly perverse monstrosities that masquerade as automated servers throughout the remainder of the universe, I trust I shall find it within my grasp to recall this moment of catharsis. Thankyou, arXiv. Fred Lunnon

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[math-fun] Curses, anticipated again.
by Bill Gosper 15 Feb '12

15 Feb '12

Neil has just uncovered fq.math.ca/Scanned/6-6/hoggatt.pdf predating my http://mathworld.wolfram.com/SquareDissection.html of a square into ten acute isosceles triangles. Neil also found in a Gardner book a reference to a Monthly paper (June-July 1962, pp550-552) claiming that *any* obtuse triangle can be cut into eight acute isosceles triangles, implying at most nine for dissecting a right isosceles, in contradiction of my round(tan(69)) solutions assertion. Has anyone a picture of this dissection? --rwg This is weird: Calculating with pen and paper for a change, it seems that sunlight falling on the diagrams and equations actually makes them *easier* to read.

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Re: [math-fun] wrong word
by Henry Baker 15 Feb '12

15 Feb '12

Thanks! <http://mathlab.mathlab.sunysb.edu/%7Escott/Papers/Newton/Published.pdf>http://mathlab.mathlab.sunysb.edu/~scott/Papers/Newton/Published.pdf has great color pictures! At 04:45 PM 2/15/2012, James Propp wrote: >http://math.jacobs-university.de/schleicher/ should have links, contact information, etc. > >(I didn't go to the talk myself, so that's all I can tell you.) > >Jim > >On Thu, Feb 16, 2012 at 11:16 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >Regardless of the wording, the talk sounds interesting. Is there a >video or a paper about some of these results? > >At 01:03 PM 2/15/2012, James Propp wrote: >>The last sentence of the abstract for Dierk Schleicher's recent talk >>"Old and new about the dynamics of iterated transcendental functions" >>at Tufts may amuse you: >> >>Abstract: Much of holomorphic dynamics in the past 30 years has >>focused on iterated rational maps and especially polynomials, but >>especially in recent years there has been quite some progress on >>iterated transcendental maps. In particular, several conjectures of >>the 1980's ("classical" in this field) have recently been settled. >>This often leads to natural and at the same time surprising and >>even counterproductive results. >> >>Jim Propp

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Re: [math-fun] wrong word
by Henry Baker 15 Feb '12

15 Feb '12

Regardless of the wording, the talk sounds interesting. Is there a video or a paper about some of these results? At 01:03 PM 2/15/2012, James Propp wrote: >The last sentence of the abstract for Dierk Schleicher's recent talk >"Old and new about the dynamics of iterated transcendental functions" >at Tufts may amuse you: > >Abstract: Much of holomorphic dynamics in the past 30 years has >focused on iterated rational maps and especially polynomials, but >especially in recent years there has been quite some progress on >iterated transcendental maps. In particular, several conjectures of >the 1980's ("classical" in this field) have recently been settled. >This often leads to natural and at the same time surprising and >even counterproductive results. > >Jim Propp

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[math-fun] The new RSA break -- request for circumspection.
by Schroeppel, Richard 15 Feb '12

15 Feb '12

I'd like to encourage Math-Funsters to postpone discussing on the list the algorithms omitted from this paper for a month, till March 15. The authors were not able to notify all the keyholders ahead of publication, so there is the potential for some nasty surprises. This request is not meant to inhibit off-list discussion; and you are welcome to let me know if you notice the cats have escaped confinement into some public place. Obviously it also doesn't apply to the algorithms disclosed in the paper. Rich -----Original Message----- From: math-fun-bounces(a)mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Warren Smith Sent: Tuesday, February 14, 2012 5:45 PM To: math-fun(a)mailman.xmission.com Subject: [EXTERNAL] [math-fun] We don't need no stinking factorizer to break RSA... duh!!!!!!!!! front page of NY Times just featured this paper http://eprint.iacr.org/2012/064.pdf which is hilarious! Just when you thought you'd seen the ultimate in human stupidity... _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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