February 2008 - mathfun@mailman.xmission.com

Re: [math-fun] functions behaving badly
by Dan Asimov 26 Feb '08

26 Feb '08

Jim writes [after my minor change, and addition, of notation]: << Say a function is locally unbounded at c if it is unbounded on every neighborhood of c. If a function f is differentiable everywhere, what can be said about the set of points LU[f'] at which the derivative f' is locally unbounded? Can LU[f'] be dense? Can it be all of R? >> Jim defines a function [ditto]: << {x^2 Sin[1/x^2] for x not equal to 0, g(x) = { { 0 for x = 0 >> I'm going to guess that there exists a function h:R -> R differentiable everywhere with LU[f'] dense in R. Using this function g and an enumeration {q_n}, n = 1,2,3,... of the rationals, define h:R -> R via h(x) := Sum_{n=1,2,3,...} g(x-q_n)/2^n I suspect this f *might* be differentiable everywhere with LU['h] dense in R. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] F^-1(G(F(x)))
by Steve Witham 25 Feb '08

25 Feb '08

>From: "Mark D. Niemiec" <mniemiec(a)interserv.com> > >In the J programming language, this is denoted by > G &. F >and is called "G under F". > >(J (http://jsoftaware.com) is the descendent of APL, >also designed by Kenneth Iverson, and it includes >functional programming operators, as well as using >ASCII characters for readability.) > >Under can also be used dyadically: > F^-1( G( F(x), F(y)) >for example: > x +&.*: y NB. Euclidean distance (*: is square) > x +&.% y NB. Electrical resistance in parallel (% is reciprocal) > x -:@+ y NB. Arithmetic mean (-: is halve; F@G(x,y) = F(G(x,y)) ) > x -:@+&.% y NB. Harmonic mean Lucky for us, J is still readable in that brisk, bracing APL sense (-: (-: is smiley) An Amazon review of the book "Bypasses" mentions exp( add( log(x), log(y) ) ) In all these examples except arithmetic mean, F is nonlinear. --Steve

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[math-fun] help needed with a definite integral
by N. J. A. Sloane 25 Feb '08

25 Feb '08

Can anyone help with following integral, which arises in computing the average distance between two random points in a unit square? (Maple and Mathematica can't do it, at least the versions that I have.) Let h = 1/2, to save space. Int_{x = 0..h} Int_{r=-h-x..h-x} Int_{y=0..h} Int_{s=-h-y..h-y} sqrt(r^2+s^2) dx dr dy ds A numerical value would be fine - and a method - there are several other integrals that are of interest. Thanks! Neil

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[math-fun] a counterexample to fermat's last theorem!
by Erich Friedman 24 Feb '08

24 Feb '08

(you'll need a fixed font to view this correctly) 70^3 = 3 4 3 0 0 0 212^3 = 9 5 2 8 1 2 8 ----------------------------- 462^3 = 9 8 6 11 1 2 8 erich

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[math-fun] Re: F^-1(G(F(x)))
by Mark D. Niemiec 23 Feb '08

23 Feb '08

On 16 Feb 2008 at 18:58, Steve Witham wrote: > Is there a common name for the idiom > F^-1( G( F( x ) ) ) ? In the J programming language, this is denoted by G &. F and is called "G under F". (J (http://jsoftaware.com) is the descendent of APL, also designed by Kenneth Iverson, and it includes functional programming operators, as well as using ASCII characters for readability.) Under can also be used dyadically: F^-1( G( F(x), F(y)) for example: x +&.*: y NB. Euclidean distance (*: is square) x +&.% y NB. Electrical resistance in parallel (% is reciprocal) x -:@+ y NB. Arithmetic mean (-: is halve; F@G(x,y) = F(G(x,y)) ) x -:@+&.% y NB. Harmonic mean -- Mark D. Niemiec <mniemiec(a)interserv.com>

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Re: [math-fun] Re: east of the south pole
by Eugene Salamin 20 Feb '08

20 Feb '08

There is an energy scam based on a crackpot theory concerning "hydrinos". These are supposed to be hydrogen atoms from which energy has been extracted, leaving them in a state below the ground state. Bob Park, author of "Voodoo Science", has said that hydrinos are like exploring the region South of the South Pole. Gene ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ

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Re: [math-fun] Re: conjugation, lifting, pull-back
by Eugene Salamin 20 Feb '08

20 Feb '08

----- Original Message ---- From: Steve Witham <sw(a)tiac.net> To: math-fun(a)mailman.xmission.com Sent: Tuesday, February 19, 2008 9:26:23 PM Subject: [math-fun] Re: conjugation, lifting, pull-back >From: James Propp <jpropp(a)cs.uml.edu> > >And then Melzak went ahead and did just that, in a book called >"Bypasses: A Simple Approach to Complexity". >From: Eugene Salamin <gene_salamin(a)yahoo.com> >They're wonderful books. And you have no excuse for not reading >them yourself, since they have been reprinted by Dover, "two volumes >bound as one". http://store.doverpublications.com/0486457818.html Neither that web site or Amazon mentions "Bypasses"--are you sure it's the second volume of that set? --Steve _______________________________________________ I'm quite certain that "Bypasses" is volume 2. Can you gamble $26.56 at Amazon, with free shipping and no tax? Gene ____________________________________________________________________________________ Never miss a thing. Make Yahoo your home page. http://www.yahoo.com/r/hs

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Re: [math-fun] Re: conjugation, lifting, pull-back
by Olivier Gerard 20 Feb '08

20 Feb '08

On Feb 20, 2008 6:26 AM, Steve Witham <sw(a)tiac.net> wrote: > > >From: Eugene Salamin <gene_salamin(a)yahoo.com> > >They're wonderful books. And you have no excuse for not reading > >them yourself, since they have been reprinted by Dover, "two volumes > >bound as one". http://store.doverpublications.com/0486457818.html > > Neither that web site or Amazon mentions "Bypasses"--are you sure it's > the second volume of that set? > > --Steve > There was a little confusion. Mr Melzak wrote several books. There is among them a series of two books now reprinted by Dover in one volume Companion to Concrete Mathematics Vol 1: Companion to Concrete Mathematics 1973 Vol 2: subtitled : Mathematical Ideas, Modeling & Applications 1976 But "Bypasses" is another: Z.A. Melzak Bypasses- A Simple Approach To Complexity 1983 Wiley and Sons ISBN 047186854X Currently out of print And very commendable. Olivier Gérard

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[math-fun] Re: east of the south pole
by Steve Witham 20 Feb '08

20 Feb '08

>From: Dan Asimov <dasimov(a)earthlink.net> ><< >IMAGING The Plato Observatory, on Dome Argus, 13,000 feet high and >about 700 miles east of the South Pole. >>> >Is this as nonsensical as I think, or is there a reasonable meaning >to the phrase "east of the South Pole" ? You have to stand on the South Pole in ruby slippers and spin really fast to get there. --Steve

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[math-fun] Re: conjugation, lifting, pull-back
by Steve Witham 20 Feb '08

20 Feb '08

> > On 2/17/08, Bernie Cosell <bernie(a)fantasyfarm.com> wrote: >> > On 16 Feb 2008 at 18:58, Steve Witham wrote: >> > >> > > Is there a common name for the idiom >> > > F^-1( G( F( x ) ) ) ? >> > > > > I think that was called "conjugation by F" when I ran across it. >From: Dan Asimov <dasimov(a)earthlink.net> >What I know as the totally standard term for F^(-1) o G o F is the >"conjugate" of G by F. Don't you mean F o G o F^(-1)? >It has equivalent interpretations as > >1) re-expressing G in terms of a new coordinate system as defined by F, and > >2) performing the transformation G in "a new place" as defined by F. >(E.g., if G is a rotation about the origin of R^2 about the origina, >and F is a translation F(v) = v - v_0, then F^(-1) o G o F expresses >the equivalent rotation of R^2 about v_0. "In a new place" seems a more general metaphor than "in a new coordinate system" to me. Sometimes it feels more like "reach in". "Rotate the top of the cube, then the side, then rotate the top back." Doesn't easily map to ...that mapping for me. It's more like, do G with F out of the way...except there is some interference you will have to clean up. Another example, in a functional language, deconstruct a complicated structure, change one element, put the structure back together with the changed element. You can think of the slot that was changed as a plane of the space of the structure, but... I don't. "Incremental change" seems to be a common theme. Also, "look at it from a different point of view," which is ironic: For instance, as I understand it, a Fourier transform is a rotation in a Hilbert space?? So if I FT, then multiply by the frequency response curve I want, then IFT, what I did was "see it in frequency terms," which we metaphorically call "another point of view," but in the Hilbert space it really *is*... >From: James Propp <jpropp(a)cs.uml.edu> > >And then Melzak went ahead and did just that, in a book called >"Bypasses: A Simple Approach to Complexity". >From: Eugene Salamin <gene_salamin(a)yahoo.com> >They're wonderful books. And you have no excuse for not reading >them yourself, since they have been reprinted by Dover, "two volumes >bound as one". http://store.doverpublications.com/0486457818.html Neither that web site or Amazon mentions "Bypasses"--are you sure it's the second volume of that set? --Steve

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