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[math-fun] Puzzles that led to new areas of science
by Scott Kim 01 Oct '07

01 Oct '07

I'm writing an article for an upcoming special issue of Discover magazine (in Dec) devoted to puzzles. The article will be about puzzles that led to new developments in math or science. For example, solving the Bridges of Könisburg led Euler to the development of topology. I prefer examples where the puzzle statement and the solution are accessible to a lay audience, and want to include some examples that have very recent implications...for instance even though Könisburg is an old problem, there might be a recent application of Euler circuits that are newsworthy now. Anyone have any thoughts? -- Scott Kim

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[math-fun] Mathpedia/Wikiproof (was Re: Math Fun-ding)
by Henry Baker 01 Oct '07

01 Oct '07

I think that the bulk of the initial part of this work would be done by computer scientists & logicians. Only later, when the bulk of the existing math needs to be shoveled in, would mathematicians be more involved. Re the diversion of effort: one could say the same thing about nearly every tool that has ever been developed -- the effort of creating the tool needs to be traded off against the alternative uses of this effort. Wikipedia presumably doesn't create any "new" knowledge, but makes existing knowledge accessible to a wider audience. Similarly, the discipline of putting every known mathematical proof into computer understandable form would make many of these proofs more accessible to a wider audience. This encyclopedia/database could include multiple proofs of the same theorems, as well as all kinds of human-oriented crutches -- diagrams, animations, movies, pointers to the literature, etc. In many cases, there are smart folks who may not be capable (at least not yet) of developing new theorems (they haven't imbibed enough coffee), but who are perfectly capable of translating math into computer understandable form. "Those who can, do; those who can't, translate proofs." I don't expect that such a project will uncover any significant gaps in existing proofs, but it will require significant clarifications in some cases. The overall goal of this effort would be to have mathematics put its money where its mouth is: mathematics has always claimed a higher/purer form of "truth" than other fields due to the rigor of its proofs. By building this logical structure, mathematics could "prove" this even more concretely. Ancient civilizations built pyramids, acqueducts, coliseums (colisea ?), etc., as monuments to their prowess; why can't we build such a logical structure ? At 11:42 AM 9/29/2007, Andy Latto wrote: >On 9/28/07, Henry Baker <hbaker1(a)pipeline.com> wrote: >> I guess there's also another meaning of "Far Side", as in the "Far Side" cartoons... >> >> I know that Mann is trying to be "open ended", but his topics seem to me to be a bit fuzzier than Hilbert's topics. >> >> I don't know if (D)ARPA is interested, but the time is right for the following project, which could conceivably take 10 or more years: >> >> --- >> Mathpedia/Wikiproof/(still looking for a good name): >> >> Computers are now ubiquitous, and now have enough memory & processing power to enable the following scenario for mathematics publishing: >> >> In order to get a paper published, you have to make all of the proofs mechanically checkable, and submit them to a web-based proof checker prior to being considered by the referees. No proof; no publish. This is arXiv with a prerequisite. > >I wonder why you think this project would be worth the huge diversion >of effort it would involve. > >Number theorists would have to devote effort currently devoted to >advances in number theory to work on formalization of existing work. > >Algebraic geometers would have to devote effort currently devoted to >advances in Algebraic geometry to work on formalization of existing >work. > >And so forth. > >Why is formalization so important that work in every other field of >mathematics should be slowed to make progress in formalization? > >The main advantage I see in formalization is that incorrect proofs are >not mistakenly accepted by the mathematical community. But this seems >to be a solution in seach of a problem; I can count on the fingers of >one hand the cases I know of where a widely accepted result turned out >to be incorrect. > >It's true that under the current system, once an important work is >published, other mathematicians devote effort to verifying its >correctness. But this work would not be eliminated; its main purpose >is not the catching of errors, but the achievement by the verifier of >a deeper understanding of the result and its proof. > >-- > Andy.Latto(a)pobox.com

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[math-fun] Math Fun-ding
by Hilarie Orman 29 Sep '07

29 Sep '07

The following challenges are summarized from a DARPA request for white papers and proposals for mathematics research, an area that DARPA has traditionally avoided, seeing it as NSF's purview. See http://www.darpa.mil/dso/personnel/mann.htm for links to more information. What do math-funners think of these topics? Mathematical Challenge One: The Mathematics of the Brain * Develop a mathematical theory to build a functional model of the brain that is mathematically consistent and predictive rather than merely biologically inspired. Mathematical Challenge Two: The Dynamics of Networks * Develop the high-dimensional mathematics needed to accurately model and predict behavior in large-scale distributed networks that evolve over time occurring in communication, biology, and the social sciences. Mathematical Challenge Three: Capture and Harness Stochasticity in Nature * Address Mumford's call for new mathematics for the 21st century. Develop methods that capture persistence in stochastic environments. Mathematical Challenge Four: 21st Century Fluids * Classical fluid dynamics and the Navier-Stokes Equation were extraordinarily successful in obtaining quantitative understanding of shock waves, turbulence, and solitons, but new methods are needed to tackle complex fluids such as foams, suspensions, gels, and liquid crystals. Mathematical Challenge Five: Biological Quantum Field Theory * Quantum and statistical methods have had great success modeling virus evolution. Can such techniques be used to model more complex systems such as bacteria? Can these techniques be used to control pathogen evolution? Mathematical Challenge Six: Computational Duality * Duality in mathematics has been a profound tool for theoretical understanding. Can it be extended to develop principled computational techniques where duality and geometry are the basis for novel algorithms? Mathematical Challenge Seven: Occam's Razor in Many Dimensions * As data collection increases can we "do more with less" by finding lower bounds for sensing complexity in systems? This is related to questions about entropy maximization algorithms. Mathematical Challenge Eight: Beyond Convex Optimization * Can linear algebra be replaced by algebraic geometry in a systematic way? Mathematical Challenge Nine: What are the Physical Consequences of Perelman's Proof of Thurston's Geometrization Theorem? * Can profound theoretical advances in understanding three dimensions be applied to construct and manipulate structures across scales to fabricate novel materials? Mathematical Challenge Ten: Algorithmic Origami and Biology * Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding. Mathematical Challenge Eleven: Optimal Nanostructures * Develop new mathematics for constructing optimal globally symmetric structures by following simple local rules via the process of nanoscale self-assembly. Mathematical Challenge Twelve: The Mathematics of Quantum Computing, Algorithms, and Entanglement * In the last century we learned how quantum phenomena shape our world. In the coming century we need to develop the mathematics required to control the quantum world. Mathematical Challenge Thirteen: Creating a Game Theory that Scales * What new scalable mathematics is needed to replace the traditional Partial Differential Equations (PDE) approach to differential games? Mathematical Challenge Fourteen: An Information Theory for Virus Evolution * Can Shannon's theory shed light on this fundamental area of biology? Mathematical Challenge Fifteen: The Geometry of Genome Space * What notion of distance is needed to incorporate biological utility? Mathematical Challenge Sixteen: What are the Symmetries and Action Principles for Biology? * Extend our understanding of symmetries and action principles in biology along the lines of classical thermodynamics, to include important biological concepts such as robustness, modularity, evolvability, and variability. Mathematical Challenge Seventeen: Geometric Langlands and Quantum Physics * How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa? Mathematical Challenge Eighteen: Arithmetic Langlands, Topology, and Geometry * What is the role of homotopy theory in the classical, geometric, and quantum Langlands programs? Mathematical Challenge Nineteen: Settle the Riemann Hypothesis * The Holy Grail of number theory. Mathematical Challenge Twenty: Computation at Scale * How can we develop asymptotics for a world with massively many degrees of freedom? Mathematical Challenge Twenty-one: Settle the Hodge Conjecture * This conjecture in algebraic geometry is a metaphor for transforming transcendental computations into algebraic ones. Mathematical Challenge Twenty-two: Settle the Smooth Poincare Conjecture in Dimension 4 * What are the implications for space-time and cosmology? And might the answer unlock the secret of "dark energy"? Mathematical Challenge Twenty-three: What are the Fundamental Laws of Biology? * Dr. Tether's question will remain front and center in the next 100 years. I place this challenge last as finding these laws will undoubtedly require the mathematics developed in answering several of the questions listed above.

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[math-fun] darpa math problem list (warning: opinions ahead)
by Chris Landauer 29 Sep '07

29 Sep '07

hihi, all - i was at darpatech last month i believe, and i saw that list of ``math'' problems - i decided that most of them are modeling problems, not exactly mathematics, and i noticed that the ``real' math problems had next to no description i thought the list was pretentious but it is right in line with some of the bad things happening there, that i hope will change also, separately, i don't see why math is in nsf's purview, not darpa's - when my colleague was there (93-97), there were severl good math projects, mostly with specific application aras, and there are still some applied math study areas that require good theoretical advances before much will come of them mroe later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org

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Re: [math-fun] El Matemático question
by Danny Hillis 29 Sep '07

29 Sep '07

-Danny (via blackberry) -----Original Message----- From: math-fun-bounces(a)mailman.xmission.com <math-fun-bounces(a)mailman.xmission.com> To: Gareth McCaughan <gareth.mccaughan(a)pobox.com> CC: math-fun(a)mailman.xmission.com <math-fun(a)mailman.xmission.com> Sent: Sat Sep 22 15:21:10 2007 Subject: Re: [math-fun] El Matemático question Many thanks Gareth! I looked on the Internet before posting here buy obviously missed the link you gave. Jim On 9/22/07, Gareth McCaughan <gareth.mccaughan(a)pobox.com> wrote: > On Saturday 22 September 2007, James Buddenhagen wrote: > > > In 1918 (or 1919?) the acclaimed Mexican artist Diego Rivera > > painted a painting called "El Matemático" (the mathematician). > ... > > I am trying the find out who is depicted in that painting, > > which, by the way, was done in Paris. > > According to http://www.vf.utwente.nl/~jagersaa/D_R.html > it was one René Paresce, who got a PhD in physics, spent > some time as a journalist and ended up as a painter. > > -- > g _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Compute a Function From its Fourier Series?
by Bill Gosper 28 Sep '07

28 Sep '07

Bob Baillie asked >>One of the first examples of Fourier series that a student encounters is something like this: f(x) = x/2, for -Pi < x < Pi. The Fourier series is Sum[ (-1)^(n+1)/n * Sin[n x] ]. A second, more complicated example is: f(x) = -Pi/2, for -Pi < x < 0, f(x) = Pi/2, for 0 < x < Pi. This Fourier series is Sum[ (1 - (-1)^n)/n * Sin[n x] ]. >>Can one work backwards, from the coefficients to the function? And I answered incompletely: >About all you can do is replace sin(x) with Im(e^ix) and treat the problem as a hypergeometric series, of which you can only do the easiest cases. [...] Indeed, I once sent this list inf t ==== Gamma(-----) \ log(n) sin(n t) 2 %pi > --------------- = %pi log(------------) / n %pi ==== n = 1 t sin(-) 2 %pi log(------) + %gamma (t - %pi) + log(2 %pi) t 2 + -------------------------------------------------, 2 (0<t<2 pi, where %gamma = Euler's constant ~ .57721566), whose derivation was given in my Fields Institute paper about integrating log Gamma. At the other (i.e., simple) extreme, one can merely iteratively integrate dx Bob's examples and get, e.g., sum(cos(n*x-%pi*k/2)/n^k,n,1,inf)=-2^k*%pi^k*bernpoly(x/(2*%pi),k)/(2*k!) inf %pi k k k x ==== cos(n x - -----) 2 %pi bernpoly(-----, k) \ 2 2 %pi > ---------------- = - -------------------------- / k 2 k! ==== n n = 1 (0<=x<=2 pi, except < when n=1, which ought to be in Mathworld someplace), and its generating function (also a Fourier series) sum(((n*sin(n*x)-t*cos(n*x))/(t^2+n^2)),n,0,inf) =((%pi*(%e)^(t*x))/(1-(%e)^(2*%pi*t)))-1/(2*t) inf ==== t x \ n sin(n x) - t cos(n x) %pi %e 1 > ----------------------- = ------------- - --- / 2 2 2 %pi t 2 t ==== t + n 1 - %e n = 0 The former gives the polylog reciprocation formula (%e)^( - ((%i * %pi * k)/2)) * li[k](z) + (%e)^((%i * %pi * k)/2) * li[k]((1/z) ) = - ((2^k * (%pi)^k * bernpoly( - ((%i * log(z))/(2 * %pi)), k))/(k!)) %i %pi k %i %pi k k k %i log(z) - -------- -------- 2 %pi bernpoly(- ---------, k) 2 2 1 2 %pi %e li (z) + %e li (-) = - -------------------------------- k k z k! (which seems also absent from Mathworld, even for k=2). At least for some part of its nightmarish Riemann surface (which Gene once worked out). But notice we only get sum(sin/n^odd) or cos/n^even. Using the hypergeometric approach, sum(((cos(n * x))/n),n,1,inf) = - log(2 * sin((x/2))) inf ==== \ cos(n x) x > -------- = - log(2 sin(-)) / n 2 ==== n = 1 But then when we try the iterated integral, we are met at step one with sum(((sin(n*x))/(n^2)),n,1,inf)=((%i*(li[2]((%e)^(-%i*x))-li[2]((%e)^(%i*x))))/2) inf - %i x %i x ==== %i (li (%e ) - li (%e )) \ sin(n x) 2 2 > -------- = --------------------------------, / 2 2 ==== n n = 1 and seem to need another special function to express it in strictly real terms. --rwg --------------------------------- Don't let your dream ride pass you by. Make it a reality with Yahoo! Autos.

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Re: [math-fun] Mathematical cosmos article by Max Tegmark
by Dan Asimov 27 Sep '07

27 Sep '07

Er, I meant to send that to Rich first, but instead sent it directly to all of math-fun. Oh, well. A friend sent me that Tegmark article from the Sept. 14 New Scientist magazine, and the idea that physics is ultimately nothing more than the underlying mathematics seemed worth sending on to math-fun. I'd be very curious to learn people's reactions to that article. --Dan

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[math-fun] Re: Collectives
by Steve Witham 27 Sep '07

27 Sep '07

>From: Bill Gosper <rwmgosper(a)yahoo.com> > > JZ>Oddly, a quick perusal uncovers no specific word for a collection of > >collective nouns. I have to assume I've just missed it. And a > >collection of those words would be...? > > Plague. A plague of collective nouns, but a "visitation" of plagues. Also, a locus of points but a plague of loci. --Steve

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[math-fun] Compute a Function From its Fourier Series?
by Bill Gosper 26 Sep '07

26 Sep '07

Bob Baillie asked >One of the first examples of Fourier series that a student encounters is something like this: f(x) = x/2, for -Pi < x < Pi. The Fourier series is Sum[ (-1)^(n+1)/n * Sin[n x] ]. A second, more complicated example is: f(x) = -Pi/2, for -Pi < x < 0, f(x) = Pi/2, for 0 < x < Pi. This Fourier series is Sum[ (1 - (-1)^n)/n * Sin[n x] ]. Can one work backwards, from the coefficients to the function? About all you can do is replace sin(x) with Im(e^ix) and treat the problem as a hypergeometric series, of which you can only do the easiest cases. Rarely one might use a bibasic series with q=e^ix and p=1. To see the hopelessness of the general case, note that the Snowflake and more general fractals have fairly nice Fourier coefficients: http://gosper.org/fst.dvi, with (poorly scanned) figures http://gosper.org/fst1.png, http://gosper.org/fst1.png, http://gosper.org/fst1.png, and http://gosper.org/fst1.png . I have even seen a fairly simple Fourier series for the boundary of the Mandelbrot set. Just imagine writing a closed form for Im(Snowflake). That paper also gives the Fourier series for an arc repeated around the edges of a regular polygon or "star", and by letting the dimension parameter d=1, the Fourier series for the polygon itself. By studying these, we can learn to recognize the series for piecewise linear functions. --rwg PS, on the Lisp Machine I once had a demo where Macsyma computed in closed form the F.s. of a point alternately tracing an equilateral triangle and its inscribed circle, which I then animated as a sum of rotating vectors, using that machine's superior color graphics primitives. --------------------------------- Take the Internet to Go: Yahoo!Go puts the Internet in your pocket: mail, news, photos & more.

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[math-fun] Re: math-fun Digest, Vol 55, Issue 12
by Steve Witham 26 Sep '07

26 Sep '07

>From: Bill Gosper <rwmgosper(a)yahoo.com> > > >) rabbits may be any of warren, nest, colony, bevy, bury, drove, or trace. > > Plague. > > JZ>Oddly, a quick perusal uncovers no specific word for a collection of > >collective nouns. I have to assume I've just missed it. And a > >collection of those words would be...? > > Plague. Yeah, as in, "a plague of lexicologists." --Steve

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