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[math-fun] amazing photo
by Thane Plambeck 16 Feb '07

16 Feb '07

>From a flickr user who's been taking photos of European churches and running them through the MathMap GIMP plugin, getting MC Escher-like results ** snip *** Saint-Etienne-du-Mont church in Paris, one of the most beautiful churches to visit. It is based on an equirectangular (see the immersive version!) panorama and it is transformed into the conformal Guyou Doubly Periodic mapping. *** snip ** click here http://www.flickr.com/photos/sbprzd/391549837/ -- Thane Plambeck tplambeck(a)gmail.com http://www.plambeck.org/ehome.htm

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[math-fun] Quantum computer solves Sudoku
by Henry Baker 16 Feb '07

16 Feb '07

FYI -- I'm glad to see that quantum computers are working on useful things: http://www.nature.com/news/2007/070212/full/070212-8.html

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[math-fun] one-line/proof without words/elegant proof that coeffs of (1+x)^n approx a Gaussian?
by Henry Baker 15 Feb '07

15 Feb '07

Is there a simple/elegant/one-line proof that the coefficients of (1+x)^n approximate a Gaussian curve? I know that it is the convolution of n boxes, and that a large convolution of non-impulses approaches a Gaussian, but is there a quick/obvious proof of this that also gives you the appropriate mean & std deviation parameters?

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Re: [math-fun] one-line/proof without words/elegant proof thatcoeffs of (1+x)^n approx a Gaussian?
by Daniel Asimov 15 Feb '07

15 Feb '07

<< Why should a discrete distribution (the binomial coefficients) approach the continuous distribution ~ exp(-(x/sigma)^2/2), . . . >> Because the limit is taken of cumulative probability functions (cdf) F_n(x) = Prob{X_n <= x) (where X_n's are random variables with the distributions in question). --Dan

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[math-fun] sections of quadratic surfaces
by R. William Gosper 15 Feb '07

15 Feb '07

Some find it counterintuitive that slicing a cone gives an ellipse and not an oval. What about slicing a paraboloid of rotation? More generally how can eliminating a quadratic mumbloid vs a linear (planar) equation yield anything but a quadratic (conic)? --rwg o o o o o o o o o o o o o o o o o o o o o o o o o kosher kosher salt

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Re: [math-fun] sections of quadratic surfaces
by Daniel Asimov 14 Feb '07

14 Feb '07

Re "ovals" Duh -- it never occurred to me before now that the word "oval" derives from the word "ovum".. --Dan

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Re: [math-fun] sections of quadratic surfaces
by Daniel Asimov 14 Feb '07

14 Feb '07

Bill G. writes: << Some find it counterintuitive that slicing a cone gives an ellipse and not an oval. What about slicing a paraboloid of rotation? More generally how can eliminating a quadratic mumbloid vs a linear (planar) equation yield anything but a quadratic (conic)? >> What happens when a plane passes through both sheets of a hyperboloid (of two sheets) in R^3, but contains the vertex of only one sheet? (Naively it's hard to believe the intersection has the symmetry of a hyperbola.) ------------------------------------------------------------------------------------------- But to escalate the question a bit, what happens when two quadratic surfaces in R^3 intersect? It appears that all kinds of bizarre space curves are possible (that I never learned about in school). --Dan

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Re: [math-fun] Matrix Calculus
by Henry Baker 13 Feb '07

13 Feb '07

One of the coolest algorithms of all time is the use of Newton iterations to compute the "polar decomposition" of a matrix: http://www.cs.wisc.edu/graphics/Courses/838-s2002/Papers/polar-decomp.pdf I believe that this algorithm is used in graphics animation systems. Re quaternions: matrix algorithms can always be used, by simply using the matrix version of a quaternion (either real 4x4 or complex 2x2). I am not aware of any Newton iterations for quaternions that aren't inherited from real/complex numbers (and therefore assume commutativity), or are inherited from more general matrices. *** Challenge *** Are there any such Newton algorithms that work _only_ for quaternions? At 09:09 AM 2/13/2007, Fred lunnon wrote: >Recent discussion of Newton-Raphson iteration led me to >reconsider the of the iterative algorithm for reciprocal of a >scalar c --- > x <- 2 x - a x^2 >--- derived via N-R applied to the scalar function of a scalar > f(x) = c - 1/x. > >This has a matrix analogue (mentioned here a while back, >in a thread discussing the shortcomings of Gauss-Seidel >iteration) for the inverse of a matrix A --- > X <- 2 X - X A X --- >which, by a rather nice twist, can actually be employed in >inverting the Jacobian matrix required to apply Newton- >Raphson to a vector function of a vector variable! > >But what matrix calculus framework might permit the >matrix iteration to be derived in a fashion analogous >to the scalar version? In particular, is it possible to restrict >the universe of matrix functions --- as is done for functions >of a complex variable, and attempted less successfully for >a quaternion variable --- in such a way as to facilitate this? > >There is indeed a recognised topic which appears relevant >--- see on the web e.g. Wikipedia; or appendix D of > John Dattoro "Convex Optimization & Euclidean Distance >Geometry" --- >but closer iinspection reveals this to be merely a reworking >of tensor notation omitting the subscripts: in particular, >the derivative of a matrix function of a matrix variable is >a rank 4 object, in effect a Jacobian. > >Fred Lunnon

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[math-fun] Quintic equation
by Christian Boyer 13 Feb '07

13 Feb '07

With the 5th-degree equation: x(x² + a)(x² + b) = c x, a, b, c real numbers > 0 c > ab what is the fastest method to get an approximated value of x? (very few steps) Christian.

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[math-fun] Matrix Calculus
by Fred lunnon 13 Feb '07

13 Feb '07

Recent discussion of Newton-Raphson iteration led me to reconsider the of the iterative algorithm for reciprocal of a scalar c --- x <- 2 x - a x^2 --- derived via N-R applied to the scalar function of a scalar f(x) = c - 1/x. This has a matrix analogue (mentioned here a while back, in a thread discussing the shortcomings of Gauss-Seidel iteration) for the inverse of a matrix A --- X <- 2 X - X A X --- which, by a rather nice twist, can actually be employed in inverting the Jacobian matrix required to apply Newton- Raphson to a vector function of a vector variable! But what matrix calculus framework might permit the matrix iteration to be derived in a fashion analogous to the scalar version? In particular, is it possible to restrict the universe of matrix functions --- as is done for functions of a complex variable, and attempted less successfully for a quaternion variable --- in such a way as to facilitate this? There is indeed a recognised topic which appears relevant --- see on the web e.g. Wikipedia; or appendix D of John Dattoro "Convex Optimization & Euclidean Distance Geometry" --- but closer iinspection reveals this to be merely a reworking of tensor notation omitting the subscripts: in particular, the derivative of a matrix function of a matrix variable is a rank 4 object, in effect a Jacobian. Fred Lunnon

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