February 2010 - mathfun@mailman.xmission.com

Re: [math-fun] cybernetics
by Dan Asimov 04 Feb '10

04 Feb '10

Steve Witham wrote: << [I wrote:] << [***comment expressing skepticism over whether the upcoming "Cybernetics" conference at RPI is actually connected to cybernetics, and wondering if cybernetics per se has any theorems.***] >> The hard results were and still are in control theory. I know Wiener's no slouch, I think what happened was people hoped to get traction by looking at feedback loops, loops controlling the parameters of loops, etc., in hierarchies... but didn't get anywhere. So it is cybernetics itself (not just this conference) that has kooks. >> The control theory I'm aware of is quite specific: It addresses the problem of how to steer most efficiently from one point to another in some phase space -- assumed locally Euclidean -- when the allowable steering options are subject to infinitesimal constraints. (Example: How do you best roll a sphere on a plane without slippage so its net motion is some prescribed rotation plus translation? Here "best" is defined by minimizing some cost function defined on all allowable itineraries.) << For instance people who want to manage economies with feedback systems, without really getting that economies already are multilayered feedback systems attempting to "route around damage" as the saying goes. >> Don't know much economics. Are there standard economics models as "multilayered feedback systems" ? << It's similar to how Complex Systems had some interesting results and then kooks. >> Despite having done research in differential dynamical systems, I never really understood what the field of "Complex Systems" or for that matter "Chaos Theory" were (and extend my skepticism to those fields as well). << The funny thing is, hierarchical feedback systems are making a comeback with people like Jeff Hawkins, only it turns out every little piece of knowledge is its own feedback loop, >> How so? << so (assuming they work) you don't get general high-level results, just general infrastructure designs and lots of specific stuff for a system to learn. >> I like the idea of hierarchical feedback systems (though maybe they can always be relabeled so as to be non-hierarchical?). In any case they're probably useful in a field I'd love to see more progress in -- causal inference. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] 2x2 matrices, eigenvectors, complex numbers
by Henry Baker 03 Feb '10

03 Feb '10

The general 2-D linear mapping over reals can be represented by a 2x2 matrix of reals: [x1 y1]M = [x2 y2] But this mapping can also be represented by 2 complex numbers A, B, which representation also has 4 real parameters: (x+iy) = A*(x+iy)+B^2*(x-iy), or more succinctly, z=Az+B^2z', where z' is the conjugate of z. (We'll see the reason for using B^2 instead of B shortly.) Now the matrix corresponding to (A,B) is symmetric iff A is real, so the eigenvalues of this mapping are real: lambda1,2 = A+-|B|^2 = A+-BB' We could go to the trouble of solving for the eigenvectors, but they are completely obvious (& obviously orthogonal): B, iB. Consider B: A(B)+B^2(B)' = AB+BBB' = AB+(BB')B = (A+BB')B = lambda1 B Consider iB: A(iB)+B^2(iB)' = A(iB)-BBiB' = A(iB)-BB'(iB) = (A-BB')iB = lambda2 B --- I'm sure that something along these lines can be done with quaternions; Cayley and/or Hamilton must have already done this, but it's difficult to decode their writings. More to come.

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[math-fun] Exponentially speaking
by Dan Asimov 02 Feb '10

02 Feb '10

In the weekly NY Times column about proper usage, a commenter writes: << A note about "exponential": technically, the term refers to numbers that are increasing by a fixed (compounded) percentage, rather than a fixed amount, per unit of time. So it is entirely possible that the number of schools teaching Chinese has increased by a steady 18% per year from 300 a decade ago to 1,600 today; that would indeed be an exponential increase, though it's unlikely the author of the article did the research to confirm this. >> Am I the only one who thinks this comment's writing style looks familiar? --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] Exponentially speaking
by Dan Asimov 02 Feb '10

02 Feb '10

Rich wrote: << . . . The remark does illustrate the need for at least three data points, for a phenomenon to be called exponential. --- [I wrote:] << A note about "exponential": technically, the term refers to numbers that are increasing by a fixed (compounded) percentage, rather than a fixed amount, per unit of time. So it is entirely possible that the number of schools teaching Chinese has increased by a steady 18% per year from 300 a decade ago to 1,600 today; that would indeed be an exponential increase, though it's unlikely the author of the article did the research to confirm this. . . . >> >> It may be a bit tricky to know just how many data points are needed, since any finite set of points (t,f(t)) fits some polynomial f(t) = P(t). But I'd also distinguish between technical and common usage of the term. I wouldn't want to encourage common usage of exponential if it merely means "very fast" -- which was the point of the guy who wrote the column at < http://topics.blogs.nytimes.com/2010/02/02/when-spell-check-cant-help-4/ > -- but to use it for something that *isn't* increasing especially fast would be confusing to many (outside of a technical context). --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] New York Times, The Math Column
by Dan Asimov 02 Feb '10

02 Feb '10

Hilarie wrote: << If you've got New York Times access, this new column is the place to watch: >From Fish to Infinity By STEVEN STROGATZ http://opinionator.blogs.nytimes.com/2010/01/31/from-fish-to-infinity/ >> It's great that the NY Times has started a math column, and when Strogatz guest-wrote a few math columns (in biologist Olivia Judson's space) last year, they were pretty interesting: http://opinionator.blogs.nytimes.com/2009/05/19/math-and-the-city/ http://opinionator.blogs.nytimes.com/2009/05/26/guest-column-loves-me-loves… http://opinionator.blogs.nytimes.com/2009/06/02/guest-column-like-water-for… . But I found his Jan. 31 math column to have very little substance. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] New York Times, The Math Column
by Hilarie Orman 02 Feb '10

02 Feb '10

If you've got New York Times access, this new column is the place to watch: >From Fish to Infinity By STEVEN STROGATZ http://opinionator.blogs.nytimes.com/2010/01/31/from-fish-to-infinity/ ---- Hilarie

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Re: [math-fun] cybernetics
by Dan Asimov 01 Feb '10

01 Feb '10

Harold writes: << I havent heard the term applied to the internet, and I think it has somewhat fallen out of favor and been forgotten. >> Maybe not cybernetics, but the prefix cyber-, is what I was saying: << I know, or think I know, Norbert Wiener coined the word cybernetics and wrote a book about it. Beyond that, I know the prefix cyber- has often been used to mean computer- or Internet-related. >> E.g., < www.m-w.com > defines cyber- as follows: << : of, relating to, or involving computers or computer networks (as the Internet) <the cyber marketplace> --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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