DON Two for the Mathematically Inclined

January 11, 2013

We’ve found a couple of treats for all you math lovers. “The Lambda Calculus,” presented to us by the Stanford Encyclopedia of Philosophy, delves into this function-representing notation. Then there’s Math ? Programming’s piece on a landmark algorithm from 1964, “The Fast Fourier Transform.”

The lambda calculus piece by post-doctoral researcher Jesse Alama is extensive, covering everything from specifics like the use of the notation in multi-argument operations to the history behind the concepts and their applications. The article describes:

“The syntax of basic ?-calculus is quite sparse, making it an elegant, focused notation for representing functions. Functions and arguments are on a par with one another. The result is an intensional theory of functions as rules of computation, contrasting with the traditional extensional approach one of function as a set of pairs of a certain kind. Despite its sparse syntax, the expressiveness and flexibility of the ?-calculus make it a cornucopia of logic and mathematics.”

The article on Fast Fourier Transform algorithm emphasizes that our modern reliance on digital signal processing rests on the legacy of this breakthrough. Writer and mathematics PhD student Jeremy Kun explains:

“It’s often said that the Age of Information began on August 17, 1964 with the publication of Cooley and Tukey’s paper, ‘An Algorithm for the Machine Calculation of Complex Fourier Series.’. . .

“Digital audio and video, graphics, mobile phones, radar and sonar, satellite transmissions, weather forecasting, economics and medicine all use the Fast Fourier Transform algorithm in a crucial way. (Not to mention that electronic circuits wouldn’t exist without Fourier analysis in general.) Before the Fast Fourier Transform algorithm was public knowledge, it simply wasn’t feasible to process digital signals.”

This thorough article takes us through the derivation of the algorithm, then describes how Kun applied it to the task of filtering white noise from a sound clip.

Both these posts are full of details that are, I’m afraid, over my head. If such calculations are your thing, though, enjoy.

Cynthia Murrell, January 11, 2013

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Written by Stephen E. Arnold · Filed Under News

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Stephen E. Arnold monitors search, content processing, text mining and related topics from his high-tech nerve center in rural Kentucky. He tries to winnow the goose feathers from the giblets. He works with colleagues worldwide to make this Web log useful to those who want to go "beyond search". Contact him at sa [at] arnoldit.com. His Web site with additional information about search is arnoldit.com.