Mathematics

A collection of 15 posts

Mathematics

Singular Value Decomposition explained circa 1976

Latent Semantic Indexing (LSI) is used widely today in Semantic Search and has many other uses in Deep Machine Learning. This is a pretty good explanation/visualization from 40 years ago.

Ring probabilities in F#
F#

Ring probabilities in F#

A few months back I took a look at Elixir. More recently I’ve been exploring F# and I’m very pleased with the experience so far. Here

Functional Programming

Purely Functional Data Structures & Algorithms : Union-Find (Haskell)

*Updated 08-23-2012 01:04:38* Replaced the use of Data.Vector with the persistent Data.Sequence which has O(logN) worst case time complexity on updates. A Haskell version of the previous codeΒ using the more efficient(access and update) persistent Data.Sequence type

Functional Programming

Purely Functional Data Structures & Algorithms : Union-Find

It’s been a while since I last posted in this series. Today we look at the disjoint-set data structure, specifically disjoint-set forestsΒ and the complementary algorithm : union-find. InΒ computing, aΒ disjoint-set data structureΒ is aΒ data structureΒ that keeps track of a

History

Codebreaker - A new film about the life of Alan Turing

CODEBREAKER tells the story ofΒ one of the most important people of the 20th century.Β  Alan Turing set in motion the computer age and his World War II codebreaking helped save two million lives. Β Yet few people have heard his name, know his tragic

Bayes's Theorem is more powerful than Jesus
History

Bayes's Theorem is more powerful than Jesus

Richard Carrier puts forward a fantastic approach to verifying history in his latest book : Proving History: Bayes’s Theorem and the Quest for the Historical Jesus “…

Functional Programming

Purely Functional Data Structures & Algorithms : Fast Fourier Transform in Qi

In this second post in this series we look at an implementation of the always useful Fast Fourier Transform. (FFT) An algorithm for computing the Fourier transform of a set of discrete data values. Given a finite set of data points, for example a

Mathematics

Chaitin Proving Darwin

White paper : To a mathematical theory of evolution and biological creativity We present an information-theoretic analysis of Darwin’s theory of evolution, modeled as a hill-climbing algorithm on a fitness landscape. Our space of possible organisms consists of computer programs, which are subjected

Artificial Intelligence / Machine Learning

Artificial Intuition

Artificial Intuition – A New Possible Path To Artificial Intelligence – by Monica Anderson Artificial Intelligence was born in Computer Science departments, and inherited their value sets including Correctness. This mindset, this necessity to be logical, provable, and correct has been a fatal roadblock

Mathematics

Ο€ in assembly (spigot algorithm)

// pi_spigot.s - calculates Pi using a spigot algorithm // as an array of n digits in base 10000. // http://mathworld.wolfram.com/SpigotAlgorithm.html // // x86-64/SSE3 with for Linux, Intel, gnu assembler, gcc // // assemble: as pi_spigot.s -o pi_spigot.o // link:

Mathematics

Happy Ο€ approximation day/night (in assembly) !

// pi_x64.s - calculates Pi using the Leibniz formula. // Each iteration prints a closer approximation to 50 digits. // This is not an optimal implementation and it runs forever. // // x86-64/SSE3 with for Linux, Intel, gnu assembler, gcc // // assemble: as pi_x64.s -o

Functional Programming

Generating Ο€ in Haskell

Haskell beats CL quite comfortably using the same algorithm : module Main( main ) where import System( getArgs ) arccot :: Integer -> Integer -> Integer arccot x unity = arccot' x unity 0 start 1 1 where start = unity `div` x arccot' x unity sum xpower n sign

Functional Programming

Generating Ο€ in CL (faster)

Thanks to metacircular for pointing out that (floor (/ x y)) can be written as (floor x y) while avoiding the intermediate rational. (defun machin-pi (digits) "Calculates PI digits using fixed point arithmetic and Machin's formula with double recursion" (labels ((arccot-minus (xsq n xpower) (let

Functional Programming

Generating Ο€ in CL

Update 2009-07-23 : Faster version in CLΒ and a Haskell version. ——————————————————————————– A trivial approximation using the Leibniz formula.