Hyperreal numbers: infinities and infinitesimals - "In 1976, Jerome Keisler, a student of the famous logician Tarski, published this elementary textbook that teaches calculus using hyperreal numbers. Now it's free, with a Creative Commons copyright!" (pdf—25mb :) [more inside]
Network Theory Overview - "The idea: nature and the world of human technology are full of networks! People like to draw diagrams of networks. Mathematical physicists know that in principle these diagrams can be understood using category theory. But why should physicists have all the fun? This is the century of understanding living systems and adapting to life on a finite planet. Math isn't the main thing we need, but it's got to be part of the solution... so one thing we should do is develop a unified and powerful theory of networks." (via ;)
...to leave a smile on your face, by Helder Guimarães: Individual vs Crowd | Chaos | Freedom | Trick [more inside]
GaMuSo is an application of BioGraph-based data mining to music, which helps you get recommendations for other musicians. Based on 140K user-defined tags from last.fm that are collected for over 400K artists, results are sorted by the "nearest" or most probable matches for your artist of interest (algorithm described here). [more inside]
Bayesian analysis shows redshirts are not most likely to die on Star Trek:TOS. Although Enterprise crew members in redshirts suffer many more casualties than crew members in other uniforms, they suffer fewer casualties than crew members in gold uniforms when the entire population size is considered. Only 10% of the entire redshirt population was lost during the three year run of Star Trek. This is less than the 13.4% of goldshirts, but more than the 5.1% of blueshirts. What is truly hazardous is not wearing a redshirt, but being a member of the security department. The red-shirted members of security were only 20.9% of the entire crew, but there is a 61.9% chance that the next casualty is in a redshirt and 64.5% chance this red-shirted victim is a member of the security department. The remaining redshirts, operations and engineering make up the largest single population, but only have an 8.6% chance of being a casualty.
The Nature of Computation - Intellects Vast and Warm and Sympathetic: "I hand you a network or graph, and ask whether there is a path through the network that crosses each edge exactly once, returning to its starting point. (That is, I ask whether there is a 'Eulerian' cycle.) Then I hand you another network, and ask whether there is a path which visits each node exactly once. (That is, I ask whether there is a 'Hamiltonian' cycle.) How hard is it to answer me?" (via) [more inside]
H _ _ _ m _ n, Y a _ _ _ e e, _ _ t t _ _ _ h i p, _ h u t _ s & L a _ _ e r _ , R _ _ k , _ _ n d y _ _ _ _ , and _ _ r t s.
The year was 1945. Two earthshaking events took place: the successful test at Alamogordo and the building of the first electronic computer. Their combined impact was to modify qualitatively the nature of global interactions between Russia and the West. No less perturbative were the changes wrought in all of academic research and in applied science. On a less grand scale these events brought about a [renaissance] of a mathematical technique known to the old guard as statistical sampling; in its new surroundings and owing to its nature, there was no denying its new name of the Monte Carlo method (PDF). -N. MetropolisConceptually talked about on MeFi previously, some basic Monte Carlo methods include the Inverse Transform Method (PDF) mentioned in the quoted paper, Acceptance-Rejection Sampling (PDFs 1,2), and integration with and without importance sampling (PDF).
An "Exciting Guide to Probability Distributions" from the University of Oxford: part 1, part 2. (Two links to PDFs)
Durango Bill's Home Page. With topics that include: 3D end-to-end tour of the Grand Canyon, the origin and formation of the Colorado River, and examples of river systems that cut through mountain ranges instead of taking easier routes around them in Ancestral Rivers of the World. [more inside]
The Monty Hall Problem has struck again, and this time it’s not merely embarrassing mathematicians. If the calculations of a Yale economist are correct, there’s a sneaky logical fallacy in some of the most famous experiments in psychology." The NY Times' John Tierney reports on new research into cognitive dissonance as examined through the famous Monty Hall Problem. [A previous MetaFilter thread about the Monty Hall Problem: Let's Make A Deal!]
Interactive mathematics miscellany and puzzles, including 75 proofs of the Pythagorean Theorem, an interactive column using Java applets, and eye-opening demonstrations. (Actually, much more.)
Know less than nothing!? What could negative knowledge possibly mean? In short, after I tell you negative information, you will know less... "In this week's issue of Nature, however, Michal Horodecki and colleagues present a fresh approach to understanding quantum phenomena that cannot be grasped simply by considering their classical counterparts." [via slashdot :]
The Logic of Diversity "A new book, The Wisdom of Crowds [..:] by The New Yorker columnist James Surowiecki, has recently popularized the idea that groups can, in some ways, be smarter than their members, which is superficially similar to Page's results. While Surowiecki gives many examples of what one might call collective cognition, where groups out-perform isolated individuals, he really has only one explanation for this phenomenon, based on one of his examples: jelly beans [...] averaging together many independent, unbiased guesses gives a result that is probably closer to the truth than any one guess. While true — it's the central limit theorem of statistics — it's far from being the only way in which diversity can be beneficial in problem solving." (Three-Toed Sloth)
An Intuitive Explanation of Bayesian Reasoning. [Page contains Java]