Project Euclid is a project of Cornell University to mathematical journals online. The site hosts a mix of free and for-pay journals, but several journals are available completely free:

• Annals of Mathematics
• Communications in Mathematical Physics
• Nagoya Mathematical Journal
• Pacific Journal of Mathematics
• Probability Surveys
• Proceedings of the Japan Academy, Series A

So for example, if you’re interesting in seeing Feit and Thompson’s proof of the odd order theorem in the Pacific Journal of Mathematics, you can find it here.

Last week, when Michael asked for a list of fundamental theorems in different branches of mathematics, Juan de Mairena suggested the Lax Equivalence Theorem as a candidate. Today on ArXiv I spotted a paper that makes the rather dramatic claim that the theorem is “wrong” — not that it is wrong in the strict mathematical sense, but that its conditions are not realistic for real-world problems. I’m not in a position to evaluate the claim (I never even heard of the result until Juan’s comment), but I thought it was interesting to see a paper on the subject so soon after we discussed it.

You knew it was only a matter of time before Flash was used for pure evil. John Tantalo, who could be using his talents to cure cancer or something, has written the Planarity Flash Game. The game generates random planar graphs and draws them to hide the fact that they are planar. You the player (or rather victim) must move the vertices around to show that it really is planar. The game is hard, but moving the little dots around is incredibly hypnotizing.

Via Eszter at Crooked Timber.

M. Lothaire is a pseudonym for a group of authors who wrote the book Combinatorics on Words. The study of words — strings of letters drawn from a fixed alphabet — is surprisingly fruitful in mathematics. For example, finite words form a basis of the free algebra. Sets of infinite words closed under shifts form dynamical systems known as symbolic dynamical systems. Many apparently more-complicated dynamical systems can be reduced to symbolic systems.

A more complicated application is that of Lyndon words. The property of being an aperiodic word is preserved under cyclic permutations. Let two aperiodic words which are cyclic permutations of each other be considered equivalent. Lyndon words are a particular method of choosing one member of each equivalence class. Surprisingly Lyndon words can be used to write down a vector space basis for the free Lie algebra.

M. Lothaire is back with a sequel, Algebraic Combinatorics on Words. The best part? It’s available online.

Wikipedia now has everything. I was looking up Green’s relations on Google, and there it was. Green’s relations are one of the key concepts in the theory of semigroups, but semigroups themselves are one of the less-fashionable areas of mathematics. So I think it’s safe to say that Wikipedia’s work is done.

Reinhard Diestel has a new edition of his graduate text, Graph Theory available on his website. The book is a dense but broad introduction to the field.

If you’re interested in recent developments in abstract algebra, an excellent place to look is the homepage of William Crawley-Boevey. He provides lecture notes covering quivers (which we’ve discussed before), the cohomological approach to central simple algebras, and invariant theory.

The NUMDAM Project is digitizing the archives of French mathematical journals, and placing them online. Some of the journals are quite venerable — the Annales Scientifiques de l’École Normale Supérieure was founded in 1864. Another highlight is the prestigious Publications Mathématiques de l’IHÉS.

I was looking at the Wikipedia entry for Harmonic number, where I spotted a rather surprising reformulation of the Riemann hypothesis.

The Riemann hypothesis was already known to be equivalent to a not-very-complicated statement about the distribution of primes. Let π be the number of primes less than n. Then the Riemann hypothesis is equivalent to:

for all ε > 0. This fact, which goes back at least to Riemann, is the main reason why the Riemann hypothesis is of interest. In 2002, Jeffrey Lagarias found an even more elementary statement.:

where H_n is the nth Harmonic number (the sum of reciprocals less than or equal to n). It almost looks you could solve it, doesn’t it?