Ars Mathematica

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.

MOND, the acronym for MOdified Newtonian Dynamics is a theory put forth by Moti Milgrom in 1983 to resolve problems with galaxy kinematics without resorting to dark matter. The theory decouples inertial and gravitational mass (breaking the equivalence principle) positing that at very small accelerations (those below an observationally determined constant a₀), the gravitation force felt by a body is actually smaller than the force predicted by the famous Newtonian equation F = ma. Aside from the bizzare nature of the change, the theory has several things going for it, not the least of which is that it made predictions that were later verified (namely the existence of low surface brightness galaxies).

Another point in its favor is that it can explain the Pioneer anomoly. When I first heard of the theory and the problems it was meant to solve, I thought that the theory was crazy, but that it might just be crazy enough to be true. I even bandied about the idea of writing a popular science book about it which itself would have several things going for it:

• MOND could actually be correct - such a book would be early to the game
• While there are many science popularization books written, there certainly aren’t many speculative hard science popularizations written - unless you count all the string theory stuff - it could jump start a whole category!

Some final food for thought; I haven’t done the calculation myself, but apparently constant acceleration at a₀ for our best guess at the age of the universe produces a velocity of - wait for it - the speed of light. I don’t know if this new world would be cool enough for Walt to have to put on shades before he glanced at it, but quite a few textbooks would need to be rewritten

And some MOND links:

• The MOND pages
• Dr Dark Matter