Ars Mathematica

The Hausdorff dimension is used to define the dimension of fractals, for example, the dimension of the Sierpinski triangle is log(3)/log(2).

To find the d-dimensional Hausdorff measure of a set: cover the set with very small balls, sum the diameter to the power of d of each ball, and take the lim inf as the balls get smaller. For integer dimensions, the Hausdorff measure is equivalent to the Lebesgue measure. The Hausdorff dimension of a set is the point where the d-dimensional Hausdorff measure changes from infinity to zero, i.e. the dimension of a set is d^* if for d < d^* its measure is infinity and for d > d^* its measure is zero.

From the abstract for a paper by Dierk Schleicher:

… we construct a set E ⊂ ℂ of positive planar measure and with dimension 2 such that each point in E can be joined to ∞ by one or several curves in ℂ such that all curves are disjoint from each other and from E, and so that their union has Hausdorff dimension 1. We can even arrange things so that every point in ℂ which is not on one of these curves is in E. These examples have been discovered very recently; they arise quite naturally in the context of complex dynamics, more precisely in the iteration theory of simple maps such as z → sin(z).

The Physics and Astronomy department at Georgia State University has an excellent online resource, HyperPhysics. It is a hyperlinked encyclopedia that provides an overview of physics at an undergraduate non-physics major level.

Sigfpe, our most prolific commenter, has the most disturbing photo ever on his weblog.

While going some boxes the other day, I found an index-card-sized piece of paper with a commutative diagram on it, but no other text. Where does it come from? I have no idea. For all I know if you leave any box alone long enough, it starts to sprout commutative diagrams.

Or, how to make a search engine.

Imagine the web is irreducible, by which I mean you could get from any page to any other by following links; pages without links (and pages no one links to) demonstrate that the web is not irreducible — but this is mathematics, so we are not going to let it worry us. Further, imagine there are millions of monkeyspigeons randomly clicking on links (forming a Markov chain). Perron-Frobenius theory can tell us the probability of these random walks through cyberspace visiting a particular page at an instance in time.

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Brad de Long, a Berkeley economist, has an interesting post on his weblog about limitations of current models in economics. The "Marshallian toolkit" means basically the kind of economics you find in a microeconomics textbook (in a more sophisticated form, of course). Brad is claiming that these models simply fail to explain what makes some economies grow and others stagnate.

An example of the kind of model Brad refers to is the Solow growth model, which has three basic inputs: the amount of labor, the amount of capital (which includes things like factories), and a third factor, productivity, which represents how efficiently capital and labor are used. Changes in capital and labor are explainable in terms of conventional economics, but productivity is basically a black box for technological change. When economists fit the model against the data, it turns out that truly dramatic economic growth comes from increase in productivity, the very factor that is beyond the reach of conventional economics.

Lieven Le Bruyn, an algebraist in Belgium, has a weblog. His particular interest is in noncommutative geometry. There are as many approaches to noncommutative geometry as there are noncommutative geometers, but he outlines his particular point of view in a three-part post, here, here, and here.

Lambda the Ultimate links to a new paper, Lambdascope, that describes an implementation of optimal reduction in the lambda calculus.

Optimal, in this case, means the minimum amount of duplication. Usually, when you apply a function in the lambda calculus, you must make a copy of the function body before you start evaluating. In his 1978 thesis, Levy showed that some of this copying could be redundant; in some cases, exponentially so. Levy did not provide an algorithm for optimal reduction, but several have since been invented.

Optimal reduction is a refinement of lazy evaluation. In lazy evaluation, arguments to functions are only copied as needed. Laziness minimizes copying on the right side of function applications, while optimal reduction minimizes it on both sides.

Implementations of optimal reduction are incredibly difficult to understand. The one in the lambdascope paper looks simpler, but I haven’t fully assimilated it.

On ArXiv there is a new survey paper on finite-dimensional algebras and quivers. The paper is rather dense, so it would be tough going for someone not already familiar with the vocabulary of quivers, but it covers some of the surprising connections with Kac-Moody Lie algebras.

Peter Woit’s weblog is an interesting source for information about the intersection of math and physics. His latest is a post on the early history of using group theory in quantum mechanics. While group-theoretic methods in physics (and chemistry) are uncontroversial these days, the original emergence of the subject was painful, with pro- and anti-group theory partisans. (Wolfgang Pauli termed group theory the “Gruppenpest”.)

About a month ago, Slashdot had linked to a completely opaque article about something called the Serre conjecture. The subsequent thread showed Slashdot at its worst, as people desperately typed “Serre conjecture” into Google to find links to stick in a comment to earn some undeserved karma. Unfortunately, Serre was an influential and productive mathematicians, so there are lots of Serre conjectures. One particularly unfortunate soul hit upon a link to the Quillen-Suslin theorem, which was conjectured by Serre in the 1950s but proven in 1976, and complained “do people not do research any more to see if their work has already been done?”

The Serre Conjecture in question is a completely separate conjecture in Galois theory. Many questions in number theory can be reduced to studying the Galois group of the algebraic closure of the rationals over the rationals. This group is gigantic and extremely complicated, so mathematicians try to understand it in pieces, but the pieces themselves are hard to come by. Serre conjectured one approach to understand certain small pieces. The paper in question, by Chandrashekhar Khare, proves a part of that conjecture. So Khare proved a part of a conjecture that only provides a small part of the solution of the general problem, but even that small part has remained unsolved for 18 years. So this is a hard problem we’re talking about. The truly macho can download the paper itself at ArXiv.