## Archive for November, 2010

### conformal cyclical cosmology

Something in the news here today, referring also to Penrose’s paper from several years ago.

In my limited understanding, Penrose suggests that the universe goes through these cycles of what can be interpreted as infinite expansions “followed by” big bangs, where the cycle renewal “happens” in a mathematical sense: in the way spacetime is metrized. He says that in the infinite future, when all massive particles will have evaporated, we will be returned to a situation without a notion of space or time (since all things are lightlike, I suppose). From this, the very large scale of the given final universe can be reinterpreted as the very small beginning of the next universe. It’s an interesting thought.

### a polynomial problem

The latest problem from fakalin. Took some wrong turns and hints to solve it…

Given a polynomial $$f: \mathbb{Z}\to \mathbb{Z}$$ with positive integer coefficients, how many evaluations of $$f$$ does it take to obtain the polynomial?

(An $$f: \mathbb{R}\to \mathbb{R}$$ polynomial with real coefficients would take the number of degrees plus 1 to specify, which, if it held in this case, would render the answer unbounded. But the correct answer in this case is surprisingly small.)

### what’s an application anyway

Stephen Boyd quips about a power allocation algorithm:

Oh by the way, this is used now, for example, in DSL and it’s used actually everywhere. Okay, and I’m not talking about used by …professors… I’m talking about, it’s used when you use DSL.

Sometimes engineers forget that to a mathematician, an “application” is another theoretical problem … only maybe in physics.

### escalator

How do escalators work? I’ve wondered for years how escalators recycled their step blocks internally. At one point I thought they slid past each other on all four faces to save on turning radius (because I thought the blocks locked along grooves). Today I see an escalator under repair. Now the answer is clear. It’s much simpler than that: the blocks just turn along a track in the most obvious way imaginable.