Archive for October, 2010

III.93 Universal Covers and and III.94 Variational Methods by Lawrence C. Evans: Pages 309 – 310.

Friday, October 8th, 2010

Universal Covers.

A cover of a space X is a space Y and a continuous surjection from Y to X such that the inverse image of a small neighborhood in X is a disjoint union of small neighborhoods in Y.

How is a cover related to a random variable, where inverse images of open sets are open?

If U is a universal cover of X and and Y is any other cover of X, then U can be made into a cover of Y in a natural way.

Variational Methods by Lawrence C. Evans.

Critical points.

One-dimensional variational problems.

Shortest distance in a plane is a straight line.

III.93 Universal Covers: Pages 309.

Thursday, October 7th, 2010

Universal covers.

Loops and simply connected.

III.92 Trigometric Functions by Ben Green: Pages 307 – 308.

Saturday, October 2nd, 2010

Trigometric Functions by Ben Green.

For real or complex z. 

sin(z) = z – z3/3! + z5/5! – z7/7! + …

cos(z) = 1 – z2/2! + z4/4! – z6/6! + …

ez = 1 + z + z2/2! + z3/3! + z4/4! + …

And the coup de grâce:

eiq = cos(q) + i sin(q).

III.91 Transforms by T. W. Körner: Pages 305 – 307.

Friday, October 1st, 2010

Transforms by T. W. Körner.

Complex values extensions, the Fourier transform and transforms in general.