A Problem

It’s like, if I’m not in the middle of a crush of tests, I’m getting ready for being in the middle of a crush of tests … however, things are not all bad. The semester is in fact almost over. 6 more weeks? Nothing.

And next semester should be fun. The way it’s shaping up, it looks to be Graph Theory and Complex Analysis for math, and Nuclear Physics and Electronics for … physics. It’ll be nice to have some actual math to talk about again.

But, I do in fact have a problem to pose. In the usual fashion, it is disturbingly general and open ended. A little more technical than my usual fare,

Given a vector space V with finite dimension, over a field … say Z₃, with a subspace S. What can we say about the set of matrices M such that for any matrix in M, S, and only S, is the kernel of that matrix?

Discuss.

An Inequality

So, poking along through thermodynamics, in comparing temperatures of different systems, my solution resulted in the following inequality. For ,

I’m pretty sure I’ve seen this in analysis, or something, years ago, but I certainly couldn’t recall the proof straight off and had to re-derive it.

Trig Sums

We are interested, and by we I mean I, and hopefully that means by extension, you, in, for an integer n,

For an arbitrary n, what is the value of that summation?