Archive for April, 2010

EDP14 — strategic questions

April 25, 2010

In this post I want to take the following attitude. Although there are several promising approaches to solving EDP, I am going to concentrate just on the representation-of-diagonals idea and pretend that that is the problem. That is, I want to pretend that the main problem we are trying to solve is not a problem about discrepancy of \pm 1 sequences in HAPs but the following question instead.

Problem. Is it true that for every positive constant C there exists a diagonal matrix with trace at least C that can be expressed as a linear combination of the form \sum_i\lambda_i P_i\otimes Q_i with \sum_i|\lambda_i|=1 and each P_i and Q_i the characteristic function of a homogeneous arithmetic progression?

There are other equivalent ways of formulating this problem, but I’ll stick with this one for now. Incidentally, P_i\otimes Q_i can be thought of as notation for the characteristic function of P_i\times Q_i.

In this post I want to try to encourage a certain stepping back. Our general problem is to construct something with some rather delicate properties. We don’t really know how to go about it. In that kind of situation, what is one to do? Does one wait to be struck suddenly by a brilliant idea? Or is there a way of searching systematically? Of course, I very much hope it will be the latter, or at least nearer to the latter. (more…)

A little experiment

April 22, 2010

For a long time, a side interest of mine has been how people think when they are doing mathematics. Two difficulties in investigating this general question are (i) that it is quite hard to examine one’s own thought processes reliably (since very often what one remembers of these processes after solving a problem is a very tidied up version of what actually happened) and (ii) that in any case I am just one mathematician with my own particular style and my own little bag of tricks.

I would therefore be grateful to anyone who was prepared to spend about 90 seconds contributing to a little experiment. What you have to do is solve a simple equation that appears just after the fold and pay close attention to your thought processes as you do so. Once you have done that, you can look at further instructions about how to record the result of your participation.

I stress once again that the whole thing should be quick and easy. But please don’t look at the equation until you are ready to start thinking about it and remembering the sequence of your thoughts, since otherwise there is a danger that the tidying-up process will take over and it will be impossible to get a reliable result. (more…)

Use of mathematics II

April 2, 2010

Today I had an experience that I have had many times before, and so, I imagine, has almost everybody (at least if they are old enough to be the kind of person who might conceivably read this blog post). I was in a queue in a chemist (=pharmacy=drugstore), and I knew that my particular item would be quick and easy to deal with. But I had to wait a while because in front of me was someone who had an item that was much more complicated and time-consuming. In this instance the complexity of the items was not due to their sizes, but a more common occurrence of the phenomenon is something that often happens to me in a local grocery: I want to buy just a pint of milk, say, and I find myself behind somebody who has a big basket of things, several of which have to be weighed, some of which don’t have their prices on, etc. etc.

Suppose you were a shopkeeper with just one till and you wanted to devise a good system that would allow people to jump the queue if they were only going to take a very short time. What might you do? And given that I don’t have a good answer to that question, what am I doing posting about it?