This was the first meeting of PSG over the summer of 2011. People around this summer include: Lucas Van Meter, Bill Huber, Kirsten Baer, Paul Gallahger, Mitchell Johnston, Chang Cao, and Nuzhat Arif.
At this meeting we discussed three problems, listed below:
1. Let (a[n]), n = 0, 1, 2, …, be an infinite sequence of positive integers such that gcd(a[n+1], a[n]) > a[n-1] for all n >= 1. Prove that a[n] >= 2^n for n >= 1. (I haven’t thought about this or checked it, so some adjustment of the index n might be needed for the result to be true.)
2. The functional equation f(2p-1)/2 = f(Sqrt(p)) – constant, 0 < p < 1, is satisfied by Arcsine. What is a “minimal” additional set of conditions (such as continuity, antisymmetry, differentiability, slope at 0, whatever) needed to make this uniquely characterize Arcsine?
We assume that the wheel has thin spokes connecting the circular rim radially towards a hub at the center. (Some performance wheels are actually built this way.) The wheel is assumed to roll in a plane parallel to the plane of the film; the camera does not pan. The shutter speed is assumed to be slow enough so that there is some blurring, but fast enough so that some parts of some spokes are without visible blur. The problem is to determine which parts of which spokes will be the most clearly imaged.
What generalizations and assumptions are needed to make this question more realistic? (It would be nice to make an empirical test of our solution!)
Here is a summation of our discussion on the first problem written by lucas. It is not complete yet but maybe lucas will finish it later. PSG 6-2-11
We talked about the bicycle problem but did not write up any math analysis of the problem. Anyone care to comment on it?
We made a lot of work on the functional equation. Some of that can be seen in the photos.
Lastly we have some photos that we took during the meeting.