I did my first class over zoom last Friday. For the San Jose State Math Circle I talked about configurations which I introduced as a problem of forming various committees from the board of directors of a corporation subject to the following conditions:

(1) Each director belongs to the same number of committees.
(2) Each committee has the same number of members.
(3) Two directors are on at most one committee together.

Here’s a sample problem from the math circle:

There are 10 directors on the board of Starlight

a. The directors would like to form five committees.
If each director is on two committees, then how many
directors must belong to each committee? Show how the
committees may be formed, or prove that it is not possible.

b. After a few years, the company realizes that it needs
more committees and wants to increase the number of
committees to 10, while reducing the size of each committee
to three. Assuming this is possible, how many committees
must each director belong to? Show how the committees may
be formed, or prove that it is not possible.

The directors and committees are the points and lines of a (combinatorial) configuration. In some case the configuration can be drawn in the plane using points and lines; if that is the case, then the configuration is realized as a geometric configuration. Both problems above have a realization as a geometric configuration (the configuration in part a. is unique, but part b. has 10 different solutions).

And more zoom ahead: I’ll be resuming several of my classes over zoom starting tomorrow.

Coronavirus Update

In this blog I have been graphing the growth of COVID-19 cases in Santa Clara county. Since I lack any background in mathematical epidemiology, and my statistics background, is in general, not very deep, I decided to do some reading in modeling epidemics, so I downloaded Modeling Paradigms and Analysis of Disease Transmission Models, Gumel & Lenhart (Eds.).

From reading just a portion of the introductory material I am changing the variable I track use to make projections. Instead of using the number of new cases as a percentage of the total cases, I will use the Basic Reproduction Number R0. I estimate R0 as Nt/Nt-5 where Nt is the number of new cases on day t. I use 5 because it is believed to be approximately the median incubation period of COVID-19; the idea being that those newly infected 5 days ago infect those detected today.

So here is the graph of the estimate of R0:

As you can see R0 is in a decreasing trend. Assuming the trend is linear, on April 26, R0 will fall below 1, a critical threshold. At that point the number of new cases per day will start to diminish!

Based on that assumption, here is a projection for new cases for next few weeks:

We can see in this graph an inflection point at April 26. It still appears that we are more than a month away from a peak in the number of cases…

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