SCU Mathematics Colloquium Series Schedule
Fall 1998
Talks will be at 4:00 Tuesdays in O'Connor 206. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.
Title: On the first digit problem
It is an empirical fact that the digit "1" occurs in nature more often than other digits. Some examples will be given, as well as a short survey of various explanations of this phenomenon.
Title: Means to an End
The arithmetic-geometric mean inequality insures that the ratio of the geometric mean of a finite list of positive numbers to the arithmetic mean of the same list is less than or equal to one, with equality holding only when all the numbers in the list are the same. [The geometric mean is the n-th root of the product of the n numbers; the arithmetic mean is just the ordinary average of the numbers.] Can we be more specific about the value of that ratio? Suppose we have a sequence of positive numbers, a(n), generated by a polynomial function, i.e., a(n)=p(n) for some fixed polynomial p(x). Let us look at the ratios, GM(n)/AM(n), of the "partial means"--means involving the first n numbers in the sequence. It turns out that if we go out to the "end"--i.e., find the limit of these ratios--we get a very nice number: (s+1)/e^{s}, where s is the degree of p(x). A consequence of this will be the amusing fact that the n-th root of n factorial, when divided by n, goes to 1/e in the limit. This lecture should be accessible to anyone who has had the calculus sequence. The tools involved are not difficult, just basic integration and the liberal use of inequalities. In particular, we'll see how the inequalities that arise in the proof of the integral test can be put to good use for increasing functions!
Title: On the partial regularity of minimizing maps.
This will be an expository talk explaining some of the tricks and difficulties for proving that "energy minimizing" maps in the calculus of variations are smooth. I will in particular discuss a fascinating such problem when there are constraints on the Jacobian of the mapping, and discuss some recent progress (with R. Gariepy) on this question.
Title: Feedback Control of linear systems and the real Schubert Calculus
Website:
http://www.msri.org/people/members/sottile/talks/98/SantaClara.html
In 1981, Brockett and Byrnes showed how the feedback laws which control a given linear system are determined through the Schubert calculus of enumerative geometry. This talk will describe that connection and propose numerical homotopy methods to solve the resulting systems of polynomials.
The related questions of finding real feedback laws and of trying to do Schubert's calculus over the real numbers are intertwined with a precise conjecture of Shapiro and Shapiro. We will discuss this connection and present some partial results and computational evidence in support of this conjecture.
Unfortunately, this presentation has been cancelled.
Title: A Sensor Management Problem.
An aircraft with an array of sensors is not able to use all sensors in all places simultaneously. Givden a configuration of sensors with certain capabilities and a set of goals (such as tracking or target recognitio), what is the best way to deploy the sensors? As usual, definition of the term 'best' is an issue. We will discuss a probabilistic model developed by Keith Kastella (then of Lockheed) and Stan Musick (of AFRL) which amounts to a large probability density function over the physical space, and define 'best' in terms of the discrimination gain; however, the author will spend most of the talk discussing different approaches to choosing the p.d.f. in order to facilitate the computations in the optimization and update phases.
Title: Non-Euclidean Wallpaper
We show how to construct wallpaper for inhabitants of the hyperbolic plane, the Poincarites. Using both the Poincare Half-Plane and Poincare Disk models of a non-Euclidean plane, we construct functions with various wallpaper symmetries. A brief, pictorial introduction to hyperbolic geometry leads to a more technical discussion of the group SL(2,Z) and our method for constructing automorphic functions. The pictures constructed give a vivid view of the world of the Poincarites.
Title: Heuristic Methods in Elementary Number Theory
The famous problems of determining whether or not there are an infinite number of twin primes, Mersenne primes, or Fermat primes seem as intractable today as ever. This makes it all the more amazing that very simple probabalistic ideas can lead to sharp conjectures for the number of primes in these sequences.
The list of talks from previous quarters are available via this archive link.
Last Updated: 2 November 1998
Maintainer: dsmolarski@scu.edu