The problem of estimating the number of unseen species in a
population based on the results of a single sample of animals
is a familiar one in the statistical literature. In a related
problem associated with genome sequencing the goal is to design
a sampling strategy for finding a specified proportion of the
total number of species. A generalized multinomial model is
applied to estimate the number of unseen species; the model also
forms the basis for a Monte Carlo simulation approach to determing
the sample size required to guarantee that a specified proportion
of the total species are collected. The methods are demonstrated
on simulated data and data from a DNA sequencing application.

We consider the problem which informally can be formulated as follows. Initially a finite set of independent trials is available. If a Decision Maker (DM) chooses to test a particular trial she receives a reward depending on the trial tested. As a result of testing a random finite set (possibly empty) of new independent trials is added to the set of available trials, and so on, but the total number of potential trials is finite. A DM knows the rewards and transition probabilities of all trials. On each step she can either stop testing or continue. Her goal is to select a testing order and a stopping time to maximize the expected total reward. This problem has a long history and is related to the Multi-armed Bandit Problem with independent arms. We prove that an index can be assigned to each possible trial, and the optimal strategy is to use on each step a trial with maximal index among those available. We give a simple procedure for constructing this index.

This is joint work with Isaac Sonin.

For an irreducible polynomial $f(T)$ in ${\\bf Z}[T]$ whose
values are not all multiples of a common prime, the sequence $\\mu(f(n))$
is not expected to have any periodicity properties. In contrast, there
can be periodicity when $f(T) \\in {\\bf F}[u][T]$ with $\\bf F$ a
finite field. That is, the sequence $\\mu(f(g))$ can be periodic as $g$
runs over ${\\bf F}[u]$. This is based on peculiarities of
characteristic p.

We will briefly discuss the case of odd characteristic, and then focus on
the extra subtleties of characteristic 2, where we make an interesting
application of the residue theorem for a certain rational differential
form. The ideas will be made explicit by treating a concrete case: as $g$
runs over ${\\bf F}_2[u]$, $\\mu(g^8+(u^3+u)g^4+u) = 1$.

Remark: Note the unusual day, time, and room for the number theory
seminar this week.

A knot is doubly slice if it is the intersection of a three sphere with a
trivially embedded two sphere in a four sphere. The resulting knot splits
the two sphere into two distinct slicing disks for the knot. Thus, the
term \"doubly slice\"".