|
|
Research
Experiences for
Undergraduates
(REU)
Sponsored by the National Science Foundation and
Trinity University Mathematics
 |
Dr. Scott Chapman -- Algebra
The study of nonunique factorizations in integral domains
and monoids is an active area of current research which has
involved experts from not only Commutative Algebra and
Algebraic Number Theory, but also Additive Number Theory,
Combinatorics, Combinatorial Algebra and Monoid Theory.
Most of the major results obtained in these investigations
occur in the case where the integral domain in question is a
Krull domain. It is well known in this case that properties
relating to lengths of factorizations of the base domain $D$
correspond to the factorization properties of certain easy to
describe monoids. It is these monoids, and their algebraic
and combinatorial structures, which are of interest to the
current project.
More ...
(pdf)
|
| |
 |
Dr. Brian Miceli -- Combinatorics
It is a well-known result that the number of 321 and 312-avoiding permutations
of {1, 2, ... n} is given by the n-th Catalan number, Cn. Recent work has shown that
an alternative notion of pattern avoidance over all
words from the alphabet {1, 2, 3, ...}, subject to certain statistics, will yield rational
generating functions. Moreover, one can give bijective proofs that choosing certain
(different) patterns to avoid will give the same generating function, and we call
such patterns "Wilf equivalent".
More ...
(pdf)
|
| |
 |
Dr. Ryan Daileda -- Number Theory
There are some very interesting and surprisingly difficult
problems dealing with the representation of an integer
n in the base q, where q >= 2 is fixed . One
example is the study of base-q Niven numbers,
which are defined to be those positive integers n that are
divisible by the sum sq(n) of their base-q
"digits". The question of the distribution of the
base-q Niven numbers was recently answered by two
independent groups of researchers, who proved that the
counting function
Nq(x) = #{0 <= n < x : sq(n) | n},
satisfies the asymptotic formula
Nq(x) = [k + o(1)] x / log(x),
where k is an explicit constant.
More ...
(pdf)
|
|
Questions and comments concerning this page are to be
addressed to
schapman@trinity.edu .
