
Research Activities
I'm always at a bit of a loss for words when someone asks
me "What is your specialty?" because I'm never sure that
I currently have one. It's simply not in my blood to dig
a deep narrow intellectual furrow and try to become one
of the "world's leading experts in xxx," even though I
know that it's in most academics' best interests to focus
her/his energies in this way. I'm constantly interested
in more mathematical topics than I can possibly master in
a finite amount of time, rather like a kid in a candy
shop.
So let me start this page by giving a (partial) list of
the areas that I've investigated to the extent of
publishing something in them, in roughly chronological
order. See my CV for a full publication list. Doing
mathematics is often a social activity, so I'm going to
mention some of the people who have been most influential
to me.
 Group theory  especially generalized
solvable and nilpotent groups as developed by the
Kurosh school  my PhD thesis, "Groups with a
category", is in this area. I was directed by two fine
mathematicians, R. E. Phillips and L. M.
Sonneborn.
 Ring theory  particularly connections
between ring structure and module structure, e.g.,
Loewy modules, torsion modules and torsion theories.
This was my first acquired interest after coming to
Nebraska, influenced by the interests of my algebra
colleagues at Nebraska, Leo Chouinard, Max Larsen, Bill
Leavitt and Jim Lewis
 Commutative algebra  related to the
above, of course. This included work on semihereditary,
Bezout and FGC rings, and was influenced by the
interests of the above colleagues along with Roger and
Sylvia Wiegand.
 Numerical analysis  especially numerical
linear algebra and numerical methods for differential
equations. This was a major switch in my interests,
fueled by my fascination with computer technology and
mathematical computing.
 Sinc methods  really a subset of the
previous item, but I find this particular methodology
for numerical differential equations, as developed by
the Stenger school, particularly intriguing. I received
some friendly guidance from two of the leading experts
in this area, Frank Stenger and John Lund.
 Mathematical modeling  particularly
hyperbolic models and models of contaminant, solute and
ion transport. These interests were particularly
stimulated by my colleages in applied math, especially
David Logan and Steve Cohn.
 Inverse theory  my most current
mathematical romance. You might even call me a
specialist in this area, but I've still got a lot to
learn. Serious work began with a PhD student of mine,
Jennifer Mueller, who wanted to work in inverse theory
in spite of my warning that I was not a specialist in
the area. We both learned a lot and she did a great
job. Subsequently, I was extremely fortunate to get to
know several first rate specialists (and really great
colleagues) in the area, Alemdar Hasanov and Alexander
Denisov, when they visited Nebraska. They taught me a
lot about the subject, and they are still doing
so.
Current and Recent Research Activities
Here are some of the problems that my colleagues and I
worked on recently or are currently working on. Copies of
the papers listed below are available in pdf form as well
as postscript.
 An inverse coefficient problem for an
integrodifferential equation  This is work that
was carried out jointly with Alexander Denisov from
Moscow State University. The problem arose in a one
dimensional mathematical model of adsorption dynamics
which can be converted to an integrodifferential
equation. This work appeared in Applicable
Analysis. For a postscript copy of our paper,
click here.
We developed and tested numerical algorithms that
confirmed the theory. The complete set of function
files can be found here. These files
will work under Matlab or Octave. I gave a colloquium
on this subject at CSU in May 2002. Click on CSU Lecture for a
copy of this slide presentation.
 Sinc Methods for a singular ODE/PDE 
Jennifer Mueller from Colorado State University and I
completed work on a problem that we felt was
particularly suited to sinc methods, namely a second
order ODE boundary value problem on the semiinfinite
interval [0,infinity) with a mixed boundary condition
at 0 modeling contaminant transport. In fact, a method
for such a problem already exists, but it involves
differentiation of the dispersion coefficient. Since we
are interested in inverse problems of coefficient
recovery, we wanted to avoid differentiating this
coefficient. So we developed a new forward solver in a
paper which has appeared in Computers and
Mathematics with Applications. For a pdf
copy of our paper, click here. We developed and
tested numerical algorithms that confirmed the theory.
The complete set of function files can be found here. These
files will work under Octave and, with a little
tweaking, Matlab (Octave allows multiple functions to
be defined in a single script file, so these have to be
separated out into their own files for Matlab.) .
 Properties of Sinc matrices  This is
joint work that was completed by Iyad AbuJeib
from SUNY at Fredonia and myself. It examines
properties of certain matrices denoted by
I^{(1)} that are used in numerical Sinc
integration and makes a few generalizations to classes
of matrices with a special structure. For a postscript
copy of our paper, which it to appear in the New
Zealand Journal of Mathematics, click here.
 Inverse coefficient problems for a singular
ODE  This is work in progress. I have completed one
phase with a paper that I presented at the CTAC 2003 meeting
which was part of the ICIAM 2003 Meetings in Sidney. For a pdf
copy of this paper, click here.
 Curve recognition  This is work in
progress.
 Diffusion as a modeling tool  This is work in
progress with Ashok Samal and Raghu Bemgal from the CS department at UNL. Matlab programs that were used in this research are available here.
 Numerical Methods for vertical sounding  This is work in
progress with V. I. Adamchuk. We are using inverse theory to estimate apparent electrical conductivity in terms of sounding profile data.
PhD Students
They began their careers at Nebraska as students but they
finished as colleagues. My students have taught me a lot
and hopefully I've returned the favor.

Name 
Graduated 
Thesis 

Bonnie Hardy 
1976 
Arithmetical Semigroup Rings 

Frederick Call 
1979 
Torsion theories with the bounded
splitting property 

Jagannadham Pakala 
1979 
Commutative torsion theories 

Kristin
Pfabe 
1995 
A problem in nonlinear ion
transport 

Kamel AlKhaled 
1996 
Theory and computation in hyperbolic
model problems 

Jennifer
Mueller 
1997 
Inverse problems for singular
differential equations 

Paul Gierke 
1999 
Discrete approximations of
differential operators by Sinc methods 

Iyad
AbuJeib 
2000 
Frames in Hilbert space and matrices
of special structure 

Brian Bockelman 
2008 
Solving partial differential equations using sinc methods 
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