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 integro-differential 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 integro-differential 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 semi-infinite 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 Abu-Jeib 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 Al-Khaled 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 Abu-Jeib 2000 Frames in Hilbert space and matrices of special structure
  Brian Bockelman 2008 Solving partial differential equations using sinc methods

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