Dear Colleagues,

We have a very unusual Colloquium in the Math Department.


Sonja Petrovic


Sonja is a graduating student from Mathematics Program who has written
an Honors Project under the direction of Professor John Graef. Other
members of the Committee are R. Smith, B. Belinskiy, and N. Ozbek.

Sonja will give a Colloquium presentation based on her paper on August
7, Thursday, at 3:30, EMCS Room 422 . The title of her paper is

Oscillation of Solutions of Dynamic Equations on Time Scales: the
Sturm-Picone Comparison Theorem

Abstract. Presented here are the basic properties of time scales and
operations on them. We consider two generalized derivatives on time
scales, namely, the delta and nabla derivatives.  We further discuss
integration on time scales with respect to both of these derivatives.
The delta derivative, an equivalent of the forward difference operator
on $\Bbb Z$, is described in more detail.  The analogous results for
nabla derivative, the equivalent of a backward difference on $\Bbb Z$,
follow similarly.  Finally, we focus on second order dynamic equations,
which are the time-scale equivalents of second order differential and
difference equations on their respective domains.

The particular problem we study in this paper is a generalized version
of the well-known Sturm-Picone Comparison Theorem.  We focus on the
second order delta-derivative self-adjoint equations of a particular
form and derive the analogous result for time scales.  The result is
proved directly, avoiding the usual Riccati type transformations.  The
possibility for further research stems from the final result which may
be considered for a more general form of self-adjoint equations that
includes not only delta, but also the nabla derivative, and also some
nonlinear  cases.





Boris P Belinskiy

Department of Mathematics, Dept. 6956

University of Tennessee at Chattanooga

Ph. (423) 425-4748