Colloquium Announcement
Department of Mathematics
West Virginia University
for
Thursday, October 23, 1997, at 3:45pm
in 324 Armstrong Hall
(Tea and cookies begin at 3:00 in coffee room.)
Professor Tomasz Natkaniec
Gdansk University, Poland
Universal Darboux-like summands for families of functions
The talk will be suitable for a general audience.
Students are strongly encouraged to participate.
Abstract
For a given family F of Darboux-like functions (i.e.,
Darboux functions D, almost continuous functions ACF, and
extendable functions Ext) we examine the
following question:
For which families F of functions from R to R
does there exist an
F-universal summand, i.e., g:R-->R
such that that f+g belongs to F
for all f from F?
This question was considered by many authors. The first result
in this direction was obtained by H. Fast in 1959:
If F is a family of functions and the cardinality of
F is less then or equal to continuum, then there
exist a D-universal summand for F.
The analogous theorem for the class of all almost continuous
functions was obtained in 1974 by K. Kellum. Those results were
generalized in 1994 by K. Ciesielski and A. Miller and, for
extendable functions, in 1995 by K. Ciesielski and I. Reclaw.
Darboux-like universal summands for families of Borel measurable
functions were studied by J. Ceder and recently by I. Reclaw
and me. We consider also an Ext-universal summands for big
"regular" families of functions. In particular, we prove
that there are Ext-universal summands for the family of all
Lebesgue measurable functions and for the family of all functions
with the Baire property.
The information on the future (and past) Colloquia can be also found on web at the
address:
http://www.math.wvu.edu/homepages/kcies/colloquium.html