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                Department of Mathematical Sciences
                   The Johns Hopkins University

                              SEMINAR

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Ann Trenk                                         November 15, 2001
Department of Mathematics                         304 Whitehead Hall
Wellesley College                                 Refreshments: 3:30 p.m.
                                                  Seminar: 4:00 p.m.

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        THE LINEAR AND WEAK DISCREPANCY OF AN ORDERED SET

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                               ABSTRACT

In this talk, we discuss criteria for assigning integer ranks to elements 
of a partially ordered set.  The \emph{weak discrepancy} of an ordered set 
$P = (X, \prec)$ is the least integer $k$ for which there exists a function 
$f: X \to Z$ satisfying (i) If $x \prec y$ in $P$ then $f(x) < f(y)$, and 
(ii) If $x \parallel y$ in $P$ then $|f(x) - f(y)| \le k$.  The 
\emph{linear discrepancy} of $P = (X, \prec)$ is the least integer $k$ for 
which there exists a 1-1 function $f: X \to Z$ satisfying (i) and (ii) above.  
Both concepts have applications in assigning ranks to elements of an ordered 
set so that the discrepancy in rank between incomparable elements is minimized.  
We discuss results for each of these parameters.