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Range of the Fractional Weak Discrepancy Function

Content Provider	Springer Nature Link
Author	Shuchat, Alan Shull, Randy Trenk, Ann N.
Copyright Year	2006
Abstract	In this paper we describe the range of values that can be taken by the fractional weak discrepancy of a poset and characterize semiorders in terms of these values. In [6], we defined the fractional weak discrepancy $wd_{F}(P)$ of a poset $P=(V,\prec)$ to be the minimum nonnegative $k$ for which there exists a function $f: V \to {\mbox{\mathbf{R}}}$ satisfying (1) if $a \prec b$ then $f(a) + 1 \le f(b)$ and (2) if $a \parallel b$ then $\|f(a) \!-\! f(b) \| \!\le\! k$ . This notion builds on previous work on weak discrepancy in [3, 7, 8]. We prove here that the range of values of the function $wd_{F}$ is the set of rational numbers that are either at least one or equal to $\frac{r}{r+1}$ for some nonnegative integer $r$ . Moreover, $P$ is a semiorder if and only if $wd_{F}(P) < 1$ , and the range taken over all semiorders is the set of such fractions $\frac{r}{r+1}$ .
Starting Page	51
Ending Page	63
Page Count	13
File Format	PDF
ISSN	01678094
Journal	Order
Volume Number	23
Issue Number	1
e-ISSN	15729273
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2006-07-07
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	weak discrepancy fractional weak discrepancy weak orders semiorders Convex and Discrete Geometry Geometry Theory of Computation
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Computational Theory and Mathematics Geometry and Topology

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