Range of the Fractional Weak Discrepancy Function

Abstract

Thalidomide does not significantly improve pulmonary function or quality of life in patients with corticosteroid-dependent sarcoidosis. Few patients can tolerate doses of greater than 100 mg/day. Thalidomide may have been corticosteroid-sparing in a subgroup of these patients.

Fractional Weak Discrepancy of Posets and Certain Forbidden Configurations

“…As an example, consider the salary assignment problem described in [8]. A manager wishes to assign a salary f (a) to each employee a in her division in a fair way.…”

“…By a theorem of Scott and Suppes [6], this class is equivalent to the class of unit interval orders, that is, posets which can be represented as follows: each element x of the ground set V is assigned a unit length interval I x on the real number line so that x ≺ y if and only if the interval I x is completely to the left of I y . In [8] we show how we can use fractional weak discrepancy to characterize the class of semiorders. In particular we establish the following two results.…”

“…Lemma 8, which appears as Proposition 9 of [8], allows us to describe optimal fractional weak labelings for forcing cycles whose "up" to "side" ratios achieve the maximum value of wd F (P ). In particular, every optimal labeling is tight on such a forcing cycle in the following sense.…”

“…Semiorders were characterized by their fractional weak discrepancy in [8]. Here we describe the range of values of wd F (P ) according to whether P contains an induced 2 + 2 and/or an induced 3 + 1.…”

“…Generalizing previous work on weak discrepancy in [5,12,13], the notion of fractional weak discrepancy wd F (P ) of a poset P = (V, ≺) was introduced in [7] as the minimum nonnegative k for which there exists a function f :Semiorders were characterized by their fractional weak discrepancy in [8]. Here we describe the range of values of wd F (P ) according to whether P contains an induced 2 + 2 and/or an induced 3 + 1.…”