This paper proposes an original exchange property of valuations.This property is shown to be equivalent to a property described by Dress and Terhalle in the context of discrete optimization and matroids and shown there to characterize the valuations for which the demand oracle can be implemented by a greedy algorithm. The same exchange property is also equivalent to a property described indepen...

let r be a ring and v be a matrix valuation on r. it is shown that, there exists a correspondence between matrix valuations on r and some special subsets ?(mvpr) of the set of all square matrices over r, analogous to the correspondence between invariant valuation rings and abelian valuation functions on a division ring. furthermore, based on malcolmson’s localization, an alternative proof for t...

This is the fourth part in the series of articles [4], [5], [6] (see also [3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations is an automorphism of the algebra of valuations. Then an integration functional on valuations wi...

HADWIGER INTEGRATION OF DEFINABLE FUNCTIONS Matthew L. Wright Robert Ghrist, Advisor This thesis defines and classifies valuations on definable functionals. The intrinsic volumes are valuations on “tame” subsets of R, and by easy extension, valuations on functionals on R with finitely many level sets, each a “tame” subset of R. We extend these valuations, which we call Hadwiger integrals, to de...

The theme of this paper is the extension of continuous valuations on the lattice of open sets of a T0-space to Borel measures. A general extension principle is derived that provides a unified approach to a variety of extension theorems including valuations that are directed suprema of simple valuations, continuous valuations on locally compact sober spaces, and regular valuations on coherent so...