Zonotopal algebra and forward exchange matroids

Abstract

Zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors X. It connects objects from combinatorics, geometry, and approximation theory. The origin of zonotopal algebra is the pair (D(X), P(X)), where D(X) denotes the Dahmen-Micchelli space that is spanned by the local pieces of the box spline and P(X) is a space spanned by products of linear forms.The first main result of this paper is the construction of a canonical basis for D(X). We show that it is dual to the canonical basis for P(X) that is already known.The second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by Ardila-Postnikov, Holtz-Ron, Holtz-Ron-Xu, Li-Ron, and others. We call the underlying combinatorial structure of those spaces forward exchange matroid. A forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom.

Zonotopic fault detection observer design for discrete‐time systems with adaptively adjusted event‐triggered mechanism

“…By Tables 1 and 2 Fig. 2, which indicates that both x 1 (k) and x 2 (k) are affected by injected fault signal (29). Fig.…”

“…A fault detection and isolation method was proposed via interval observer and invariant set methods in [28]. Zonotopal algebra, which appears in mathematics literature, has great potential applications in systems and control theory [29]. Recently, a zonotope-based fault detection approach has attracted much attention from researchers.…”

Classification of External Zonotopal Algebras

“…The first definition was used by F. Ardila and A. Postnikov [2]; it originates from the algebras generated by the curvature forms of tautological Hermitian linear bundles [4,29], see also papers [5,6,15,16,17,23,24,27,28,30], where the quotient algebras by these ideals were discussed by details. At the same time a similar definition and the term were established by O. Holtz and A. Ron [13]; Their definitions originates from Box-Splines and from Dahmen-Micchelli's space [1,9,11], see also the papers [10,12,14,19,20,21,22,31].…”

Classification of external Zonotopal algebras

“…We will work with the definition from F. Ardila and A. Postnikov [2]; it comes from algebras generated by the curvature forms of tautological Hermitian linear bundles [3,28], see also papers [4,5,14,15,16,22,23,26,27,29], where people work with quotients algebras by these ideals. At the same time the definition and the name was established by O. Holtz and A. Ron [12]; it comes from Box-Splines and from Dahmen-Micchelli space [1,8,10], see also the papers [9,11,13,18,19,20,21,30].…”