A continuous linear operator T , on the space of entire functions in d variables, is PDE-preserving for a given set P ⊆ C[ξ1, ..., ξd] of polynomials if it maps every kernel-set ker P (D), P ∈ P, invariantly. It is clear that the set O(P) of PDE-preserving operators for P forms an algebra under composition. We study and link properties and structures on the operator side O(P) versus the corresp...

In this short paper we answer a question posed by Chmieliński in [Adv. Oper. Theory 1 (2016), no. 1, 8–14]. Namely, we prove that among normed spaces of dimension greater than two, only inner product spaces admit nonzero linear operators which reverse the Birkhoff orthogonality.

In the general theory of quantum measurement, one associates a positive-semidefinite operator on $d$-dimensional Hilbert space to each $n$ possible outcomes an arbitrary measurement. special case projective these operators are pairwise Hilbert-Schmidt orthogonal, but when $n>d$, orthogonality is restricted by positivity. This restriction allows us more precisely state adage: information gain...

We address the problem of simplifying two-dimensional polygonal partitions that exhibit strong regularities. Such are relevant for reconstructing urban scenes in a concise way. Preserving long linear structures spanning several partition cells motivates point-line projective duality approach which points represent line intersections, and lines possibly carry multiple points. propose simplificat...

Definition 1. An inner product on a complex vector space V is a map 〈., .〉 : V × V → C such that (i) 〈., .〉 is linear in the first slot: 〈c1v1 + c2v2, w〉 = c1〈v1, w〉+ c2〈v2, w〉, c1, c2 ∈ C, v1, v2, w ∈ V, (ii) 〈., .〉 is Hermitian symmetric: 〈v, w〉 = 〈w, v〉, with the bar denoting complex conjugate, (iii) 〈., .〉 is positive definite: v ∈ V ⇒ 〈v, v〉 ≥ 0, and 〈v, v〉 = 0⇔ v = 0. A vector space with ...

Alexandrov topologies are the topologies induced by relations. This paper addresses the properties of Alexandrov topologies as the extensions of strong topologies and strong cotopologies in complete residuated lattices. With the concepts of Zhang’s completeness, the notions are discussed as extensions of interior and closure operators in a sense as Pawlak’s the rough set theory. It is shown tha...

We have proposed a complete set of basis Euler operators for updating cell complexes in arbitrary dimensions, which can be classified as homology-preserving and homology-modifying. Here, we define the effect of homology-preserving operators on the incidence graph representation of cell complexes. Based on these operators, we build a multiresolution model for cell complexes represented in the fo...