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Compactness properties of weighted summation operators on trees-the critical case

Mikhail Lifshits, Werner Linde (2011)

Studia Mathematica

The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [Studia Math. 202 (2011)] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator to those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques....

Compactness properties of weighted summation operators on trees

Mikhail Lifshits, Werner Linde (2011)

Studia Mathematica

We investigate compactness properties of weighted summation operators V α , σ as mappings from ℓ₁(T) into ℓ q ( T ) for some q ∈ (1,∞). Those operators are defined by ( V α , σ x ) ( t ) : = α ( t ) ∑ s ⪰ t σ ( s ) x ( s ) , t ∈ T, where T is a tree with partial order ⪯. Here α and σ are given weights on T. We introduce a metric d on T such that compactness properties of (T,d) imply two-sided estimates for e ₙ ( V α , σ ) , the (dyadic) entropy numbers of V α , σ . The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights with α(t)σ(t)...

Comparison of subset systems

Evelyn Nelson, Jiří Adámek, Andreas Jung, Jan Reiterman, Andrzej Tarlecki (1988)

Commentationes Mathematicae Universitatis Carolinae

Compatibility and central elements in pseudo-effect algebras

Paolo Vitolo (2010)

Kybernetika

An equivalent definition of compatibility in pseudo-effect algebras is given, and its relationships with central elements are investigated. Furthermore, pseudo-MV-algebras are characterized among pseudo-effect algebras by means of compatibility.

Complemented ordered sets

Ivan Chajda (1992)

Archivum Mathematicum

We introduce the concept of complementary elements in ordered sets. If an ordered set S is a lattice, this concept coincides with that for lattices. The connections between distributivity and the uniqueness of complements are shown and it is also shown that modular complemented ordered sets represents “geometries” which are more general than projective planes.

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