Set-valued Set Research Articles

Role Of ψ-Operator in the Study of Minimal Open Sets

The operator ψ has been introduced as an associated set-valued set function. Although it has importance for the study of minimal open sets as well as minimal I-open sets. As a result of this study, we introduce minimal I^*-open sets . In this study, several characterizations of minimal I^*-open sets are also investigated. This study also discusses the role of minimal I^*-open sets in the *-locally finite spaces. In an aspect of topological invariant, the homeomorphic images of minimal I^*-open set has been discussed here.

Topologically Transitive and Mixing Properties of Set-Valued Dynamical Systems

We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two properties for set-valued functions and generalize some results from a single-valued case to a set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.

A Set-Valued Approach to Observability

We restate observability in a set-valued setting which generalizes the standard output equation, then a close connection to viability kernels is established. Both global and local cases are studied by means of a set of single-valuedness results. Among these, we highlight the results which are derived by monotone set-valued mappings theory and the ones using contingent and paratingent derivatives. Moreover, this new setting gives rise to the notions of maximal observability domains and minimal unobservability domains, which we characterize in the framework of antitone mappings and their fixed points.

Set-valued assessments of solutions to stochastic differential equations with random set parameters

We consider stochastic differential equations depending on parameters whose uncertainty is modeled by random compact sets. Several approaches are discussed how to construct set-valued stochastic processes from the solutions. The induced lower and upper probabilities are compared to a set-valued set function and a set of probability measures constructed from the distributions of the solutions and the selections of the random set.

A Topological Approach to Chemical Organizations

Large chemical reaction networks often exhibit distinctive features that can be interpreted as higher-level structures. Prime examples are metabolic pathways in a biochemical context. We review mathematical approaches that exploit the stoichiometric structure, which can be seen as a particular directed hypergraph, to derive an algebraic picture of chemical organizations. We then give an alternative interpretation in terms of set-valued set functions that encapsulate the production rules of the individual reactions. From the mathematical point of view, these functions define generalized topological spaces on the set of chemical species. We show that organization-theoretic concepts also appear in a natural way in the topological language. This abstract representation in turn suggests the exploration of the chemical meaning of well-established topological concepts. As an example, we consider connectedness in some detail.

Set-valued extension of operators via Steiner selections(II)—Applications to approximation

On the basis of Part (I) of this series some applications to the approximation of set-valued functions are obtained: Korovkin type theorems, a method to extend classical approximation operators to the set-valued setting and a Jackson type estimate.

ON THE EXTENSION OF SET-VALUED SET FUNCTIONS

We extend additive set-valued set functions and normal multimeasures defined on a ring of subsets. We also prove a Carathéodory-Hahn-Kluvanek-type theorem for additive set-valued set functions. Finally, we establish results on the extension of transition multimeasures.

A calculus for set-valued maps and set-valued evolution equations

A definition of differentiability of a set-valued map is offered. As derivatives, which are called directives in the set-valued setting, unions of affine maps are used; these are called multiaffines. A multiaffine is a directive if it is a first-order approximation of the set-valued map. One application is a necessary condition for maximin optimality of constrained decisions. A distance among multiaffines permits the development of set-valued evolution equations along the lines of ordinary differential equations in a vector space. The theory is displayed along with some comments on applications.

Set-valued set theory. II.

Set-valued set theory. II.

Set-valued set theory. I.

Set-valued set theory. I.

On set correspondences into uniformly convex Banach spaces

It is proved that the values of a set-valued set function, the total variation of which is an atomless finite measure, are conditionally convex.

Continua for which the set function $T$ is continuous

The set-valued set function T has been studied extensively as an aid to classifying metric and Hausdorff continua. It is a consequence of earlier work of the author with H. S. Davis that T, considered as a map from the hyperspace of closed subsets of a compact Hausdorff space to itself, is upper semicontinuous. We show that in a continuum for which T is actually continuous (in the exponential, or Vietoris finite, topology) semilocal connectedness implies local connectedness, and raise the question of whether any nonlocally connected continuum for which T is continuous must be indecomposable.