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Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
In the same way we define the intersection of an arbitrary family of sets: A⋂A∈AA=defx:∀A∈Ax∈A $$ A\bigcap _{A \in \mathcal A } A \ \mathop {=}\limits ^\mathrm{def}\ \left\{ x \ : \ \forall A \in \mathcal A \ \ x \in A \right\} $$
Introduction to Sets and Relations
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
The notions of union and intersection of two sets can be generalized to unions and intersections of more than two sets after introducing the notion of indexed family of sets. Let ∆ be a set and assume that with each element γ ∈ Δ there is associated a subset Aγ of a given set S. The collection of all such sets Aγ is called an indexed family of subsets of S with Δ as the index set and it is denoted {Aγ}γ∈Δ
Sets
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
We have defined above intersection and union for finite collections of sets A1 A2, ...,An. We now turn our attention to more general collections or families of sets, including infinite families. Firstly we need to make clear what we mean by, and how we can specify, an infinite collection of sets. By a family or collection of sets, we really mean a set of sets, although the terms ‘family of sets' and ‘collection of sets' are both in widespread use.
Pullback exponential attractors in nonlocal Mindlin's strain gradient porous elasticity
Published in Applicable Analysis, 2023
A family of sets is said to be a pullback absorbing family of the process , if for any and every bounded subset , there exists a such that for . In addition, the family is said to be pullback -absorbing, if for any , there exists a such that
Signed ring families and signed posets
Published in Optimization Methods and Software, 2021
Kazutoshi Ando, Satoru Fujishige
The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff (see [5,6]). This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic applications bring us distributive lattices as a ring family of sets which is closed with respect to the two binary operations of set union and intersection. Typically we have such a ring family of sets as a family of minimizers of a submodular set function (see [11]). When it comes to a ring family of sets, the underlying set is partitioned into subsets (or components) and we have a poset structure on the partition. This decomposition with a poset structure on the set of components plays important and crucial roles in many practical problems related, for example, to the decomposition of a directed graph into strongly connected components, the Dulmage-Mendelsohn decomposition of a bipartite graph [29], etc. This is a set-theoretical variant of the original Birkhoff theorem, to be called the Birkhoff-Iri decomposition, revealing the correspondence between finite ring families and finite posets on partitions of the underlying sets, which was intensively pursued by Masao Iri around 1978, especially concerned with the problem of what is called the principal partition of discrete systems [13,22–25,28].
An iterative method for solving the strong vector equilibrium problem with variable domination structure
Published in Optimization, 2018
San-hua Wang, Jin-xia Huang, Chuan-xi Zhu
Let E and Z be two real Hausdorff topological vector spaces and X be a nonempty subset of E. Let be a set-valued mapping such that, for each , C(x) is a proper, closed and convex cone. The set-valued mapping or the family of sets is called a domination structure on Z (see [1]). Let and be two vector-valued mappings. In this paper, we shall study the following strong vector equilibrium problem with variable domination structure (VSVEP): find such that