Journal•ISSN: 0167-8094
Order
Springer Science+Business Media
About: Order is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Partially ordered set & Total order. It has an ISSN identifier of 0167-8094. Over the lifetime, 1239 publications have been published receiving 16179 citations.
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1,324 citations
TL;DR: It is proved the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution.
Abstract: We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets.
1,159 citations
TL;DR: In this article, the incidence relation of a graph G=(V. E) was viewed as an order relation on its vertices and edges, i.e. if and only of a is a vertex and b is an edge incident on a.
Abstract: We view the incidence relation of a graph G=(V. E) as an order relation on its vertices and edges, i.e. a
309 citations
TL;DR: The problem of counting the number of linear extensions of a partially ordered set is #P-complete as discussed by the authors, which is the state-of-the-art algorithm for this problem.
Abstract: We survey the problem of counting the number of linear extensions of a partially ordered set. We show that this problem is #P-complete, settling a long-standing open question. This result is contrasted with recent work giving randomized polynomial-time algorithms for estimating the number of linear extensions.
305 citations
TL;DR: A complete and reliable data base for all possible order types of size n=10 or less is established and substantiate the usefulness of the data base by applying it to several problems in computational and combinatorial geometry.
Abstract: Order types are a means to characterize the combinatorial properties of a finite point configuration. In particular, the crossing properties of all straight-line segments spanned by a planar n-point set are reflected by its order type. We establish a complete and reliable data base for all possible order types of size n=10 or less. The data base includes a realizing point set for each order type in small integer grid representation. To our knowledge, no such project has been carried out before. We substantiate the usefulness of our data base by applying it to several problems in computational and combinatorial geometry. Problems concerning triangulations, simple polygonalizations, complete geometric graphs, and k-sets are addressed. This list of applications is not meant to be exhaustive. We believe our data base to be of value to many researchers who wish to examine their conjectures on small point configurations.
137 citations