Proof. As is known, a Bayesian network is equivalent to a topological ordering (Chapter 8, Section 8.1 on Page 362 in [1]). Therefore, we prove Proposition 1 by showing that i) Eqn. (1a ∼ 1d) lead to a topological ordering (the necessary condition), and ii) a topological ordering from a DAG can meet the requirements in Eqn. (1a ∼ 1d) (sufficient condition). First, we prove the necessary conditi...

The aim of this paper is to investigate partially ordered real linear topological spaces in which directed sets admit a supremum in their closure. In particular, we point out that this property is intimately related to the normality of the ordering cone and also to the Scott continuity of functionals belonging to the nonnegative polar of the ordering cone. Mathematics Subject Classification (20...

A graph is outer-cylindrical if it embeds in the sphere so that there are two distinct faces whose boundaries together contain all the vertices. The class of outer-cylindrical graphs is closed under minors. We give the complete set of 38 minor-minimal non-outer-cylindrical graphs, or equivalently, an excluded minor characterization of outercylindrical graphs. We also give the obstruction sets u...

We study the topological space of left-orderings of the braid group, and its subspace of Nielsen-Thurston orderings. Our main result is that no Nielsen-Thurston ordering is isolated in the space of braid orderings. In the course of the proof, we classify the convex subgroups and calculate the Conradian soul for any Nielsen-Thurston ordering of Bn. We also prove that for a large class of Nielsen...

Changes in Young's modulus were measured during structural relaxation of FeB, FeNiB, FeNiP and FeNiPB amorphous alloys. The behaviour of the Young's modulus in the four alloys is found to be qualitatively the same. The results are discussed in terms of relaxation consisting of two processes: Compositional Short Range Ordering, described by AES-model /2/ and Topological Short Range Ordering, des...

The ordering dynamics of the Higgs field is studied, using techniques inspired by the study of phase ordering in condensed matter physics, as a first step to understanding the evolution of cosmic structure through the formation of topological defects in the early universe. The common feature of these different physical processes is scaling. A fully analytical approximate scheme − the linear-gau...