Topological Entities Research Articles

The topological foundations of electrical networks — complete models

AbstractThe aim of this paper is to identify the topological entities underlying the construction of a complete (not reduced) network model. Consider an active network as a pair of physically different structures, i.e. a ‘dead network’ and an ‘excitation’. First, we define the general structure of the excitation which can be applied to a dead network; then the graphs Ga and G of the active and dead networks, respectively, are introduced. It is shown that, if a complete network model is to be constructed, the topological relations among the voltages and those among the currents of the network elements must be written choosing maximal independent sets of loops and cut‐sets of Ga. It is successively pointed out that the graph G can be obtained from Ga by means of a reduction operation, and the topological entities of G corresponding to the loops and to the cut‐sets of Ga are singled out. It has been found that a complete model of an electric network implies the use of four topological entities defined on the set of G‐edges, i.e. the closed path, the open path, the cut‐set and the pseudo‐cut, together with two topological entities defined on a maximal set of independent vertex couples of G, i.e. the ‘junction pair’ extreme of an open path and the ‘family of independent node couples’ split by a cut‐set.Furthermore the different reference frames and the relative topological matrix transformations which allow a complete model of the network to be built up are singled out and discussed.

An Edge‐Face Relational Scheme for Boundary Representations

AbstractWe propose a relational scheme for representing and modelling regular objects, which is based on the adjacency relations between faces and edges. In this structure, called edge‐face graph, the nodes represent the faces and the arcs the edges of the corresponding object. Other topological entities, such as vertices, loops of edges, and shells, can be obtained from this relational scheme. We give a formal description of the edge‐face graph, and the relationships between its properties and the topological entities of the object are analyzed in detail. Furthermore, a set of basic Euler operators based on the edge‐face adjacency relation is denned, which allow the incremental manipulation of boundary representations of solid objects.

Morphometric study of the development of Purkinje cell dendritic trees in the mouse using vertex analysis.

Vertices are the points in an arborescence which interconnect segments and comprise terminal or pendant vertices (Vp), nodal or branching points and the root point. Branching points may be dichotomous (Vd) or trichomtomous (Vt), etc., and are subdivided into distinct two-dimensional topological entities according to the number of terminal vertices they drain, i.e. Vds comprise primary vertices (Va), connecting two Vps; secondary vertices (Vb), connecting one Vp and one Vd or one Vt; and tertiary vertices (Vc), connecting either two Vds, two Vts or one Vt and one Vd. The four types of Vt (Va', Vb', Vc', Vd') similarly connect three, two, one and zero Vps respectively. Each Vt may be transformed into two Vds thus, Va' = Va + Vb; Vb' = Va/3 + 4Vb/3 + Vc/3; Vc' = Vb + Vc and Vd' = 2Vc. Analysis proceeds by transforming mixed trees containing varying proportions of Vds and Vts into entirely dichotomous branching structures. The topology is then defined by the Va/Vb ratio which has a unique value according to the mode of growth and the frequency of Vts. Vertices are ordered by a centrifugal technique. The frequency distribution of vertices of different order allow the changes in growth characteristics and in remodelling to be detected within particular regions of the tree. Metrical parameters are readily incorporated into the analysis since all vertices are interconnected by segments of finite length and are given the same order magnitude as the vertex they drain. The analytical capabilities of the method are exemplified by its application to the study of growth and plasticity in the dendritic trees of Purkinje cells in the mouse. Growth is defined in metrical and topological terms and sites of reorganization within the mature tree are identified.

On vector spaces associated with a graph

Four topological entities are necessary for a complete description of a network graph, as required by orthogonal network theory: the seg, the circ and two others, introduced in this paper, as complementary to the seg and the circ respectively. A vector space and its dual are associated with a nonoriented graph. Both spaces are decomposed in two complementary subspaces. The vectors of each subspace share the topological properties of one of the four entities singled out. The results obtained are extended to a directed graph by associating two dual Z-modules with it. The given description of the network graph is perfectly functional to the orthogonal network theory.

A Partition of Finite T0 Topologies

The aim of this paper is to study a decomposition of finite T0 spaces into topological entities called chains and cells. These objects behave like complete units under homeomorphisms and they appear to be useful in investigating certain aspects of finite spaces. As an elementary illustration of how these entities can be used, the concept of an A2 space is introduced (in the next paragraph) and it is demonstrated that the order of the automorphism group of an A2 space is expressible as 2t, for some t ≧ 0.