Topological notions

We present in this section the topological notions for the surfaces which will be used. Topology is concerned with continuous structures and invariants under continuous 

No proofs are given but we hope to compensate for this by giving some geometrical examples. More thorough explanations are available in basic algebraic topology textbooks, for example, [HatOl] and [God7l]. 

A closed map, cell-complex of a polyhedron. It is a 5Rq, 4/ 2 plane graph (see Chapter 9) 

If M is a closed map, then we can define its dual map M by interchanging faces and vertices. See Section 4.1 for some related duality notions for non-closed maps. A map is called a cell-complex if the intersection of any two faces, edges, or vertices 

A reader who is interested only in plane graphs, our main subject, can move now directly to Section 1.4. But, for foil understanding of the toroidal case, we need maps in all their generality. For reference on Map Theory see, for example, [BoLi95] and [MoThOl]. 

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The second difference is related to the structure of the lamellar phase. The Euler characteristic has been assumed zero in the whole lamellar phase by Gompper and Kraus [47], whereas we show that it fluctuates strongly in the lamellar phase between the transition line and the topological disorder fine. The notion of the topological disorder line has not appeared in their paper. We think that the topological disorder line is much closer to the transition... 

A discrete topology is a collection of all subsets of X. To define the product topology wc first introduce the notion of a basis for a topology a basis / for a topology T on X is a collection of subsets of X such that... 

In order to define the metric topology, we must first introduce the notion of metric, a function cl AxA->7 isa metric on a set A if and only if it satisfies... 

An arbitrary endpoint can also be marked as "root". A tree with a root will be called a planted tree the vertices different from the root are nodes. If no root is marked, the tree is called an unrooted or free tree. From a topological point of view, two trees with the same structure are identical the exact definition of this and some similar, less familiar notions, will be discussed in Sections 34-35. In the sequel, we use the following notations ... 

The isotropic chord length distribution (CLD) is of limited practical value if soft matter with only short-range order is studied. Nevertheless, the related notions have been fruitful for the development of new methods for topology visualization from SAXS data. 

Students of topology wiU recognize that approximation is also closely related to the notion... 

Let us start by stating the definitions and theorems we use from topology. We will use the notion of local homomorphisms. 

Readers who are uncomfortable with the notion of a fourth dimension may refer to an alternative interpretation. The source term may be viewed as a transfer of energy and momentum from one region of 3D space to another, under the assumption that our 3D space may be topologically disconnected, as in Gribov [105]. 

After a brief discussion of the notion of molecular topology and the analogy principle as related to topology/reactivity relationships more recent developments in the field of reactivity indices for polynuclear benzenoid hydrocarbons are reviewed. Reaction mechanisms and correlations of reactivity indices with rates of electrophilic substitution and Diels-Alder reactions, thermally induced polymerization, and biochemical transformations of benzenoid hydrocarbons are discussed. 

In the present article, we will try to go beyond the above intuitive view and treat in a more rigorous fashion the concept of chirality, starting with the classical definitions and going further by discussing in detail the more recent notion of topological chirality. 

From the notion that animals form an overall representation of an experienced spatial environment, at least three implications have been drawn. One is that animals use multiple landmarks to locate important places by computing their distance and direction from these landmarks. A second implication is that animals can use the cognitive map to infer new routes or shortcuts through space that would be to their advantage. The third implication to be discussed is the suggestion that by exploring a spatial environment, an animal can form a topological map of that environment. 

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