Euler-Poincare characteristic

Proof, (i) The relation 2e = kv allows us to rewrite the Euler-Poincare characteristic as ... 

From topological point of view similar structure is not any other than 3D-sphere with p + q holes, Euler-Poincare characteristic of which is equal to... 

On the other hand Euler-Poincare characteristic of the any cage structure can be evaluated as the alternating sum of numbers of vertexes, verges and faces in the structure ... 

The Betti numbers bp are related to the Euler-Poincare characteristic %, ... 

The Euler-Poincare characteristic % is an important topological invariant of the object. 

This property holds because (F-E+V) is a topological characteristic, dependent only on the topology of the facetted surface. Since all polyhedra are topologically equivalent to the sphere (Fig 1.9), F-E+V) is conserved. The value of this integer is known as the Euler-Poincare characteristic x x) ... 

This result implies that the ring size and connectivity of a network determine the topology of the surface which contains that network. This allows for simple characterisation of cage, sheet and framework nets, distinguishable by the value of their Euler-Poincare characteristic (Table 1.2). 

Another topological characteristic, the genus, g(x), of a surface, is a measure of its connectedness. It is equal to the number of holes or handles in the surface and simply related to the Euler-Poincar characteristic by... 

The integral curvature of a surface is linked to the Euler-Poincare characteristic of that surface (x) by eq. (1.12). This allows the average geometry of orientable surfaces to be related to the number of holes or handles, characterised by the surface genus, g, and the area of the surface. A, by the relation ... 

The Euler-Poincare characteristic x, which is related to the number of holes (or missing points ), with and without boundary, of the surface. For example, a sphere with perforations has x = 2 — whereas x = —for a punctured torus, with s 0. 

Note that one molecular surface can be a collection of more than one closed geometric surface. In this case, a vector of Euler—Poincare characteristics can be used for the description. 

The integral over the Gaussian curvature in Eq. [27] is a topological invariant.i "i85 For a closed orientable 2D surface (i.e., one without boundary), the Gauss—Bonnet theorem ties the value of this invariant to the genus g or the Euler—Poincare characteristic x of the surface ... 

An additional limitation of the index theorem is that in higher dimensions, there may often be different time scales corresponding to fast and slow reactions, so that the dynamics rapidly relax to a manifold that is embedded in the N-sphere. In this case, the Euler-Poincare characteristic of the submanifold may be different from that of the iV-sphere and (14) would have to be modified appropriately. A very general mathematical approach in which the geometric and topological properties chemical kinetic equations is treated has recently been developed. Although the formulation is in principle capable... 

Besides the surface curvatures that define the metric relationships of objects, there are some fundamental aspects of shapes which are preserved if the structures are made of stretchable rubber sheet —topology. Suppose a surface is subdivided into a number of faces, edges, and vertices. A topological characteristic, called the Euler-Poincar characteristic, depends only on the topology of the facetted surface and is defined ... 

The Euler-Poincare characteristic, nd the genus, g, can be estimated from the total number of junctions, N, and total number of branches, B, through the following equation [49] ... 

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