Topological object

Summing up the results Depending on whether the corona of edges is treated as a configuration in two- or three-dimensional space or as a topological object, the admissible permutations form the "associated" groups Hj, respectively. 

We have found recently the topological interpretation of property (34). The stoichiometric constraints (24) can be interpreted in terms of the topological object, the circuit. Existence of the circuit "explains" the appearance of the cyclic characteristic in the constant term of kinetic polynomial. Thus, we can say that in some sense the correspondence between the detailed mechanism and thermodynamics is governed by pure topology. 

The simulation of a dynamic system using cellular automata requires several parts that make up the process. The cell is the basic model of each ingredient, molecule, or whatever constitutes the system. These cells may have several shapes as part of the matrix or grid of cells. The grid may have boundaries or be part of a topological object that eliminates boundaries. The cells may have rules that apply to all of the edges, or there may be different rules for each edge. This latter plan may impart more detail to the model, as needed for a more detailed study. 

The Betti numbers are important topological invariants which can be used for shape characterization. The p-th Betti number bp is the rank of homology group HP. For topological objects we encounter in this book, the Betti numbers bp can be... 

As it happens, entropy really relates to something that allows us to look at the drug design problem at another way, as a kind of puzzle like a Rubik Cube with a (typically at least in a sense) specific solution. To solve many kinds of problem by computer simulation, there is usually some implied very complicated function surface that is searched. Not only does this have many hills and valleys in which one can become trapped, but it is an exploration puzzle in many dimensions, mathematically speaking (and maybe involving more complex topological objects than higher dimensional counterparts of hills and valleys). 

Tlie combination of branched DNA molecules and sticky ends creates a powerful mo-leculai assembly kit for stmctural DNA nanotechnology. Polyhedra, complex topological objects, a nanoniechanical device and two-dimensional aiiays with programmable surface features have already been produced in tliis way. Future applications range from macro-molecular ciystallography and new materials to moleculai electronics and DNA-based computation. ... 

Singularities in liquid crystals are not merely interesting topological objects. Their presence is often necessary in many applications, and they probably play a major role in biological processes. And we should not forget the aesthetic pleasure they bring each time we observe liquid crystals. 

Examples of basic topological objects (a) knot (b) link (c) braid (d) tangle. 

The most severe topological objects of a PES are the stationary points minima and saddle points of any index between 1 and (n-1). Here, the gradient of E is zero, and this property overrides all regular coordinate transformations. 

Fig. 2.6 Simple system of two ordinary differential equations, which shows a separatrix (dashed line) and two critical points (pink). This topological object separates the basin dominated by the attractor critical point (0,1) (top). The second critical point shown is a saddle-t5rpe critical point, at which the separatrix trajectories (dashed line) terminate. The collection of trajectories (phase flow) can be seen as the paths followed by imagined particles travelling in time. The superscripted dots in the equations signify differentiation with respect to time...

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