Elementary collapse

A simplicial collapse of A is the removal of all simplices 7 such that T C 7 C (T. if additionally, we have dimr = dim[Pg.94]

Definition 6.17. Two generalized simplicial complexes are said to have the same simple homotopy type if there exists a sequence of elementary collapses and expansions leading from one to the other. Such a sequence is called a formal deformation. 

Since stx(o ) is a cone, in particular collapsible. Step 1 can be performed as a sequence of elementary expansions. Furthermore, Step 2 can be performed as a sequence of elementary collapses as follows the set of simplices that are to be deleted can be written as a disjoint union of sets A and B, where B is the set of all simplices that contain both a and v. Clearly, adding to a simplex is a bijection jx A B. Let ti. .., r be a reverse linear extension order on A then (ri, /u(ti),. .., (n, fx(.n)) is an elementary collapsing sequence. 

Sometimes such a collaprse is called an elementary collapse. Note that a simplicial collapse is possible if and only if there exists a simplex t whose link in A consists of a single vertex the simplex cr is then given by the span of r and v. For a general CW complex one has to take care of some additional technicalities see Definition 11.12. 

Definition 11.12. LetX be a topological space and letY be a subspace of X. We say that Y is obtained from X by an elementary collapse if X can be represented as a result of attaching a ball B to Y along one of the hemispheres. In other words, if there exists a map (p —> Y such that... 

Unfortunately, there are so many different ways to create universes by compactify-ing the six dimensions that string theory is difficult to relate to the real universe. In 1993, researchers suggested that if string theory takes into account the quantum effects of charged mini black holes, the thousands of 4-D solutions may collapse to only one. Tiny black holes, with no more mass than an elementary particle, and strings may be two descriptions of the same object. Thanks to the theory of mini black holes, physicists now hope to mathematically follow the evolution of the universe and select one particular Calabi-Yau compactification—a first step to a testable theory of everything. ... 

There remains one objection—of a less precise kind but felt by many chemists. It is that third-order kinetics as embodied in the representation of step (1) are intrinsically objectionable. If the equations had to be interpreted as representing elementary steps, this would be a weightier consideration, but it has also been asserted that the oscillatory properties of certain other model schemes collapse completely (King, 1983 Gray and Morley-Buchanan, 1985) if the third-order steps therein are replaced. Accordingly it is most desirable to establish whether oscillations and other exotic behaviour arising from a cubic rate-law of the form k ab2 can also arise from a series of successive second-order or bimolecular steps. Similar interests have been expressed previously by Tyson (1973) and Tyson and Light (1973). 

Einstein pointed out that the collapse, which he assumed took place on a t = constant hypersurface means that the influence of the appearance of an elementary quantum leads to an instantaneous (faster that light)... 

Separating the total rate of bubble expansion into its constituents, i.e. the rates of the elementary processes, is a complex issue that can only be solved in some special cases. For example, if the foam films are very stable, the average bubble size will increase mainly as a result of diffusion. If the films are very unstable, the internal collapse will be caused by coalescence. 

We turn now to the mechanism of orbital collapse, which is not immediately obvious from the numerical calculations of the various authors quoted above. It has been explained by Connerade [210, 211] using analytic potentials and elementary quantum theory. The key feature to note is the difference in nature between different kinds of potential in quantum mechanics. This is illustrated in fig. 5.9(a). First, we have the familiar Coulomb well or long range potential. This, as we have seen in chapter 2, gives rise to Rydberg series containing an infinite number of... 

Random collapsing interactions have been used to model the effect of the environment (understood in a broad sense that could include the measurement apparatus) on the decaying particle [13,113]. For example, it is argued that as an unstable elementary particle decays in a bubble chamber, each bubble is a measurement indicating that the particle has not yet decayed (has survived), so that a reduction takes place, resetting the system into the initial undecayed state. The decay law that should be observed in experiment will be therefore an environment-affected F(t) rather than P(t). The probability that the system is not subjected to any measurement in a time interval St is taken to be exp (-kSf). The survival probability f(f) resulting from these measurements satisfies [13]... 

Why does not the void collapse unto itself The architecture of the void is shaped by the behavior of this tiny entity called electron. In general, all chemical matter and its various interaction modes abide by the rules of the electronic behavior. In this sense, chemical matter is unique because the electron is an elementary particle and hence chemical patterns cannot be further reduced or reconstructed from bottom up (i.e. from more elementary constituents). 

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