Critical points, Morse theory

Morse, M. and Cairns, S.S. (1969) Critical Point Theory in Global Analysis and Differential Topology An Introduction, Academic Press, New York, London. 

It is well known that if a smooth function / with nondegenerate critical points, i.e., a Morse function, is given on a smooth manifold Q, then knowing these points and their indices allows us to say much about the topology of the manifold Q. It will be shown in the present chapter that an analogue of this theory exists also in the case where on a symplectic manifold a set of independent functions in involution is given, the number of which is equal to half the dimension of the manifold. 

An important stage in the construction of the ordinary Morse theory is the well-known Morse lemma. It asserts that in some open neighbourhood of a nondegenerate critical point Xq of a Morse function /, there always exist local regular coordinates yi,..., ym such that the function / is written in the form... 

The Euler-Poincare formula invokes the use of Betti numbers [10] which may be calculated as the count of the number of critical points, of various types, associated with the geometrical structure of nanotori. The theory of Morse flmctions [11] relates critical points to topological structure. We shall show, an alternating sum of Betti numbers defines the Euler characteristic of a torus to be zero. This connects the topology of a nanotorus, nanotube, and plan sheet, which have the same Euler characteristic. We show that for every possible carbon nanotorus there is a geometrical dual boron nanotorus. 

Wall interfaces have also been found to alter the phase separation of nearby symmetric block copolymer chains. Fredrickson used SCF theory to demonstrate that a block copolymer melt in the vicinity of a solid wall or free surface (one with selective attraction) possessed a modified Flory-Huggins interaction parameter. Due to the connectivity of the blocks and the incompressibility of the material (an assumption of the calculation), the calculated interaction parameters have an oscillatory component with period 2njdo, normal to the wall plane, which decays exponentially from the interface. Milner and Morse also predirted this oscillatory profile normal to the surface for bulk-cylindrical morphology as well (corresponding to thickness commensurability), though they also observed that the decay length is longer closer to the mean-field critical point. 

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