Topological singularity

Topology is a branch of mathematics investigating relations between objects and object s properties pertinent to continuous transformations of one object into another [8]. These transformations may involve considerable deformations of the objects. However, no cutting of them or gluing their points together are allowed by the transformations. Topological singularity of such molecules as those... 

Topological singularities from spontaneous symmetry breaking at phase transitions, according to Kirzhnits and Linde in 1972 (42). 

Figure II. Density of states (DOS) of vibrational and electronic excitation in condensed matter systems and in molecular systems. The DOS for electronic and vibrational excitation in condensed matter systems is continuous, and is characterized by van Hove topological singularities for ordered structures and by exponential Mott tails for disordered materials. The DOS for electronic excitation of molecular systems is discrete below the first ionization potential. In molecular systems the vibrational DOS is discrete, while in large molecules a quasicontinuum of vibrational states exists at high energies.

In our pursuit of a theoretical description of such problems, we are led to the following ubiquitous situation In order to solve for the bulk flow field (governed by the Navier-Stokes equations, for instance), the position of the free surface and boundary conditions there, need to be known the flow field, and other interfacial forces, act to move the interface to a new position which in turn affects the flow field. In general, this coupling is nonlinear and in many problems of practical interest no steady state exists. In fact, severe flow regimes can arise as in the breakup of Newtonian liquid jets where a topological singularity is encountered in finite time. 

Topological Configurations.—It may be noticed from the preceding that there is a certain relation between limit cycles and singular points. 

Suppose we have a certain topological configuration, say, SUS in our previous notation this means that the singular point is stable and the nearest cycle is unstable. The bifurcation of the first kind can be represented by the scheme ... 

For a locally compact topological space X, let Ftl X) denote the homology group of possibly infinite singular chains with locally finite support (Borel-Moore homology). The... 

The purpose of the lectures was to discuss various properties of the Hilbert schemes of points on surfaces. Although it was not noticed until recently, the Hilbert schemes have relationship with many other branch of mathematics, such as topology, hyper-Kahler geometry, symplectic geometry, singularities, and representation theory. This is reflected to this note each chapter, which roughly corresponds to one lecture, discusses different topics. 

Statement 2. Substitution into the concentration dynamics (equations (8.2.12) and (8.2.13)) of the reaction rate K — K(Na, Nb), dependent on the current concentrations, changes the nature of the singular point. In particular, a centre (neutral stability) could be replaced by stable or unstable focus. This conclusion comes easily from the topological analysis its illustrations are well-developed in biophysics (see, e.g., a book by Bazikin [30]). 

The pancake theory today is perceived by mathematicians as a chapter contributed by Ya.B. to the general mathematical theory of singularities, bifurcations and catastrophes which may be applied not only to the theory of large-scale structure formation of the Universe, but also to optics, the general theory of wave propagation, variational calculus, the theory of partial differential equations, differential geometry, topology, and other areas of mathematics. 

Figure 1 2 1. The different types of 2.5 Lifshitz electronic topological transition (ETT) The upper panel shows the type (I) ETT where the chemical potential EF is tuned to a Van Hove singularity (vHs) at the bottom (or at the top) of a second band with the appearance (or disappearance) of a new detached Fermi surface region. The lower panel shows the type (II) ETT with the disruption (or formation) of a neck in a second Fermi surface where the chemical potential EF is tuned at a vHs associated with the gradual transformation of the second Fermi surface from a two-dimensional (2D) cylinder to a closed surface with three dimensional (3D) topology characteristics of a superlattice of metallic layers...

The interband pairing term enhances Tc [93-97,102] by tuning the chemical potential in an energy window around the Van Hove singularities, z =0, associated with a change of the topology of the Fermi surface from ID to 2D (or 2D to 3D) of one of the subbands of the superlattice in the clean limit. 

Figure 1 2 6. The Fermi surface of the second (red) and third subband (black) of a 2D superlattice of quantum wires near the type (III) ETT where the third suhhand changes from the one-dimensional (left panel) to two-dimensional (right panel) topology. Going from the left panel to the right panel the chemical potential EF crosses a vHs singularity at Ec associated with the change of the Fermi topology going from EF>EC to EF

Quite apart from their singular topology, the fullerenes are distinguished from other conjugated hydrocarbons by their non-planarity. The geometrical aspects of fullerene formation as it relates to pyramidalization of the constituent carbon atoms has been recognized for some time (Haddon et al. 1986 Haddon 1988). Here we consider the effect of non-planarity on the electronic structure of the carbon atoms as it arises in the fullerenes (Haddon et al. 1986 Haddon 1992). 

As can be seen from Fig. 4.7, the kinetic tangent pinch point at the critical Damkohler number Dar = 0.166 has an important role for the topology of the maps. This is also reflected by the feasibility diagrams given in Fig. 4.8(a-c). In Fig. 4.8(c), the stable node branch at positive Damkohler numbers are collected from the singular point analyses of the reactive condenser (Fig. 4.8(a)) and the reactive reboiler... 

As demonstrated by means of residue curve analysis, selective mass transfer through a membrane has a significant effect on the location of the singular points of a batch reactive separation process. The singular points are shifted, and thereby the topology of the residue curve maps can change dramatically. Depending on the structure of the matrix of effective membrane mass transfer coefficients, the attainable product compositions are shifted to a desired or to an undesired direction. 

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