Differential topology

A special technique, the Alexandrov one-point compactification method, often used by topologists within a differential-topological framework, has been applied in the proof of the Holographic Electron Density Fragment Theorem [159-161],... 

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

Guillemin, V. and Pollack, A. (1974) Differential Topology, Prentice Hall, Englewood Cliffs. 

Vol. 1545 J. Morgan, K. O Grady, Differential Topology of Complex Surfaces. VIII, 224 pages. 1993. 

Topology is the discipline within mathematical science dealing with the intuitive concepts of continuity and limits. This discipline is itself composed of several distinct fields of interest. Within the scope of this particular chapter, we shall investigate differential topology, with the understanding that this discipline aims... 

It is helpful to differentiate topologically peripheral positions (2, 3, 7, etc.) from meso positions (5, 10, 15, 20). Thus, two fundamental types of hydroporphyrins may be deduced. 

The Piecewise Linear Reasoner (PLR) [5] takes parameterized ordinary differential equations and produces maps with the global description of dynamic systems. Despite its present limitations, PLR is a typical example of a new approach that attempts to endow the computer with large amounts of analytical knowledge (dynamics of nonlinear systems, differential topology, asymptotic analysis, etc.) so that it can complement and expand the capabilities of numerical simulations. 

The definition of structure and structural stability for a molecular system as presented above is an example of the application of a general mathematical theory of structural stability. This work has evolved, under the general headings of differential topology and qualitative dynamics, a theory of... 

The ELF is a scalar function of three variables, and in order to obtain more information from it, it is necessary to use a mathematical approach called differential topology analysis. This was first done by Silvi and Savin,11 and later on extended by them and co-workers.45,46 Unfortunately, one cannot visualize in a global way a three-dimensional function. Usually, one resorts to isosurfaces like the ones in Figure 1, or to contour maps. A three-dimensional function has a richer structure than a one-dimensional function, and their mathematical characterization introduces some new words which are necessary to understand in order to go further. It is the purpose of this section to explain this new terminology in a manner as simpler as possible. Let us begin with a one-dimensional (ID) example, a function f(x) like the one in Figure 3. The function has three maxima and two minima characterized by the sign of the second derivative. In three dimensions (3D) there are more possibilities, for there are nine second derivatives. Hence, one does not talk about maxima but about attractors. In ID, the attractors are points, in... 

The algebraic and differential topological similarity measures required much simpler mathematical and computational apparatus than the direct comparisons of the original, complex quantum mechanical objects. 

Many of the well-established additional methods of algebraic and differential topology could be adapted and employed for the inherently topological molecular problem. 

Method 4. Index theory approach.. This method is based on the Poincare-Hopf index theorem found in differential topology, see, e.g., Gillemin and Pollack (1974). Similarly to the univalence mapping approach, it requires a certain sign from the Hessian, but this requirement need hold only at the equilibrium point. 

Munkres, J. Elementary Differential Topology Annals of Math. Studies, 54, Princeton Univ. Press Princeton, 1963. 

Guillemin, V. Pollack, A. Differential Topology Prentice Hall Englewood Cliffs, 1974. 

Besides, the mathematical properties of 7 and as dynamical systems constitute an object of investigation per se quite appealing from the purely mathematical point of view. The structure of these vector fields can be studied by the tools of differential topology. The phase portraits giving a geometric representation of the trajectories in the vicinity of points at which the modulus of the current density vanishes are particularly interesting. 

Fig. 8 Integrator-differentiator topology offering noise-less capacitive amplification and requiring a reset mechanism to avoid the saturation of the first stage with a dc signal.

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