Theory of catastrophes

From the early advances in the quantum-chemical description of molecular electron densities [1-9] to modem approaches to the fundamental connections between experimental electron density analysis, such as crystallography [10-13] and density functional theories of electron densities [14-43], patterns of electron densities based on the theory of catastrophes and related methods [44-52], and to advances in combining theoretical and experimental conditions on electron densities [53-68], local approximations have played an important role. Considering either the formal charges in atomic regions or the representation of local electron densities in the structure refinement process, some degree of approximate transferability of at least some of the local structural features has been assumed. 

Determination of the statistical characteristics of natural catastrophes in their historical aspect, selecting categories and determining spatiotemporal scales of catastrophic changes in habitats. Analysis of the history of disasters is important for understanding the present dependences of crises both in nature and in society. The statistical characteristics of the dynamics of natural disasters enable formulation of the basis for the mathematical theory of catastrophes and to determine top-priority directions of studies. 

Papers by Zeeman and a paper by Steward reveal the perspectives of a development of the generalized theory of catastrophes. 

Vol. 525 Structural Stability, the Theory of Catastrophes, and Applications in the Sciences. Proceedings 1975. Edited by P. Hiltoa VI, 408 pages. 1976. 

Proldinich V.P. Management and decision-making under the control of die operational strength of the vessel on die basis of the modem theory of catastrophes // Marine intelligent technologies, J fsfrlO), 2013, p. 7 - 14. 

Thom s theory of catastrophes has acquired much popularity for its simple yet valuable description ofthe system-environment interaction that includes phenomena such as steady state equilibrium and life cycles (Thom, 1973). In particular, biological systems come first under catastrophe modeling because they display a causal action-reaction response to various natural or imposed constraining limits. As an example, the reactions of oiganisms to vital toxicological threats were developed into the survival attractor... 

This relation is consistent with previous results observed experimentally [49]. Although the kinetic theory of Lauritzen and Hoffman predicts the same law as Eq. 16, it predicts a divergence in L at lower undercoolings. The simulations do not show any evidence for such a catastrophe. 

J.L. Salager Phase Transformation and Emulsion Inversion on the Basis of Catastrophe Theory. In P. Becher (ed). Encyclopedia of Emulsion Technology. Vol. 3. Basic Theory. Measurement. Applications. Marcel Dekker, New York (1988). 

The context of Uppal, Ray, and Poore s work was the revival of interest in Hopf bifurcation at the end of the 1960s and the soon-to-be-transmogrified catastrophe theory of Zeeman. Ray has followed up this type of analysis in studies of polymerization and other important processes. 

In the modern mathematical theory of Lagrange singularities the metamorphosis of saucer formation is the first in a long list (related to the classification of Lie groups, catastrophe theory, etc.). But Ya.B. s pancake theory was constructed two years prior to these mathematical theories and, thus, Ya.B. s work anticipated a series of results in catastrophe theory and the theory of singularities. Many later mathematical studies in the theory of singularities and metamorphoses of caustics and wave fronts were performed under the influence of Ya.B. s pioneering work in 1970 on the pancake theory [34 ]. 

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. 

By the end of the nineteenth century it was realized that the wave theory of electromagnetic radiation could not by itself explain the form of the black-body radiation shown in Fig. 1.9. In particular, it was predicted that the energy distribution should rise indefinitely as the wavelength became shorter, rather than falling to zero as found. This alarming prediction, known as the ultraviolet catastrophe, gave the first serious clue that the theory was in need of modification,... 

This concept has been borrowed from the "catastrophe theory . Nowadays this theory has been extensively developed [220]. Strictly speaking, it is the theory of the peculiarities of differentiable mappings [221]. 

We here summarize the results of Bader et a/.118 which are concerned with the definition of molecular structure and with the extension of this concept, together with the associated concept of a bond, to the dynamic case. A precise description and physical interpretation of the making and breaking of chemical bonds is presented by these workers in a quantitative analysis of the evolution of molecular structure. The topological analysis of the dynamic system, as pointed out by Collard and Hall,119 falls naturally into the realm of an existing and elegant mathematical theory, the catastrophe theory of Thom.120... 

The author wishes to offer his thanks to Professor Zeeman for bringing to his attention the theorem on structural stability by Palis and Smale and for a stimulating and most useful discussion of our use of catastrophe theory, all of which occurred during a visit to the University of Warwick in the spring of 1980. 

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