Geometric approach, development

The geometric approach develops methods of conceptual design calculation of simple and complex distillation columns (i.e., methods of determination of optimal values of the main process parameters, of the numbers of theoretical trays at... 

The basis of molecular modeling is that all important molecular properties, i. e., stabilities, reactivities and electronic properties, are related to the molecular structure (Fig. 1.1). Therefore, if it is possible to develop algorithms that are able to calculate a structure with a given stoichiometry and connectivity, it must be possible to compute the molecular properties based on the calculated structure, and vice versa. There are many different approaches and related computer programs, including ab-initio calculations, various semi-empirical molecular orbital (MO) methods, ligand field calculations, molecular mechanics, purely geometrical approaches, and neural networks, that can calculate structures and one or more additional molecular properties. 

HCR involving complex polyatomic molecules usually occur via a large number of steps and result in the parallel formation of several products. In this case, the reaction rate and selectivity may easily be affected by the geometric details of nm-sized catalyst particles. To simulate such reactions, one can use (with proper modifications) general approaches developed [58] to describe adsorption of complex... 

The breakthrough came with the work of Poincare in the late 1800s. He introduced a new point of view that emphasized qualitative rather than quantitative questions. For example, instead of asking for the exact positions of the planets at all times, he asked Is the solar system stable forever, or will some planets eventually fly off to infinity Poincare developed a powerful geometric approach to analyzing such questions. That approach has flowered into the modern subject of dynamics, with applications reaching far beyond celestial mechanics. Poincare... 

Chapters 3 and 4 introduce the concepts, techniques, and computational tools of nonlinear dynamics. Professor Robert Q. Topper focuses on applications to conservative systems, while Professors Raima Larter and Kenneth Showalter cover the uses of nonlinear dynamics in dissipative systems (they dissipate rather than conserve energy). The major application to chemistry of conservative nonlinear dynamics is in the realm of molecular dynamics, whereas chemical kinetics is the main focus of the chapter on dissipative systems. In both of these chapters, a geometrical approach to understanding dynamics is introduced. As Dr. Topper clearly explains in Chapter 3, much work has gone into developing a better understanding of molecular dynamics by... 

Thus, there are basically three options for the development of geometrical models. One is to assume a purely geometrical approach based on experimentally observed shape of the threads. This is defined by several fixed points and their piece-wise approximation by suitable functions... 

The theoretical developments based on the effects of geometry on molecular aggregation have shown that physical characteristics of surfactants such as cmc, aggregate size and shape, and micellar size distribution (polydispersity) can be quantitatively described without relying on a detailed knowledge of the specific energetic components of the various molecular interactions. It is also useful in that it applies equally well to micelles, vesicles, and bilayer membranes the latter lie outside the normal models of association processes. For that reason, the geometric approach warrants a somewhat closer look. 

Full structure search can be developed by using similar approaches to those employed in the case of 2D structure search. Thus, some topological indices can be modified in such a way that they include geometrical information. For example, the global index given by Eq. (4) can be modified to Eq. (11), where are real interatomic distances. 

The geometrical flexibility of the VOF scheme can be significantly improved if in its formulation, instead of using a fixed framework, a combination of a Lagrangian-Eulerian approach is adopted. The most common approach to develop such a combined framework is the application of the Arbitrary... 

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