Topological methods

Topological Methods. The topological properties of the stirred-reactor equations (37) can be used to predict the occurrence of multiple states and to determine their stability. More sophisticated tediniques can be enqiloyed in determining the oscillatory nature and limit-cycle bdiaviour of such systems. The introduction of topological methods in the study of chemical reactors was made by Oavalas in 196S fixed-point methods were introduced in the study of thermodynamically 

The operating state for low flow rates corresponds to the Sonenov stationary state. Boreskov and Slinko provide a modified result for the stable temperature excess attainable in the reactor. Assuming that the reaction rate depends upon reactant concentration as /(c) and that the feed tonpraature Tt equals the ambient temperature F the requirement for stability in terms of Fean be expressed by the inequality  

This is exactly analogous to the Sonenov result in the zero-order case, /(c) = constant. 

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Blood-brain barrier permeation of 7, among other drugs, was predicted from its three-dimensional molecular structure by a computational method (0OJMC2204). The combination of molecular topological methods using 137 quinolones, including 7 provided an excellent tool for the design of new... 

What is really difficult is to ascertain whether a given differential equation has a limit cycle solution. In fact, aside from a few well-known equations (of van der Pol, Rayleigh, Li6nard, etc.) we hardly know anything at all about the existence of cycles directly, that is, on the basis of the topological methods with which we are concerned here. 

Theory of Bifurcations.—In the preceding sections we have reviewed a few more important points of the existing topological methods in the theory of oscillation, assuming that the topological configuration or the phase portrait remains fixed. 

From topological methods. This method is best restricted to steric effect and polarizability parameters. 

Simple branching equation (SB) A topological method for describing steric effects which takes into account the order of branching by using as parameters the number of atoms other than H that are bonded to the i-th atoms of the substituent. 

Vol. 1537 P. Fitzpatrick, M. Martelli, J. Mawhin, R. Nussbaum, Topological Methods for Ordinary Differential Equations. Montecatini Terme, 1991. Editors M. Furi, P. Zecca. VII, 218 pages. 1993. 

Vol. 1560 T. Bartsch, Topological Methods for Variational Problems with Symmetries. X, 152 pages. 1993. 

Lipophilicity in particular, as reflected in partition coefficients between aqueous and non-aqueous media most commonly water (or aqueous buffer) and Z-octanol,has received much attention [105,141,152,153,176,199,232,233]. Logic )W for the octanol-water system has been shown to be approximately additive and constitutive, and hence, schemes for its a priori calculation from molecular structure have been devised using either substituent tt values or substructural fragment constants [289, 299]. The approximate nature of any partition coefficient has been frequently emphasized and, indeed, some of the structural features that cause unreliability have been identified and accommodated. Other complications such as steric effects, conformational effects, and substitution at the active positions of hetero-aromatic rings have been observed but cannot as yet be accounted for completely and systematically. Theoretical statistical and topological methods to approach some of these problems have been reported [116-119,175,289,300]. The observations of linear relationships among partition coefficients between water and various organic solvents have been extended and qualified to include other dose-response relationships [120-122,160,161,299-302]. 

Muecia-Soler, M., Perez-Gimenez, F., GAECiA-MARCH, F.J., Salabert-Salvador, M.T., Diaz-Villanueva, W., and Castro-Bleda, M.J. Dmgs and nondmgs an effective discrimination with topological methods and artificial neural networks. /. Chem. Inf. Comput.. Sci. 2003, 43, 1688-1702. 

The difficulty can be avoided by the use of topological methods which represent steric effects as the result of contributions of all atoms other than H in the group Various topological methods have been proposed. The discussion here will be restricted to the branching equations (A 8.o). The simple branching (SB) equation is defined by... 

R. E. Merrifield and H. E. Simmons, Topological Methods in Chemistry, Wiley-Interscience, New York, 1989. 

L. B. Kier and L. H. HaU, Molecular Connectivity in Chemistry and Drug Research, Academic Press, New York, 1976 R. E. Merrifield and H. E. Simmons, Topological Methods in Chemistry, Wiley-Interscience, New York, 1989 A. T. Balaban (Ed), From Chemical Topology to Three-Dimensional Geometry, Plenum Press, New York, 1997. 

HMO calculations, based on localized polyenes instead of isolated alkenes, can account for the heats of atomization of furan (41.64 observed, 41.69 eV) and of dibenzofuran (109.09 observed, 108.92 eV). For resonance energies (quoted as resonance energy per electron, REPE) they give furan, 0.007 oxepin, -0.006 benzo[6]furan, 0.036 benzo[c]furan, 0.002 and dibenzofuran, 0.047 /3 (72T3657). That furan emerges as hardly more aromatic than a diene while pyrrole (REPE 0.039/3) is clearly aromatic is in line with other results, including those from MINDO/3 and topological methods (see Section... 

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