Singularity structure

Kosolapoff, G.M., Isomerization of alkyl phosphites. VII. Reaction with chlorides of singular structure, /. Am. Chem. Soc., 69, 1002, 1947. 

The excellent solvent and migration fastness which is typical of all benzimid-azolone pigments probably results from this singular structural principle, which has not yet been discovered in any other azo pigment class. 

Rhizomes of Helminthostachys zeylanica (Ophioglossaceae) yielded a series of complex fla-vone and flavonol derivatives with singular structures. They were named ugonins A-I. Whereas some flavones of this medicinally used plant are known for a long time," new data include also a series of flavonols." The complexity of these compounds is remarkable. Some of the structures are depicted in Figure 12.11. 

Main-group metal alkyls and aryls are valuable tools in preparative organic and inorganic chemistry, and many industrial applications are known for these reactive substances. In addition, they have attracted considerable attention from a more fundamental and theoretical point of view, because the high Lewis acidity at the metal center is the source of singular structures and bonding (1-3). 

B. M. Deb, P. K. Chattaraj, in Solitons Introduction and Applications , Ed. M. Lakshmanan, Springer Verlag, Berlin, pp. 392-398, 1988 P. K. Chattaraj, in Symmetries and Singularity Structures Integrability and Chaos in Nonlinear Dynamical Systems , Eds. M. Lakshmanan, M. Daniel, Springer-Verlag, Berlin, pp. 172-182, 1990. 

U. Frisch and G. Parisi, On the singularity structure of fully developed turbulence, in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, H. Chil, R. Benzi and G. Parisi, eds., North-Holland, New York, 1985, p. 84. 

In a system with finite Peclet number the singular structure is smoothed out at small scales by diffusion. Thus, below the diffusive scale Id = y/D/X the concentration field is smooth, nevertheless the... 

The singular structure of manzamine has also attracted enormous attention from a synthetic point of view. The formidable synthetic challenge coupled with the significant biological activity has stimulated many synthetic studies, culminating in the synthesis of several fragments and subunits of this alkaloid. 

Singularity Structure of M0ller-Plesset Perturbation Theory... 

Extracting singularity structure from perturbation series... 

EXTRACTING SINGULARITY STRUCTURE FROM PERTURBATION SERIES... 

Other kinds of approximants can also be used. For example, Olsen et al. [20] analyzed MP series convergence using a 2 X 2 matrix eigenvalue equation [38,39], which implicitly incorporates a square-root branch point. It is of course possible simply to explicitly construct an approximant as an arbitrary function with the singularity structure that E z) is expected to have. We suggest, for example, approximants of the form ... 

The value of the gap at z = 0 is simply 2Pa(0). Since z = 0 corresponds to the HF solution, this value is easily calculated. By thus constraining Pa, we can include in the analysis additional information about the singularity structure beyond the information we obtain from the perturbation series. The quadratic approximant, equation (5), can be similarly constrained. 

Molecules such as N2, C2, and CN, and in general molecules with bonds stretched from the equilibrium geometry, do not fit well into the A/B classification scheme [5,6,13,20,40]. They show irregularities in the sign patterns and poor convergence. This has been attributed to the presence of significant singularity structure in both half-planes [7,9,10]. 

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