Exploiting Singularities

However, in practice, the reliance on the complete (or an extremely large) basis set is avoided by exploiting the analytical cancellation of the Coulomb singularities by the correlation factor in evaluating the necessary integrals of the Hamiltonian ... 

Irvine s pioneering activities covered a wide range of problems in carbohydrate chemistry. His publications are characterized by the clear and precise thinking and by the elegance of style which are evident in all his writings. The ideas he put forward have been singularly fruitful and the exploits of the small band of research workers in St. Andrews half a century ago inspired an ever-increasing volume of important work which has been carried out since those days in many different laboratories in all parts of the world. 

Carry out a review of the literature on diffusion in the vicinity of critical points and spinodal lines. Your review should concentrate on summarizing the data that has been measured as well as on contrasting the various theories that have been proposed to explain diffusion phenomena near critical conditions. Consider the question Can the fact that the matrix of Fick diffusion coefficients is close to being singular near the spinodal be exploited in a commercially useful separation process ... 

The QED theory of the light atoms, apart from the a-expansion exploits also the expansion in parameters aZ where Z is the charge of the nucleus. Thus it is valid only for aZ < 1. This condition does not hold for the inner electron shells in heavy atoms. Moreover it does not hold even for the valence electrons in heavy atoms due to the singularity of QED operators. For the evaluation of the matrix elements of such operators the small distances of the electron from the nucleus become important. At such distances the effective electron chOTge Zejf for the valence electrons may not be small. Therefore the QED theory without aZ expansion appears to be necessary. Such theory was first... 

We recall now that at p = p, (4.68) is exact for a lattice gas hence, on the basis of widely held notions of universality, p can also be assumed in a continuum-fluid computation without doing violence to the structure of the dominant singularities that emerge at the fluid critical point as t— 0, p = p. Since (4.71) follows without further assumptions from (4.64) and (4.68), (4.71) appears to be an appropriate expression for the study of critical behavior of S2 in the constant-polarizability model. It is especially useful if we note that there is gross similarity between the function O2 in (4.71) and the attractive part of a typical pair potential. To exploit this, consider the potential... 

These are the Cauchy-Riemann conditions, and when they are satisfied, the derivative dw/ds becomes a unique single-valued function, which can be used in the solution of applied mathematical problems. Thus, the continuity property of a complex variable derivative has two parts, rather than the one customary in real variables. Analytic behavior at a point is called regular, to distinguish from nonanalytic behavior, which is called singular . Thus, points wherein analyticity breaks down are referred to as singularities. Singularities are not necessarily bad, and in fact their occurrence will be exploited in order to effect a positive outcome (e.g., the inversion of the Laplace transform ). 

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