Singularities essential

Interest in the pressure dependence of structural relaxation in fluids has been stimulated by recent applications [175, 176] of a simple pressure analogue of the VFTH equation for the relaxation time x at a constant pressure P to the analysis of experimental data at variable pressures. Specifically, x(P) for both polymer and small molecule fluids has been found to extrapolate to infinity at a critical pressure Pg, and this divergence takes the form of an essential singularity,... 

P. W. Brumer Prof. Mukamel has emphasized that in examining objects as they approach the classical limit one sees essential singularity behavior or not depending upon the object. 

As emphasized by Berry and others [58], the formal limit h —- 0 presents an essential singularity in the mathematical expressions involving the quantum phases exp(iSp/h) such that the analyticity of the relevant quantities is lost near h - 0. For this reason, we may expect a variety of different behaviors as this limit is approached, depending on the type of systems considered and, especially, on die type of observables. Motivated by this remark, we would like to point out the existence of a regime different from the semiclassical one, in which the quantization of the energy levels is not the dominant feature. 

The description of the local solution and other details of selecting the Bromwich contours are given in Sengupta et al. (1994) and Sengupta Rao (2006). The near-held response created due to wall excitation is shown in these references as due to the essential singularity of the bilateral Laplace transform of the disturbance stream function. While the experiments of Schubauer Skramstad (1947) verihed the instability theory, the instability theory is incapable of explaining all the aspects of the experiments or... 

Dirichlet function, which is an approximation of Delta function, S x). Various approximate representations of Dirac delta function are provided in Van der Pol Bremmer (1959) on pp 61-62. This clearly shows that we recover the applied boundary condition at y = 0. Therefore, the delta function is totally supported by the point at infinity in the wave number space (which is nothing but the circular arc of Fig. 2.20 i.e. the essential singularity of the kernel of the contour integral). 

The theoretical analysis here in the present section clearly indicates that the localized delta function excitation in the physical space is supported by the essential singularity (a —> oo) in the image plane. This is made possible because 4> y, a) does not satisfy the condition required for the satisfaction of Jordan s lemma. As any arbitrary function can be shown as a convolution of delta functions with the function depicting the input to the dynamical system. The present analysis indicates that any arbitrary disturbances can be expressed in terms of a few discrete eigenvalues and the essential singularity. In any flow, in addition to these singularities there can be contributions from continuous spectra and branch points - if these are present. 

SO on. (Even this might not cover all possibilities, but the only counterexamples so far noted have involved essentially singular functions [120].) Thus in general we expect to require nonlinear functions 1F< > such that... 

The specific heat exhibits an unobservable essential singularity at... 

As before, the specific heat has only an essential singularity,... 

Fig. 36. Schematic temperature variation of intcrfacial stiffness kn I K and interfacial free energy, for an interface oriented perpendicularly to a lattice direction of a square a) or simple cubic (b) lattice, respectively. While for tl — 2 the interface is rough for all non zero temperatures, in d — 3 il is rough only for temperatures T exceeding the roughening transition temperature 7r (see sect. 3.3). For T < 7U there exists a non-zero free energy tigT.v of surface steps, which vanishes at T = 7 r with an essential singularity. While k is infinite throughout the noil-rough phase, k Tic reaches a universal value as T - T . Note that k and fml to leading order in their critical behavior become identical as T - T. ...

In molecular property calculations the same mutual interplay of electron correlation, relativity and perturbation operators (e.g. external fields) occurs. For light until medium atoms relativistic contributions were often accounted for by perturbation theory facilitating quasirela-tivistic approximations to the Dirac-Hamiltonian [114-117]. It is well-known that operators like the Breit-Pauli Hamiltonian are plagued by essential singularities and therefore are not to be used in variational procedures. It can therefore be expected that for heavier elements per-turbational inclusion of relativity will eventually become inadequate and that one has to start from a scheme where relativitiy is included from the beginning. Nevertheless very efficient approximations to the Dirac equation in two-component form exist and will be discussed further below in combination with their relevance for EFG calculations. In order to calculate the different contributions to a first-order property as the EFG, Kello and Sadlej devised a multiple perturbation scheme [118] in which a first-order property is expanded as... 

This is the heat capacity of a one-dimensional oscillator according to Einstein. The heat capacity deviates at low temperatures. It is not possible to expand into a Taylor series around T 0. In other words, the function has a pole at zero, which emerges as an essential singular point. A more accurate formula is due to Debye, n... 

A third type, called essential singularity, arises infrequently, but should be recognized. A classic case of this type is the function... 

For an essential singularity, it is not possible to remove the infinite discontinuity. To see this for the classic example mentioned, expand expCl/ ) in series. 

If we tried to remove the singularity at the origin, by multiplying first by s, then s, then s, and so forth, we would still obtain an infinite spike at s = 0. Since the singularity cannot be removed, it is called an essential singularity. 

Knezevic and Vannimenus also studied the branched polymer problem on the GM3 fractal [42,44]. In this case, they found the unexpected result that unlike the case on the GM2 fractal ( for which the behavior is the same as on the n = 3 simplex), for the 6 = 3 case, the number of animals of size n grows as /i" exp Kn ), where 0 Kip < 1. This corresponds to an essential singularity in the generating function of branched polymers G x, u = 1) exp(p ), with p = tp/ l - tp). 

The calculation of I offers, in the general case, some rather subtle mathematical difficulties, as can be understood by considering the essentially singular character of the limit t - 0, and the non-linearity induced by the requirement of chemical equi librium. In this lecture, we focus attention on a rather simple special case, the analysis of which, however, reveals the essential features of the instantaneous reaction regime. 

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