Aftereffect function

In the particular application to dielectric relaxation, fit) is the aftereffect function following the removal of a constant field [8]. The solution of Eq. (93) rendered in the frequency domain yields the Cole-Davidson equation [Eq. (10)] [28],... 

P (z) are the Legendre polynomials [51] which now constitute the appropriate basis set), Eq. (132) may be solved to yield the corresponding results for rotation in space, namely, the aftereffect function [Eq. (123)] and the complex susceptibility [Eq. (11)], with x and Xo from Eqs. (81) and (84), respectively. Apparently as in normal diffusion, the results differ from the corresponding two-dimensional analogs only by a factor 2/3 in Xo and the appropriate definition of the Debye relaxation time. 

We now present the solution of Eqs. (204) and (205) in terms of matrix continued fractions. The advantage of posing the problem in this way is that exact formulae in terms of such continued fractions may be written for the Laplace transform of the aftereffect function, the relaxation time, and the complex susceptibility. The starting point of the calculation is Eqs. (204) and (205) written as the matrix differential recurrence relation... 

Notice that the factor is a constant in the first integral, which allowed us to write it as part of the integrand. The second integral is obtained from the first by making the substitution s = t — tf si often referred to as the time lapse). As seen from equation (15), the response of the system is completely characterized by the function i/r(t) = (C/t)g /, which is therefore called the response function (or sometimes the aftereffect function to highlight the fact that the response is delayed). Moreover, it is seen that the (delayed) response may be expressed as a convolution or hereditary integral. 

Following the explosion at the World Trade Center (WTC) on September 11, 2001, various public health concerns arose regarding the air quality. Researchers believe that the explosion may account for adverse health effects in the workers and residents in the environment around the WTC. As a result, researchers from Johns Hopkins University, New York University, and Columbia University have monitored truck drivers exposed to the dust, fires, and air pollutants in the WTC aftermath. Phase I focused on the exposure of truck drivers from the disaster site, and Phase 11 focused on the respiratory health of the workers at the disaster site. Interviews and lung function tests were conducted to evaluate changes in lung function or symptoms (Community Update, 2002). Current research is under way addressing the aftereffects of exposure to dust in the air from the World Trade Center disaster. 

The principal result of our calculation is that the Debye theory (based on the Smoluchowski equation), when extended to fractional dynamics via a onedimensional noninertial fractional Fourier-Planck equation in configuration space, can explain the Cole-Cole anomalous dielectric relaxation that appears in some complex systems and disordered materials. A further result of our calculation is that the aftereffect solution [Eq. (66)] is, with slight modifications, the moment generating function of the configuration space distribution function. Hence the mean-square angular displacement of a dipole, and so on, may be easily calculated by differentiation. We must remark, however, that the fractional Debye theory can be used only at low frequencies (got < 1) just as... 

Here we wish to obtain the aftereffect solution for an assembly of fixed axis rotators. Recalling that for rotation, the probability density function must be periodic in cj), we expand W(< ).t) in the Fourier series ... 

In order to calculate dielectric response functions, we suppose that a uniform field F (having been applied to the assembly of dipoles at a time t = — oo so that equilibrium conditions prevail by the time t 0) is switched off at t = 0. In addition, we suppose that the field is weak (i.e., pF linear response condition [8]). Thus the initial distribution function W(<)>, 0) is the Boltzmann distribution function and is given by Eq. (57). Now one can readily obtain the corresponding aftereffect solution... 

System responses—actions taken to correct the effects and anticipate the aftereffects of an adverse outcome. Following each event, there is a system response that also needs to be analyzed. How did the system for incident response function How did the management act to improve safety Was an exposed worker properly treated Were communities notified appropriately How did the plant return to a normal state How rapidly did it return Finally, how was the system changed in light of the incident This stage of analysis is considered in Chapter 4. 

Using the numerical-analytic method, we can find periodic solutions of systems with aftereffect in the form of uniformly convergent sequences of periodic functions. Moreover, by employing the functions of this sequence we can determine whether these solutions exist. We first introduce some notations and prove some auxiliary statements, and then proceed to the presentation of the method itself. 

For each of the aforementioned cell samples, there were two EMS components corresponding to two chemical forms of high-spin Co". (The presence of the third component with the parameters typical for high-spin nucleogenic iron(lll), stabilized after nuclear decay of the parent Co + ion, is evidently a result of aftereffects of the Co Fe nuclear transformation, as Co " is not expected to appear in such systems.) The presence of at least two major cobaltous forms (with different 5 and A q values, see Table 17.1) revealed in the spectra of cells maybe related to the availability of different functional groups (also with possibly different donor atoms) as ligands at the bacterial cell surface (see, for example. Refs 31,32 and references therein). 

FIGURE 3.31 Shear deformation, e, as a function of time during the elastic aftereffect at the stages of fast and slow elastic deformations. 

According to the second postulate of the Boltzmann adopted in his theory of the elastic aftereffect, and the underlying Boltzmann-Volterra model that describes the relaxation phenomena, using a function of heredity [13] action occurred in the past few strains on the stresses caused by deformation of the body at any given time, do not depend on each other and therefore algebraically added. This position has received also the name of a principle of the Boltzmann s superposition. It should be noted that the polymer body superposition principle holds in the upper-bounded the range of deformation, stress and rate of change. 

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