Mathematical Origin

The development presented here for the complex impedance, Z = Zr+ jZj, is general and can be applied, for example, to the complex refractive index, the complex viscosity, and the complex permittivity. The derivation for a general transfer function G follows that presented by Nussenzveig. The development for the subsequent analysis in terms of impedance follows the approach presented by Bode.  

Remember 22.1 The Kramers-Kronig relations apply for systems that are linear, causal, and stable. The condition of stationarity is implicit in the requirement of causality. 

The fundamental constraints for the system transfer functions associated with the assumptions of linearity, casuality, and stability are described in this section. 

Theorem 22.1 (Linearity) The output is a linear function of the input. A general output function x t) can be expressed as a linear function of the input f (t) as 

Theorem 22.2 (Time-Translation Independence) The output depends only on the input. This means that, if the input signal is advanced or delayed by a time increment, the output will be advanced or delayed by the same time increment. Thus, x(t + r) corresponds to f(t + r) such that g t,t ) can depend only on the difference between t and t. Equation (22.1) can be written 

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A simple model that illustrates the mathematical origins of irreversibility was first introduced by Kac in 1956 [kac56] and has come to be known as Kac s Ring Model. 

One could view the occurrence of the metric terms in the equations of motion as an annoying complication, but we hold a more positive view. First they assure that whatever the choice of parameters to be used as dynamical variables, that choice will not introduce unphysical artifacts. Second, the metric terms are another component of the theory with potential for providing guiding principles for development of XC models. Those terms also allow the mathematical origin of physical affects to be assigned. 

The development of the methods described in Section 9.2 was an important step in modeling polarization because it led to accurate calculations of molecular polarizability tensors. The most serious issue with those methods is known as the polarization catastrophe since they are unable to reproduce the substantial decrease of the total dipole moment at distances close to contact as obtained from ab initio calculations. As noted by Applequist et al. [49], and Thole [50], a property of the unmodified point dipole is that it may originate infinite polarization by the cooperative interaction of the two induced dipoles in the direction of the line connecting the two. The mathematical origins of such singularities are made more evident by considering a simple system consisting of two atoms (A and B) with isotropic polarizabilities, aA and c b. The molecular polarizability, has two components, one parallel and one perpendicular to the bond axis between A and B,... 

Despite PHI s seemingly mystical mathematical origins, Langdon explained, the truly mind-boggling aspect of PHI was its role as a fundamental building block in nature. Plants, animals, even human beings all possessed dimensional properties that adhered with eerie exactitude to the ratio of PHI to 1. 

Mathematical methods which place the MESH equations into a homotopy equation of purely mathematical origin... 

The mathematical origin of these concentration discontinuities or weak solutions of Eq. 7.1 has been explained by Courant and Friedrichs [24] and by Lax [25]. The mathematical backgroimd has been reviewed in cormection with the discussion of the numerical solution of the equilibrium-dispersive model given by Rou-chon et al. [26]. In the traditional theory of partial differential equations, a solution should be continuous. Lax [27] generalized the concept of solution to include weak solutions, which are not continuously differentiable. A solution of Eq. 7.1 that includes a continuous part, or diffuse bormdary, and a concentration shock is a weak solution of this equation [1,26-28]. A serious problem then arises, since there is no unique weak solution of Eq. 7.1. It is necessary to define the weak solution that is acceptable for the physical problem in order to achieve the determination of the band profile. This solution must make physical sense and prevent the crossing of the characteristics. Oleinik has suggested a selection rule that can... 

The mathematical origin of the weak-line limit lies quite obviously in the expansion exp(—r) 1 — r for small r which necessarily fails as the product fNLH and, therefore, r increases. The physical origin of the failure is readily understood. Atoms in the lower state L wherever they are in the cloud have an equal propensity to remove photons from the beam of starlight impinging on the atoms. When fN H is very small, all atoms see the same intensity Iq and remove the same number of photons. As fN H is increased, atoms furthest from the entrypoint of the starlight into the cloud see a reduced intensity (I < Io) and, hence, remove fewer photons than they did when fNLH was smaller. The effect on the CoG is obvious W increases at a less than linear rate with increasing fNLH. 

The notation Dq has mathematical origins in crystal field theory. We prefer the use of Ao because of its experimentally determined origins (see Section 20.6). 

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