Laplace exponent

Jacobian matrix memory kernel Laplace exponent infinitesimal generator Laplace transform Laplacian matrix monodromy matrix... 

In this section we use the idea of subordination to obtain the space-fractional transport equation. Since r(t) is a nonnegative Levy process, the Laplace exponent l(s) defined in (3.161) can be written as... 

This density is the Green function for the space-fractional equation (3.174) with Q = 1. The characteristic exponent (A ) of the new process Y(t) can be obtained as a composition of the Laplace exponent l(s) with the characteristic exponent i-e.,... 

Since 0 < a < 1 the exponent in Eq. (137) is 1 — a > 0. The mathematical implication is that M(p) (137) is a multivalued function of the complex variable p. In order to represent this function in the time domain, one should select the schlicht domain using supplementary physical reasons [135]. These computational constraints can be avoided by using the Riemann-Liouville fractional differential operator oDlt a [see definitions (97) and (98)]. Thus, one can easily see that the Laplace transform of... 

Bowman and Stroud (1989) solved numerically the Laplace equation at all the sites of the lattice made of insulators (with probability I — p) and conductors (probability p) and got the values of the potential across each bond. This is solved by taking into account the condition that no potential difference can be maintained within a conducting cluster. They found that Eh (defined as the field giving the breakdown of the first bond) decreases towards 0 as p approaches pc, with an exponent equal to 1.1 0.2 in d = 2 and equal to 0.7 di 0.2 in d = 3 for both site and bond percolation. These values are consistent with the above predictions, namely that the exponent of Eh must be equal to the correlation length exponent z/. We recall that... 

This book contains tables of the properties of water and steam from 0 to 800 and from 0 to 1000 bar which have been calculated using a set of equations accepted by the members of the Sixth International Conference on the Properties of Steam in 1967. Properties which are tabulated include the pressure, specific volume, density, specific enthalpy, specific heat of evaporation, specific entropy, specific isobaric heat capacity, dynamic viscosity, thermal conductivity, the Prandtl number, the ion-product of water, the dielectric constant, the isentropic exponent, the surface tension and Laplace coefficient. Also see items [43] and [70]. 

Here subscript f refers to the fasf mode the other mode is the slow mode. 6 and Of are relaxation pseudorates p and pf are stretching exponents Af is the amplitude fraction of the fast mode. In some systems, pr <0.5 the Laplace transform of the fast mode then contains a wide range of decay times, including very short times, justifying the appellation fast for this mode. Hie breadth of the fast mode is such that it may finish decaying only at times later than the times at which the slow mode decays. 

For physically realistic systems the power of s of the denominator P should be higher or at least equal to the power of 5 of the numerator Q. The power in an exponent of a delay term is always negative, since physically predictive systems cannot be realized One of the mathematical advantages of Laplace transformation is, that it enables the input-output description of serial and parallel subsystems and description of signals easily. 

Until recently, most of the PAL data were analyzed in a finite-term lifetime approach. A computer program PATFIT, which represents annihilation lifetime distribution in a discrete manner, i.e. as a sum of several exponents, is en )loyed for this purpose. As an alternative, Gregory and Jean (10,11) proposed using a continuous lifetime analysis. In this approach the Laplace inversion program CONTIN, originally developed by Provencher (12) for analysis of fluorescence spectra, is used to obtain a continuous probability density function of annihilation lifetimes from PAL spectra. In this way one can obtain size distributions of FV in polymers. 

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