Laplace-Fourier space

To characterize diffusion on these structures, we calculate the MSD using the CTRW formalism with the waiting time PDFs (6.29), (6.31), (6.34), (6.36), and the dispersal PDF (6.27). In Laplace-Fourier space, the MSD reads... 

Thanks to the space and time convoluted nature of Eq. (170) and to the Laplace-Fourier convolution theorem, we get the counterpart of Eq. (62), which reads... 

Evaluation of the long-range part of the electrostatic potential, which results from the long-wavelength limit (fc = 0) of the corresponding Fourier expansion, is the trickiest part in the derivation of the Ewald expression for the electrastatic potential of a Coulombic system. The problem is iminediately apparent from Laplace s equation in Fourier space [sec Eq. (F.21)] which, when solved for k) directly at fc = 0, yields a divergent result because of the factor Fortunately, we are not really... 

The derivation of the electrostatic properties from the multipole coefficients given below follows the method of Su and Coppens (1992). It employs the Fourier convolution theorem used by Epstein and Swanton (1982) to evaluate the electric field gradient at the atomic nuclei. A direct-space method based on the Laplace expansion of 1/ RP — r has been described by Bentley (1981). 

As neutron scattering does not distinguish between the molecular levels, one wants to know the probability density in space Gs(r, t) = P r, t) + P2(r, t). Its Fourier-Laplace transform is... 

Note that here z is not the usual Laplace frequency (used in rest of the review), but it is a Fourier frequency. Equation (226) can be written in the Fourier-Laplace space (z) as... 

The formulas of Appendix A are easily transformed to time-dependent forms via the Fourier-Laplace transform. For the purpose of upcoming extensions to the nonself adjoint case, we introduce the self-adjoint Hamiltonian H = FT = Htt. Furthermore the function 0 considered above is said to belong to the domain of the operator H, i.e.,

self-adjoint problems D(W) = D(H) and for bounded operators, the range R(H) and the domain D(H) are identical with the full Hilbert space however, in general applications they will differ as we will see later. 

The physical meaning of the assumption of Eq. (112) is evident. We are setting the condition that in this case persistence refers to velocity rather than position, while the division by (f), setting the correct physical dimensions, establishes the time scale of the process. The GME of Eq. (57) implying convolution in both space and time, makes it easy to evaluate the Fourier-Laplace transform of p(x, t), p(k, u), which reads [49]... 

A wave may be viewed as a unit of the response of the system to applied input or disturbances. These responses could be in terms of physical deflections, pressure, velocity, vorticity, temperature etc., those physical properties relevant to the dynamics, showing up in general, as function of space and time. Any arbitrary function of space and time can be written in terms of Fourier-Laplace transform as given by,... 

Here we are interested in the asymptotic behavior of the exact solution to Eq. (57) and we follow the analysis of Bologna et al. [52]. The most direct way to determine these properties is to take the Laplace transform in time and Fourier transform in space to obtain the Fourier-Laplace transform of the Liouville density... 

The description of imaging experiments in reciprocal space is not restricted to k space, the Fourier conjugate space of physical space. The modification of the spin density by other parameters like resonance frequencies, coupling constants, relaxation times, etc., can be treated in a similar fashion [Miil4]. For the frequency-dependent spin density, the Fourier transformation with respect to 2 is already explicitly included in (5.4.7). Introduction of a Ti-dependent density would require the inclusion of another integration over T2 in (5.4.7) and lead to a Laplace transformation (cf. Section 4.4.1). 

The distribution for the distance from the origin that results from a random walk comprising N steps is the A-fold convolution of the probability density with itself. This is easily obtained in the space of the Fourier variable u or Laplace variable s, as being the characteristic or moment generating function for a single step raised to the power N ... 

The complex variable z (Im z < 0) is homogenetic to a frequency. The resolvent l/(z — L) is the Fourier-Laplace transform of the evolution operator (see Appendix A). Expression (93) shows that the dynamics is reduced to the determination of the matrix element of the resolvent between two observables. Therefore only a reduced dynamics has to be investigated. For that purpose we shall define more precisely the observables and the operators of interest. The theory is formulated in the framework of the Liouville space of the operators and based on hierarchies of effective Liouvillians which are especially convenient to study reduced dynamics at various macroscopic and microscopic timescales (see Appendix B). 

Theret et al. [1988] analyzed the micropipette experiment with endothelial cell. The cell was interpreted as a linear elastic isotropic half-space, and the pipette was considered as an axisymmetric rigid ptmch. This approach was later extended to a viscoelastic material of the cell and to the model of the cell as a deformable layer. The solutions were obtained both analytically by using the Laplace transform and numerically by using the finite element method. Spector et al. [ 1998] analyzed the application of the micropipette to a cylindrical cochlear outer hair cell. The cell composite membrane (wall) was treated as an orthotropic elastic shell, and the corresponding problem was solved in terms of Fourier series. Recently, Hochmuth [2000] reviewed the micropipette technique applied to the analysis of the cellular properties. 

Equation (3.141) takes the following form in Fourier-Laplace space ... 

Complete information about the specimen would be available only by tomographic methods with a stepwise rotation of the sample (see e.g. Schroer, 2006) or using inherent symmetry properties of the sample. Under the assumption of fibre symmetry of the stretched specimen around the tensile axis, from the slices through the squared FT-structure the three-dimensional squared FT-structure in reciprocal space can be reconstructed and hence also the projection of the squared FT-structure in reciprocal space. The Fourier back-transformation of the latter delivers slices through the autocorrelation function of the initial structure. Stribeck pointed out that the chord distribution function (CDF) as Laplace transform of the autocorrelation function can be computed from the scattering intensity l(s) simply by multiplying I(s) by the factor L(s) = prior to the Fourier back-... 

Special linear systems arise from the Poisson equation, d uldx + d uldy = f x, y) on a rectangle, 0 Laplace equation of Section II.A is a special case where fix, y) = 0.] If finite differences with N points per variable replace the partial derivatives, the resulting linear system has equations. Such systems can be solved in 0(N log N) flops with small overhead by special methods using fast Fourier transform (FFT) versus an order of AC flops, which would be required by Gaussian elimination for that special system. Storage space also decreases from 2N to units. Similar saving of time and space from O(N ) flops, 2N space units to 0(N log N) flops and space units is due to the application of FFT to the solution of Poisson equations on a three-dimensional box. 

The general method for solving Eqs. (11) consists of transforming the partial differential equations with the help of Fourier-Laplace transformations into a set of linear algebraic equations that can be solved by the standard techniques of matrix algebra. The roots of the secular equation are the normal modes. They yield the laws for the decays in time of all perturbations and fluctuations which conserve the stability of the system. The power-series expansion in the reciprocal space variables of the normal modes permits identification of relaxation, migration, and diffusion contributions. The basic information provided by the normal modes is that the system escapes the perturbation by any means at its disposal, regardless of the particular physical or chemical reason for the decay. 

The investigation of relaxation times and diffusion coefficients requires the determination of the eigenvalues of the matrix corresponding to the system of algebraic equations obtained from Eqs. (18) after Fourier-Laplace transformation (s, Laplace transform of time q, Fourier transform of the space coordinate). The roots of the secular equation are... 

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