Wiener Hopf

The solution was first obtained independently by Wertheim [32] and Thiele [33] using Laplace transfonns. Subsequently, Baxter [34] obtained the same solutions by a Wiener-Hopf factorization teclmique. This method has been generalized to charged hard spheres. 

Duduchava R., Wendland W. (1995) The Wiener-Hopf method for system of pseudodifferential equations with applications to crack problems. Integr. Eqs. and Oper. Theory 23, 294-335. 

There are several ways of detecting peaks in such noisy signals. The Wiener-Hopf filter minimizes the expectation value of the noise power spectrum and may be used to optimally smooth the original noisy profile [19]. An alternative approach described by Hindeleh and Johnson employs knowledge of the peak shape. It synthesizes a simulated diffraction profile from peaks of known width and shape, for all possible peak amplitudes and positions, and selects that combination of peaks that minimizes the mean square error between the synthesized and measured profiles [20], This procedure is illustrated... 

Noble, B., Methods Based on The Wiener-Hopf Technique for the Solution of Partial Differential Equations, London Chelsea Publishing, 1988. 

The Wiener-Hopf Technique An Alternative to the Singular Eigenfunction Method... 

The mathematical solution follows the steps outlined for the case of neutral hard spheres. There is, however, one problem in using the Wiener-Hopf factorization [1] if we take the Fourier transform of Eq. (2.3), we get... 

SYMMETRICALLY CHARGED ELECTROLYTES WIENER-HOPF FACTORIZATION... 

Waismann and Lebowitz (1970, 1972) solved the mean spherical approximation (MSA) for the RPM electrolyte using Laplace transform techniques. We give the detailed solution to this approximation using the modification of Baxter s Wiener-Hopf factorization discussed in the section on SYMMETRICALLY CHARGED ELECTROLYTES WIENER-HOPF FACTORIZATION and in the Appendix. This will serve as the starting point for the solutions of this approximation for the sticky electrolyte model (SEM). 

APPENDIX Wiener-Hopf factorization of the Ornstein-Zernike equation for a symmetrically charged electrolyte (Thompson, 1978 Lee, 1985). 

Thompson, N., 1978, Wiener-Hopf Factorization of the Ornstein-Zernike Equation for the Restricted Primitive Model of Electrolytes", University of Maine, Chemistry Department, Report No. 2001. 

The Landman-Montroll method, based on the Wiener-Hopf technique and requiring a knowledge of 9(0 in the physical region only [3]. 

Baxter discussed the Wiener-Hopf factorization of the inverse structure factor for disordered fluids [182]. He was able to split OZ2 Eq. (Ill) into a pair of equations, which provide a route to determine c (R. Dixon and Hutchinson complemented Baxter s developments by stating conditions for proper R values (in simulation work with cubic boxes / co T / 2) and proposed a highly accurate minimization method to solve Baxter s equations [183]. These R proper values will be denoted by R and termed zeros hereafter. As proven by the present author in extensive calculations [96,103,155], there is always (at least) one zero when physically significant functions are analyzed. By following this procedure (BDH), except for very low densities, one normally obtains more than one R [Pg.111]

The Mode III problem has received most attention essentially because it is the simplest case. Mode I and II problems are considerably more difficult, but have been considered in a limiting case by Atkinson (1979), and again using an exact, partly numerical application of the Wiener-Hopf method, by Popelar and Atkinson (1980). But the physical Importance of the Mode III problem is cast into doubt by experimental observations of Knauss (1970 b) indicating that such cracks do not grow rectilinearly, at least in brittle materials. 

Alili, L. and Doney, R. A. (1999). Wiener-Hopf Factorization Revisited and Some Applications, Stoc. Stoc. Rep. 66, pp. 87-102. 

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