Neumann series

It is well known that the conventional Born approximation can be applied iteratively, generating A -th order Born approximations. This approximation can be treated as the sum of N terms of the Born (or Neumann) series. However, the convergence of the Born series is questionable and depends on the norm of the integral equation (Green s) operator. It seems to be very attractive to construct similar series on the basis of the QL and QA approximations. 

Pankratov, O. V., A. V. Kuvshinov, and D. B. Avdeev, 1997, High-performance three-dimensional electromagnetic modeling using modified Neumann series. Anisotropic case J. Geomagn. Geoelectr., 49, 1541-1547. 

The method of solution described in this subsection will be referred to as the Neumann series method, or the method of successive generations. The term successive approximation is reserved for the solution method described below. 

The Neumann series or the successive-approximation methods provide formulations that, theoretically, should yield the particular solution of Eq. (252). Practical implementation of those methods calls for special care in order to avoid a contribution from the fundamental mode [the term of Eq. (254)]. A fundamental-mode contamination can result, for example, if the source condition of Eq. (253) is not satisfied exactly. Numerical round-... 

A typical regularization method for the Fredholm integral equation of the first kind is to add a term Q(f> y) (with q a fixed number) to the left-hand side of Eq. (66), thus obtaining a Fredholm integral equation of the second kind, whose solution is stable. The difiieulty of this method is related to the fact that the formula expressing such a solution (the Neumann series ) does converge only for q > A, where A is the norm of the operator A [45]. 

Here we replaced the inverse matrix by its Neumann series. This can be done for /X being sufficiently large. Thus... 

This is the Neumann series expansion (Appendix 4) for the Volterra integral equation ... 

This chapter has been used to illustrate the method discussed in Chap. 2, which may be used to solve thermoviscoelastic boundary value problems involving temperature fields simultaneously varying with position and time. The method relies upon the solution of integral equations in terms of Neumann series expansions. 

Once the random field involved in the stochastic boundary value problem has been discretized, a solution method has to be adopted in order to solve the boundary value problem numerically. The choice of the solution method depends on the required statistical information of the solution. If only the first two statistical moments of the solution are of interest second moment analysis), the perturbafion method can be applied. However, if a. full probabilistic analysis is necessary, Galerkin schemes can be utilized or one has to resort to Monte Carlo simulations eventually in combination with a von Neumann series expansion. 

Li and Neumann sought an equation of state of interfacial tensions of the form 7 l = /(Tlv. TSv). Based on a series of measurements of contact angles on polymeric surfaces, they revised an older empirical law (see Refs. 216, 217) to produce a numerically robust expression [129, 218]... 

D. Marx and J. Hiitter, in Modern Methods and Algorithms of Quantum Chemistry, J. GrotendorsC, ed., John von Neumann Institute for Computing, Julich, Germany, 2000, pp. 301-449. See http //www.fz-juelich,de/nic-series. 

Later, after von Neumann died, I saw his memoirs, and he had written a long series of discussions about putting a semiconductor near an atomic pile, producing neutrons which would excite the electrons in the semiconductor. He hoped to get very intense light out of the excited electrons, and he talked about stimulated emission. Apparently, he had sent this proposal to Edward Teller saying, Don t you think we should try this Edward Teller never bothered to answer. And so von Neumann dropped it. But he did not mention a resonant cavity for feedback, and he didn t mention coherence. He was just getting an intense light. 

For a review see M. Dolg, in Modern methods and algorithms of quantum chemistry, J. Grotendorst (Ed.), NIC Series Vol. 1, John von Neumann-Institute for Computing, Forschungszentrum Jiilich, Germany (2000). 

Roy R (1997) Phase transfer catalysis in carbohydrate chemistry. In Sasson Y, Neumann R (eds) Handbook of phase transfer catalysis. Chapman and Hall, Glasgow, (in press) Roy R, Tropper FD, Cao S, Kim JM (1997) ACS Symposium Series, (in press)... 

The names of von Neumann and Turing lead into the held of artihcial life, a subject that falls outside the scope at this book. Freitas and Merkle (2004) discuss the subject beginning from von Neumann s cellular automahon model, and usefully describe a large series of self-replicahon models presented in the artihcial life literature. 

P. Knowles, M. Schiitz, H.J. Werner, Ab initio Methods for Electron Correlation in Molecules in Modern Methods and Algorithms of Quantum Chemistry, Edited by Grotendorst, John von Neumann, Institute for computing, NIC series 88, 69 (2002). J.F. McCann Thesis, Queen s University Belfast (1984). 

Frank-Neumann has described a series of double acylations followed by oxidative cleavage reactions to yield l,6-dione-2,4-dienes (Schemes 9 and 10).21... 

A.W. Neumann, J.K. Spelt, Applied Surface Thermodynamics, Surfactant Science Series 63, Marcel Dekker, New York, 1996. 

Neumann, E.P. and Norton, J.L., "Application of Sonic Energy to Commercial Aerosol Collection Problems", Chemical Engineering Symposium Series No. 1, Vol. 47, 4 -10 (1951). 

Measurement of the inactivation rates of AMDH were performed under various conditions of salt types, salt concentrations, temperatures, and buffers (Mevarech and Neumann, 1977 Pundak et al, 1981 Zaccai et al., 1986b, 1989 Hecht andjaenicke, 1989a). It was found that (1) the inactivation process is of first order (which means that only one active form of the enzyme exists), (2) the rate constant for inactivation increases as the salt concentration decreases, (3) the temperature dependence of the rate constants of inactivation depends on type of salt, and (4) the dependence of the rate constants on salt type follows the Hofmeister series (von Hippel and Schleich, 1969), being lower for salting-out salts. The different models for the role of the salts in the stabilization of the AMDH will be discussed in Section IV,G. 

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