Kirsti Andersen: Optical Illusions in Rome – A Mathematical Travel Guide

Mathematische Semesterberichte -     

X-Band Pulsed ENDOR Study ofFe-Substituted Sodalite— The Effect of the Zero-Field Splitting

X-band (∼9.3 GHz) pulsed ENDOR measurements were carried out on⁵⁷Fe-substituted sodalite (FeSOD) which contains only one type of Fe(III) (S=) located at a framework site. The ENDOR spectrum recorded atg= 2 shows three doublets corresponding to the sixM_Smanifolds. The assignment of these signals was confirmed by hyperfine-selective and triple ENDOR experiments. The components of each of the doublets had different intensities, reflecting the different populations of the EPR energy levels at the measurement temperature, 1.8 K. ENDOR spectra were recorded at magnetic fields within the EPR powder pattern, and the field dependence observed showed an anisotropic behavior, unexpected from the isotropic character of the⁵⁷Fe(III) hyperfine coupling. This dependence was attributed to the high-order effects of the zero-field splitting (ZFS) interaction on the ENDOR frequencies. Three different theoretical approaches were used to account for the dependence of the ENDOR spectrum on the ZFS interaction. The first involves the exact diagonalization of the total spin Hamiltonian, the second uses third-order perturbation approximations, and the third employs an effective nuclear Hamiltonian for each of theM_Smanifolds. The simulations showed that the ENDOR signals of theM_S= ±5/2 (ν_±5/2) manifold are the least sensitive to the magnitude of the ZFS parameterDand are therefore the most appropriate for the determination ofa_iso. It is shown that at X band anda_isovalues of about 30 MHz, the perturbation approach is valid up toDvalues of 500 MHz if all three doublets are concerned. However, if only the ν_±5/2doublet is considered, then this approach is valid forD< 1000 MHz. The third approach was found inappropriate fora_isovalues of ∼30 MHz. Using the method of exact diagonalization together with orientation selectivity, the trends observed in the experimental spectra could be reproduced. The ENDOR spectra of the⁵⁷Fe-substituted zeolites ZSM5, L, and mazzite showed broad and ill-defined peaks since the ZFS of Fe(III) in these zeolites is significantly larger than that of FeSOD. Because this broadening is a high-order effect, it can be significantly reduced at higher spectrometer frequencies.     

Evolutionary insights on the willingness to communicate

While in previous models of pre-play communication players are forced to communicate, we investigate what happens if players can choose not to participate in this cheap talk. Outcomes are predicted by analyzing evolutionary stability in a population of a priori identical players. If the game following the communication rewards players who choose the same action then an efficient outcome is only guaranteed when participation in the pre-play communication is voluntary. If however players aim to coordinate on choosing different actions in the underlying game and there are sufficiently many messages then the highest payoff is selected when players are forced to talk to each other before playing the game. Received: September 2000/Revised: March 2003     

The Lanczos Method with Semi-Definite Inner Product

The spectral transformation Lanczos method is very popular for solving large scale real symmetric generalized eigenvalue problems. The method uses a special inner product so that the symmetric Lanczos method can be used. Sometimes, a semi-definite inner product must be used. This may lead to instabilities and break-down. In this paper, we suggest cures for breakdown by use of implicit restarting and the pseudo-Lanczos method.This revised version was published online in October 2005 with corrections to the Cover Date.     

Subspace methods for three‐parameter eigenvalue problems

We propose subspace methods for three‐parameter eigenvalue problems. Such problems arise when separation of variables is applied to separable boundary value problems; a particular example is the Helmholtz equation in ellipsoidal and paraboloidal coordinates. While several subspace methods for two‐parameter eigenvalue problems exist, their extensions to a three‐parameter setting seem challenging. An inherent difficulty is that, while for two‐parameter eigenvalue problems, we can exploit a relation to Sylvester equations to obtain a fast Arnoldi‐type method, such a relation does not seem to exist when there are three or more parameters. Instead, we introduce a subspace iteration method with projections onto generalized Krylov subspaces that are constructed from scratch at every iteration using certain Ritz vectors as the initial vectors. Another possibility is a Jacobi–Davidson‐type method for three or more parameters, which we generalize from its two‐parameter counterpart. For both approaches, we introduce a selection criterion for deflation that is based on the angles between left and right eigenvectors. The Jacobi–Davidson approach is devised to locate eigenvalues close to a prescribed target; yet, it often also performs well when eigenvalues are sought based on the proximity of one of the components to a prescribed target. The subspace iteration method is devised specifically for the latter task. The proposed approaches are suitable especially for problems where the computation of several eigenvalues is required with high accuracy. MATLAB implementations of both methods have been made available in the package MultiParEig (see http://www.mathworks.com/matlabcentral/fileexchange/47844-multipareig ).