Jack van Lint (1932-2004): A survey of his scientific work

When Jack van Lint was appointed as full professor at the Eindhoven University of Technology at the age of 26 he combined a PhD in number theory with a very open scientific mind. It took a sabbatical visit to Bell Laboratories in 1966 to make him understand that a new and fascinating field of applied mathematics was emerging: discrete mathematics. It fascinated and inspired him for the rest of his life. When he passed away on September 28, 2004, he left behind a legacy of 18 books and 177 articles, covering many aspects of coding theory, combinatorics, and finite geometry.van Lint was also a strong international advocate of the role that discrete mathematics ought to play in modern applied mathematics curricula. Quite a few departments sought his advice. Years later, four different universities showed their appreciation by awarding him an honorary degree.This overview is an homage to van Lint's academic achievements and can serve as an introduction to his work for younger generations.     

NUMERICAL LOCALIZATION OF ELECTROMAGNETIC IMPERFECTIONS FROM A PERTURBATION FORMULA IN THREE DIMENSIONS

This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated in electric field. Typically, the domain contains a finite number of unknown inhomogeneities of small volume and the inverse problem attempts to localize these inhomogeneities from a finite number of boundary measurements. Our localization approach is based on a recent framework that uses an asymptotic expansion for the perturbations in the tangential boundary trace of the curl of the electric field. We present three numerical localization procedures resulting from the combination of this asymptotic expansion with each of the following inversion algorithms： the Current Projection method, the MUltiple Signal Classification （MUSIC） algorithm, and an Inverse Fourier method. We perform a numerical study of the asymptotic expansion and compare the numerical results obtained from the three localization procedures in different settings.     

Boundary voltage perturbations caused by small conductivity inhomogeneities nearly touching the boundary

We establish an asymptotic expansion of the steady-state voltage potentials in the presence of a diametrically small conductivity inhomogeneity that is nearly touching the boundary. Our asymptotic formula extends those already derived for a small inhomogeneity far away from the boundary and is expected to lead to very effective algorithms, aimed at determining location and certain properties of the shape of a small inhomogeneity that is nearly touching the boundary based on boundary measurements. Viability of the asymptotic formula is documented by numerical examples.     

Spin dynamics in crystalline and amorphous DyAg

Crystalline (cr-) Dyag [CsC] structure] orders antiferro-magnetically with T_N≅60K; amorphous (am-) DyAg ferro-magnetically with T_C≈-18K. We measured the longitudinal field (LF) μ⁺SR relaxation functions G_ZZ(t) for 5K<T<300K using surface muons. In the paramagnetic state. cr-DyAg gives an exponential G_ZZ (t) in the relaxation rate rising first slowly then more rapidlynear T_N; no decoupling is observed in LF up to 0.4T. In the ordered state we see a Lorentzian Kubo-Toyabe G_ZZ(t), becoming nearly static at the lowest temperatures. Its static width is very narrow (Δ≈-7 MHz), and full dceoupling is achieved here in 0.1 T. On approaching T_N, the fluctuation rate and the static width increase mootonically bt the field distribution remains Loratzian. A LF of 0.4T is then insufficient to quench the fast exponential relaxation. In paramagnetic am-DyAg, the μ⁺ depolarization is always much faster then in cr-DyAg. At lower temperatures it is better described by a root-exponential than an exponential G_ZZ(t). Below T_C an exponentially relaxing signal with 1/3 amplitude is seen. The decoupling effect of LF up to 0.4T was negligible at all temperatures.     

μSR in UAs

UAs has the NaCl structure and undergoes a first order transition into a type I (single k) antiferromagnetic state at 123 K, followed by a second first order transition at 62 K into a type IA (double k) antiferromagnetic structure. μSR spectra of a powder sample were taken in zero and transverse fields up to 0.3 T. They cover the paramagnetic and the two antiferromagnetic states. The most significant features of our data are: i) a first increase of relaxation rate below T=180 K; ii) a sudden jump in both, relaxation rate and frequency shift at T=123 K, together with a small decrease in initial asymmetry (≈15%); iii) no μ⁺ spin rotation in zero field in the type I state; iv) an overlay, of 3 spectra in the type IA state. Two of these spectra show spin rotation in zero field. Their frequencies are clearly temperature dependent. In a transverse field of 5 and 10 mT the external field adds nearly fully to the internal field. Work supported in part by the Bundesministerium fur Forschung und Technologie, Federal Republic Germany.     

Itô's lemma without non-anticipatory conditions

Summary A stochastic integral (with respect to Brownian motion) which extends Itô's integral to anticipatory integrands is constructed and investigated. This stochastic integral is different from the Skorokhod integral. The Itô lemma is proved for this integral.     

Electron structure and spin dynamical properties of crystalline and amorphous rare-earth intermetallics from Mössbauer and μSR experiments

A comparative study of crystalline and amorphous DyAg using¹⁶¹Dy Mössbauer spectroscopy is presented as a way to gain insight into the changes in electronic structure and spin dynamical properties of the rare-earth ion when its local surrounding is randomly distorted. It can be seen, especially in the paramagnetic regime, that the influence of the lowering of local symmetry is significant in both the electronic ground state of Dy³⁺ and the fluctuation rate of the local magnetic moment. At high-spin fluctuation rates, Mössbauer spectroscopy rapidly becomes insensitive, while muon spin relaxation (μSR) has still a high power of resolution. μSR measurements on Laves phase intermetallics are used to demonstrate the presence of paramegnetic spin correlations far above the Curie temperature. First measurements of this kind on crystalline and amorphous DyAg will also be discussed.