On completely free elements in finite fields

Letq>1 be a prime power,m>1 an integer,GF(q ^m) andGF (q) the Galois fields of orderq ^m andq, respectively. We show that the different module structures of (GF(q ^m), +) arising from the intermediate fields of the field extensionGF(q ^m) overGF (q), can be studied simultaneously with the help of some basic properties of cyclotomic polynomials. The results can be generalized to finite cyclic Galois extensions over arbitrary fields.In 1986, D. Blessenohl and K. Johnsen proved that there exist elements inGF(q ^m) which generate normal bases inGF(q ^m) overany intermediate fieldGF(q ^d) ofGF(q ^m) overGF(q). Such elements are called completely free inGF(q ^m) overGF(q). Using our ideas, we give a detailed and constructive proof of the most difficult part of that theorem, i.e., the existence of completely free elements inGF(q ^m), overGF(q) provided thatm is a prime power. The general existence problem of completely free elements is easily reduced to this special case.Furthermore, we develop a recursive formula for the number of completely free elements inGF(q ^m) overGF(q) in the case wherem is a prime power.     

A mathematical condition for a sublattice of a propositional system to represent a physical subsystem,with a physical interpretation

We display three equivalent conditions for a sublattice, isomorphic to aP , of the propositional systemP() of a quantum system to be the representation of a physical subsystem (see [1]). These conditions are valid for dim 3. We prove that one of them is still necessary and sufficient if dim <3. A physical interpretation of this condition is given.Wetenschappelijke medewerkers bij het Interuniversitair Instituut voor Kernwetenschappen (in het kader van navorsingsprogramma 21 EN).     

Local order in polycarbonate glasses by 13C{19F} rotational‐echo double‐resonance NMR

Nearest‐neighbor chain packing in a homogeneous blend of carbonate ¹³C‐labeled bisphenol A polycarbonate and CF₃‐labeled bisphenol A polycarbonate has been characterized using a shifted‐pulse version of magic‐angle spinning ¹³C{¹⁹F} rotational‐echo double‐resonance (REDOR) NMR. Complementary NMR experiments have also been performed on a polycarbonate homopolymer containing the same ¹³C and ¹⁹F labels. In the blend, the ¹³C observed spin was at high concentration, and the ¹⁹F dephasing or probe spin was at low concentration. In this situation, an analysis in terms of a distribution of isolated heteronuclear pairs of spins is valid. A comparison of the results for the blend and homopolymer defines the NMR conditions under which higher concentrations of probe labels can be used and a simple analysis of the REDOR results is still valid. The nearest neighbors of a CF₃ on one chain generally include a carbonate group on an adjacent chain. A direct interpretation of the REDOR total dephasing for the polycarbonate blend indicates that at least 75% of carbonate‐carbon ¹³C ··· F₃ nearest neighbors are separated by a narrow distribution of distances 4.7 ± 0.3 Å. In addition, analysis of the variations in REDOR spinning‐sideband dephasing shows that most of the ¹³C ··· F₃ dipolar vectors have a preferred orientation relative to the polycarbonate mainchain axis. This combination of distance and orientational constraints is interpreted in terms of local order in the packing of the carbonate group of one polycarbonate chain relative to the isopropylidene moiety in a neighboring chain. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2760–2775, 2006     

The random planar graph process

We consider the following variant of the classical random graph process introduced by Erd?s and Rényi. Starting with an empty graph on n vertices, choose the next edge uniformly at random among all edges not yet considered, but only insert it if the graph remains planar. We show that for all ε > 0, with high probability, θ(n²) edges have to be tested before the number of edges in the graph reaches (1 + ε)n. At this point, the graph is connected with high probability and contains a linear number of induced copies of any fixed connected planar graph, the first property being in contrast and the second one in accordance with the uniform random planar graph model. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Simulation of femtosecond pulse propagation in air

We present numerical simulations of the propagation of intense spatio-temporal fields propagating in air. Characteristic parameters of the propagation like peak intensities, plasma densities and diameters of the plasma channels are obtained. The evolution of the spectral content of the pulses is investigated and, in agreement with recent experiments, white-light supercontinuum generation is observed. We show in which areas of the pulse the supercontinuum is generated and conclude that self-phase modulation is responsible for the spectral broadening. Furthermore we investigate azimuthal instabilities of intense rotationally symmetric pulsed beams propagating in air. Although the spatial-temporal evolution of the field is strongly influenced by the onset of plasma generation, the instabilities are basically caused by the Kerr effect. We conclude that calculations assuming rotational symmetry become unrealistic due to the fast growth of azimuthal instabilities in the focal region.