On combinatorial and projective geometry

Cross ratios constitute an important tool in classical projective geometry. Using the theory of Tutte groups as discussed in [6] it will be shown in this note that the concept of cross ratios extends naturally to combinatorial geometries or matroids. From a thorough study of these cross ratios which, among other observations, includes a new matroid theoretic version and proof of the Pappos theorem, it will be deduced that for any projective space M= _n (K) of dimension n2 of M over some skewfield K the inner Tutte group is isomorphic to the commutator factor group K ^*/[K ^*, K ^*] of K ^*K{0}. This shows not only that in case M= _n (K) our matroidal cross ratios are nothing but the classical ones. It can also be used to correlate orientations of the matroid M= _n (K) with the orderings of K. And it implies that Dieudonné's (non-commutative) determinants which, by Dieudonné's definition, take their values in K ^*/[K ^*, K ^*] as well, can be viewed as a special case of a determinant construction which works for just every combinatorial geometry.Research supported by the DFG (Deutsche Forschungsgemeinschaft).     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

Methodological reflections on a three-step-design combining observation,stimulated recall and interview

In this paper a tripartite qualitative design combining abservation, stimulated recall and interview is presented and discussed. This three-step-design makes it possible to get insight into the interaction of internal and external processes when solving mathematical tasks. The data analysis depends on the research question and the methodological approach. In the light of two research projects in mathematics education two different methods of data analysis are presented and methodologically reflected.     

Analytic calculation of scaling dimensions: Tricritical hard squares and critical hard hexagons

The finite-size corrections, central chargesc, and scaling dimensionsx of tricritical hard squares and critical hard hexagons are calculated analytically. This is achieved by solving the special functional equation or inversion identity satisfied by the commuting row transfer matrices of these lattice models at criticality. The results are expressed in terms of Rogers dilogarithms. For tricritical hard squares we obtainc=7/10,x=3/40, 1/5, 7/8, 6/5 and for hard hexagons we obtainc=4/5,x=2/15, 4/5, 17/15, 4/3, 9/5, in accord with the predictions of conformal and modular invariance.     

The use of polarization effects in electromagnetic inverse scattering problems

We consider the scattering of plane, time-harmonic electromagnetic waves by a perfect conductor D. We first show that the set ?_λ consisting of the span of a fixed linear combination of the electric and magnetic far-field patterns is dense in the space of square-integrable tangential vector fields defined on the unit sphere for all values of the wave number k ≠ 0 provided that Im k ≥0. We next consider the affine hull ??_λ of ?_λ and characterize the set ? in terms of electric Herglotz fields. This result is then used to derive an optimization scheme for solving the inverse scattering problem of determining D from a knowledge of ?_λ and it is shown that this (unconstrained) optimization scheme has zero as its greatest lower bound.     

Effective primality tests for integers of the formsN=k3 n +1 andN=k2 m 3 n +1

Using third roots of unity Proth's theorem for primality testing is generalized to integers of the formN=k3 ⁿ +1, avoiding the use of Lucas sequences which are more suitable ifN+1 is factored instead ofN–1. This approach has the advantage of being easily combined with Proth's test and gives polynomial time algorithms for testing integers of the formN=k2 ^m 3 ⁿ +1.     

Femtosecond laser-induced damage of gold films

Single- and multi-shot ablation thresholds of gold films in the thickness range of 31-1400 nm were determined employing a Ti:sapphire laser delivering pulses of 28 fs duration, 793 nm center wavelength at 1 kHz repetition rate. The gold layers were deposited on BK7 glass by an electron beam evaporation process and characterized by atomic force microscopy and ellipsometry. A linear dependence of the ablation threshold fluence F_th on the layer thickness d was found for d ≤ 180 nm. If a film thickness of about 180 nm was reached, the damage threshold remained constant at its bulk value. For different numbers of pulses per spot (N-on-1), bulk damage thresholds of ∼0.7 J cm⁻² (1-on-1), 0.5 J cm⁻² (10-on-1), 0.4 J cm⁻² (100-on-1), 0.25 J cm⁻² (1000-on-1), and 0.2 J cm⁻² (10000-on-1) were obtained experimentally indicating an incubation behavior. A characteristic layer thickness of L_c ≈ 180 nm can be defined which is a measure for the heat penetration depth within the electron gas before electron-phonon relaxation occurs. L_c is by more than an order of magnitude larger than the optical absorption length of α⁻¹ ≈ 12 nm at 793 nm wavelength.