Extending Kotzig’s theorem

The weight of a graphG is the minimum sum of the two degrees of the end points of edges ofG. Kotzig proved that every graph triangulating the sphere has weight at most 13, and Grünbaum and Shephard proved that every graph triangulating the torus has weight at most 15. We extend these results for graphs, multigraphs and pseudographs “triangulating” the sphere withg handlesS _g ,g≧1, showing that the corresponding weights are at most about and 24g−9, respectively; if a (multi, pseudo) graph triangulatesS _g and it is big enough, then its weight is at most 15.     

Phase transitions of p-type (Pb,Sn,Ge)Te-based alloys for thermoelectric applications

p-type (Pb,Sn,Ge)Te-based alloys for thermoelectric applications were prepared using Bridgman technique. Second-order, rhombohedral to cubic phase transitions are involved, as evaluated from anomalies in the temperature dependence values of Seebeck coefficient, electrical conductivity, heat flow and the elongation, in the vicinity of the phase transition temperature, T_c. The correlation between these anomalies in both the electronic and thermodynamic properties was interpreted by means of the relationship of Fermi energy to the chemical potential (or to the molar Gibbs free energy).     

Approximating the traffic grooming problem

The problem of grooming is central in studies of optical networks. In graph-theoretic terms, this can be viewed as assigning colors to the lightpaths so that at most g of them (g being the grooming factor) can share one edge. The cost of a coloring is the number of optical switches (ADMs); each lightpath uses two ADMs, one at each endpoint, and in case g lightpaths of the same wavelength enter through the same edge to one node, they can all use the same ADM (thus saving g−1 ADMs). The goal is to minimize the total number of ADMs. This problem was shown to be NP-complete for g=1 and for a general g. Exact solutions are known for some specific cases, and approximation algorithms for certain topologies exist for g=1. We present an approximation algorithm for this problem. For every value of g the running time of the algorithm is polynomial in the input size, and its approximation ratio for a wide variety of network topologies—including the ring topology—is shown to be 2lng+o(lng). This is the first approximation algorithm for the grooming problem with a general grooming factor g.     

Optimal first- to sixth-order accurate Runge-Kutta schemes

An optimal choice of free parameters in explicit Runge-Kutta schemes up to the sixth order is discussed. A sixth-order seven-stage scheme that is immediately ahead of Butcher’s second barrier is constructed. The study is performed in the most general form, and its results are applicable to both autonomous and nonautonomous problems.     

Identification of firearms handling by the [Fe(PDT)(3)](2+) complex: Chemical and time-dependent factors

Handling of a gun results in the formation of invisible impressions, caused by transfer of iron traces to the skin surface. Visualization of these impressions is possible by spraying the palms with a solution of 3-(2-pyridyl)-5,6-diphenyl-1,2,4-triazine (PDT), which forms a magenta complex with iron(II) residues. Hence, mark intensity is directly related to the amounts of iron transferred to the palm. Palmar sweat plays a major role in iron transfer from the firearm to the hand. More factors, however, are involved in this process. Three time-dependent factors have been studied with relation to their effect on the developed mark: the gripping duration of the weapon; the time elapsed from the contact; and the rate of iron dissolution in aqueous solutions containing sweat components in physiological concentrations.We found that the amounts of iron transferred to the palm depend on both, the gripping period and the levels of palmar moisture. Thus, only a few seconds of gripping were required for developing good marks (corresponding to 80 ng cm⁻² of iron) on highly-moistured hands, much longer gripping periods were necessary for developing marks of similar intensity on relatively dry hands. Experiments that aimed at studying the effect of sweat components on metallic iron dissolution were carried out in aqueous solutions. It was found that chloride ions in physiological concentrations remarkably enhanced the dissolution, while l-serine, the major amino acid in palmar sweat, had a detrimental effect on this process. Urea, another sweat component, had only a minor effect on the dissolution rate.     

Photorefractive solitons and light-induced resonance control in semiconductor CdZnTe

We demonstrate the formation of (1+1) - and (2+1) -dimensional solitons in photorefractive CdZnTe:V, exploiting the intensity-resonant behavior of the space-charge field. We control the resonance optically, facilitating a 10-mus soliton formation times with very low optical power.     

Nonlinear dynamics of a ferrofluid pendulum

A ferrofluid torsion pendulum in an oscillating magnetic field exhibits a rich variety of nonlinear self-oscillatory regimes. The dynamics is governed by the system of coupled differential equations for the in- and off-axis components of the fluid magnetization and the pendulum angular deflection. In the limiting case of high driving frequency, the system reduces to the sole Rayleigh-type equation. Much more complicated temporal patterns arise when the field frequency and the pendulum eigen frequency are of the same order.     

On mappings of \mathbb{Q}^d to \mathbb{Q}^d that preserve distances 1 and \sqrt 2 and the Beckman-Quarles Theorem

Benz proved that every mapping that preserves the distances 1 and 2 is an isometry, provided d ≥ 5. We prove that every mapping that preserves the distances 1 and is an isometry, provided d ≥ 5.