The Steiner ratio of several discrete metric spaces

Steiner's Problem is the “Problem of shortest connectivity”, that means, given a finite set of points in a metric space (X,ρ), search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which are to be connected. Such points are called Steiner points. If we do not allow Steiner points, that means, we only connect certain pairs of the given points, we get a tree which is called a Minimum Spanning Tree (MST). Steiner's Problem is very hard as well in combinatorial as in computational sense, but, on the other hand, the determination of an MST is simple. Consequently, we are interested in the greatest lower bound for the ratio between the lengths of these both trees:

which is called the Steiner ratio (of (X,ρ)). We look for estimates and exact values for the Steiner ratio in several discrete metric spaces. Particularly, we determine the Steiner ratio for spaces of words, and we estimate the Steiner ratio for specific graphs.     

Development of a new fluorescence decay measurement system using two-dimensional single-photon counting

A new fluorescence decay measurement system has been developed. The system consists of a spectrograph and a new two-dimensional photon counter. The combination enables measurements to be made of the fluorescence decay as a function of time and wavelength simultaneously. The time resolution is better than 5 ps with deconvolution processing, and the wavelength resolution is approximately 0.15 nm with 1200 grooves mm⁻¹ gratings. The dynamic range is 10⁵. The instrument response function (IRF) of the system is nearly gaussian, and has no tail or “after pulses” which are commonly observed using a photomultiplier in a time-correlated photon counting (TCPC) system. Therefore fast fluorescence decay of several tens of picoseconds can be measured accurately. In addition, the two-dimensional single-photon counting can be performed without wavelength scanning, so that the wavelength-dependent fluorescence decay can be easily and direcly observed with a fast throughput and a high signal-to-noise ratio. The principle of two-dimensional photon counting is discussed together with characteristics including linearity and statistical behavior.     

Metallkomplexe eines 1-Oxo-1,2-diphosphacyclopentanons

Coordination Compounds with an 1-Oxo-1,2-diphosphacyclopentanone 1-Oxo-1,2-diphosphacyclopentanone ( 2 ) is a ligand with three different sites for typical coordination. Thus seven types of structural different coordination compounds as well as salts can be received: 2 -W(CO)₅; ( 2 -SbCl₃)₂; 2 ^(–)-Cu⁽²⁺⁾- 2 ^(–); 2 ^(–)-Co⁽³⁺⁾(H₂O) · (OH)- 2 ^(–); (CO)₅W- 2 ^(–)-Cu⁽²⁺⁾- 2 ^(–)-W(CO)₅; (CO)₅W- 2 -SbCl₃; Cl₃Sb- 2 ^(–)-Cu⁽²⁺⁾- 2 ^(–)-SbCl₃; Cl₃Sb((CO)₅W)- 2 ^(–)-Cu⁽²⁺⁾-2^(–)-(W(CO)₅)SbCl₃; [Cu(en)₂(H₂O)₂]²⁺ ( 2 ^(–)); [Cu(en)₂(CH₃OH)₂]²⁺ [ 2 ^(–)-W(CO)₅]₂; NR₄⁽⁺⁾ 2 ^(–)     

Steiner Ratio for Manifolds

The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As a corollary, the Steiner ratio for flat tori, flat Klein bottles, and projective plane of constant positive curvature are computed.     

Internal partition function for the NO molecule; Integrated line absorption cross-sections for the A2Σ−X2∏ γ(1−0) and γ(2−0) bands

Internal partition functions for the NO molecule have been calculated for the range 100–1000 K at 10 K intervals using a direct summation over all the individual levels involved. Relative populations of the v = 0 level of the fundamental X²II state have been deduced.The integrated line absorption cross sections of the γ(1-0) and γ(2-0) bands have been determined for the same range of temperature.     

Excited State Properties of Lanthanide(III) Complexes with a Nonadentate Bispidine Ligand

Eu^III, Tb^III, Gd^III and Yb^III complexes of the nonadentate bispidine derivative L² (bispidine=3,7-diazabicyclo[3.3.1]nonane) were successfully synthesized and their emission properties studied. The X-ray crystallography reveals full encapsulation by the nonadentate ligand L² that enforces to all Ln^III cations a common highly symmetrical capped square antiprismatic (CSAPR) coordination geometry (pseudo C_4v symmetry). The well-resolved identical emission spectra in solid state and in solution confirm equal structures in both media. As therefore expected, this results in long-lived excited states and high emission quantum yields ([Eu^IIIL²]⁺, H₂O, 298 K, τ=1.51 ms, ϕ=0.35; [Tb^IIIL²]⁺, H₂O, 298 K, τ=1.95 ms, ϕ=0.68). Together with the very high kinetic and thermodynamic stabilities, these complexes are a possible basis for interesting biological probes.     

Laser-induced,second-order optical susceptibilities of YAB:Cr3+ single crystals

Complex temperature studies of optically stimulated second harmonic generation (SHG) at 1064?nm for pulsed 10?ns Nd:YAG laser radiation under illumination of two coherent laser beams (at 1064?nm and its 532?nm second harmonic) were carried out by means of the Maker fringe method. The bicolour coherent process allowed detection of some additionally induced non-centrosymmetry. The effect was studied at different temperatures and experimental geometries in YAB:Cr³⁺ single crystals. We found that Cr³⁺ ions play a principal role in the observed effects. The output of optically stimulated SHG was generally non-spherical in the sequence profile, contrary to the traditional non-stimulated SHG. Moreover, we also established that the optimal conditions for obtaining SHG corresponded to the fundamental/writing beam intensities ratio 6:1. A specially performed control of the photo-thermal sample heating showed that the increase of temperature did not exceed 1.2?K, which allowed the influence of photo-thermal heating to be neglected. The optimal input–output laser geometry corresponded to s-p polarisation and to the angle of about 32° between the photo-inducing and fundamental beams. Additionally in the studied temperature range (77–295?K) there was observed an enhancement of the output second-order susceptibilities from 0.72?pm/V up to 1.02?pm/V.     

Statistics of the envelope of ultrasonic backscatter from human trabecular bone

The paper describes the investigations intended to compare the results of experimental measurements of backscattering properties of the trabecular bone with the results of computer simulations. Ultrasonic RF echoes were collected using two bone scanners operating at 0.58 and 1.3 MHz. The simulations of the backscattered RF echoes were performed using the scattering model of the trabecular bone that consisted of cylindrical and spherical elements uniformly distributed in water-like medium. For each measured or simulated RF backscatter the statistical properties of the signal envelope were determined. Experimental results suggest deviations of the backscattering properties from the Rayleigh distribution. The results of simulation suggest that deviation from Rayleigh distribution depends on the variation of trabeculae diameters and the number of thin trabeculae. Experimentally determined deviations corresponded well to the deviations calculated from simulated echoes assuming trabeculae thickness variation equaled to the earlier published histomorphometric study results.