The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).     

Untersuchung von Textilien und Textilrohstoffen

Ohne Zusammenfassung     

Correction to: On Webs of Maximum Rank (Geom. Dedicata 31 (1989), 19–35)

We correct an error in Example (3.4) in Geom. Dedicata 31 (1989), 19–35.     

Photochemical expulsion of the neutral monodentate ligand L in Ru(terpy*)(diimine)(L)2+: a dramatic effect of the steric properties of the spectator diimine ligand

A series of photoreactive complexes of the type Ru(terpy*)(N-N)(L)(2+), where terpy* is 4'-(3,5-ditertiobutylphenyl)-2,2':6',2' '-terpyridine, N-N is the bidentate chelate phen or dmp (phen = 1,10-phenanthroline, dmp = 2,9-dimethyl-1,10-phenanthroline), and L is the monodentate ligand dms, MeBN, or MeOBN (dms = dimethyl sulfide, MeBN = 2,6-dimethyl benzonitrile, MeOBN = 2,6-dimethoxybenzonitrile), has been synthesized and fully characterized by proton NMR spectroscopy, electrospray mass spectrometry, and UV-vis spectroscopy. The X-ray structures of four complexes were also obtained. In neat pyridine, the quantum yields for the photosubsitution of L by pyridine were measured and showed dramatic variations depending on the steric interactions between the spectator bidentate ligand and the leaving monodentate ligand L. The use of dmp instead of phen multiplied the photosubstitution efficiency by a factor of 20-50, depending on L. This effect could be qualitatively correlated to the distortions observed in the X-ray structures of the corresponding complexes. The highly distorted structure of Ru(terpy)(dmp)(dms)(PF(6))(2) showed a very high photosubsitution quantum yield phi = 0.36 in neat pyridine. The high photoreactivity of some of the compounds makes them particularly promising as components of future light-driven molecular machines.     

Axially symmetric rotating body consisting of a perfect fluid

An alternative method of obtaining the equilibrium configurations of a rotating body consisting of a perfect fluid is outlined. Basically, the method involves recasting the gravitational hydrodynamic equations into a set of partial differential equations of first order in the radial direction such that a center-outward integration can be performed. Specifically, with suitable initial conditions at the origin of anr, grid, a numerical integration is performed outward along a number of selected-rays, with the required derivatives at each step being determined numerically from the values of the functions on the different rays. Applicable to both Newtonian and relativistic formulations, the technique is similar to that often used to obtain equilibrium configurations in spherically symmetric models.     

Vibrational spectra of nitrido and oxo complexes

The vibrational spectra of a number of transition-metal complexes containing terminal or bridging nitrido (N^3?) and oxo (O^2?) ligands are reported. Full assignments of fundamental modes are given for (OsO₃¹⁴N)^?, (OsO₃¹⁵N)^?, (Os¹⁴NX₄)^?, (Os¹⁵NX₄)^?, (Ru¹⁴NX₄)^?, (Os¹⁴NX₅)^2?, (Os¹⁵NX₅)^2? and (Ru¹⁴NX₅)^2? (X = Cl, Br), and also for the oxo complexes (Mo¹⁶OCl₄, (Mo¹⁸OCl₄)^?, (Mo¹⁶OCl₅)^2? and (Mo¹⁸OCl₅)^2?. Force constants have been evaluated for the four- and five-coordinate complexes. The significance of the results is discussed in terms of the metalligand bonding involved in these species.     

The mixed-aldehyde synthesis of difunctional tetraarylporphyrins

The synthesis is reported of nine unsymmetrical, meso-substituted porphyrins. Among the compounds prepared are the following 5-(R)-10,15,20-tri-p-tolylporphyrins; R = 2,6-dinitrophenyl, 4-hydroxy-3-ethoxy-phenyl, 4-hydroxy-3-methoxy-5-nitrophenyl, 5-hydroxy-2-nitrophenyl and 4-hydroxy-3-nitrophenyl. Other porphyrins reported include 5-(2-(1-butoxy)phenyl)-15-(2-nitrophenyl)-10-15-di-p-tolylporphyrin and the two 5-(R)-10-15,20-tripropylporphyrins in which R = 2-nitrophenyl and 2-hydroxyphenyl. The disubstituted porphyrins offer a rational route to the synthesis of difunctional “tailed-porphyrins”.     

Mechanistic Aspects of Carbon Nanotube Nucleation and Growth

The discovery, synthesis, characterization, and applicability of carbon nanotubes have produced tremendous excitement and interest among scientists and engineers. In particular, the use of these unique tubular nanostructures for new strong lightweight materials, nanoelectronics, fuel storage and cells, electron emitters and bio, scanning probe microscopy, and chemical sensing devices has created an intense effort to advance the synthesis so as to mass produce carbon nanotubes with control over diameter and helicity. The massive and controlled synthesis of this heralded nanostructure has been a great challenge. Although significant progress has advanced the preparation, more synthetic development is required. The syntheses have so far involved three main approaches: arc discharge vaporization, laser vaporization, and catalytic chemical vapor deposition. The synthetic trend has progressed to a point where further advancement with these techniques will require a better understanding of the mechanism of nucleation and growth. The mechanics of carbon nanotube nucleation and growth involve very complex and diverse phenomena occurring under extreme conditions and on the mesoscopic scale. As yet the detail mechanism is unknown. Difficulties with experimental probing and computational simulation have increased the mystery of this mechanism. This review presents an account of research on the synthesis of carbon nanotubes and the mechanism of formation. This overview includes all three mentioned synthetic approaches and hybrids thereof. On the basis of this broad account a comprehensive mechanism for carbon nanotube nucleation and growth naturally arises. This mechanism is qualitative and it hopes to inspire more quantitative exploration and synthetic advancement.