On a conjecture of wang and williams

Wang and Williams defined a threshold assignment for a graph G as an assignment of a non-negative weight to each vertex and edge of G, and a threshold t, such that a set S of vertices is stable if and only if the total weight of the subgraph induced by S does not exceed t - 1. This is always possible with zero vertex-weights, unit edge-weights, and t = 1. By definition, a threshold graph is a graph having a threshold assignment with zero edge-weights. The threshold weight of G is the minimum total edgeweight of a threshold assignment of G. A graph G = (V,E) is called heavy if its threshold weight is |E|. A clique C of G is called big if |E(C)| > |E(G - C)|. Wang and Williams showed that graphs with a big clique are not heavy, and conjectured the converse. They verified the conjecture for triangle-free graphs. We disprove the conjecture in general, but prove it for perfect graphs.     

Martin's axiom and \Delta^2_1 well-ordering of the reals

Assuming an inaccessible cardinal , there is a generic extension in which holds and the reals have a well-ordering. Received May 6, 1993     

Free sets for nowhere-dense set mappings

We give some consistency results on the existence of uncountable free sets for nowhere-dense set mappings. For example, we prove that it is relatively consistent with ZFC and Martin’s Axiom that any nowhere-dense set mapping defined on the reals has an uncountable free set.     

NMR study of solid and liquid aluminium

Knight shift and spin-lattice relaxation were accurately measured in solid and liquid aluminium from room temperature to 1100°C. The Knight shift in the solid exhibits a strong explicit temperature dependence. K(a), the reciprocal enhancement factor of the Korringa relation, is found to be temperature dependent, an effect which is unaccounted for by the present methods of calculating the susceptibility.     

Enumeration of labelled threshold graphs and a theorem of frobenius involving eulerian polynomials

We enumerate labelled threshold graphs by the number of vertices, the number of isolated vertices, and the number of distinct vertex-degrees and we give the exact asymptotics for the number of labelled threshold graphs withn vertices. We obtain the appropriate generating function and point out a combinatorial interpretation relating its coefficients to the Stirling numbers of the second kind. We use these results to derive a new proof of a theorem of Frobenius expressing the Eulerian polynomials in terms of the Stirling numbers.     

Industry structure,performance and stability

This paper looks at the issue once more of whether there is a statistically significant relationship between industry structure and economic performance. Tempered by the results of a test for structural stability, it is suggested that judicious choice of the industries included in the sample used to estimate the hypothesized relationship will lead to any desired result. As a consequence, one must be cautious in accepting the results of any study that unequivocally purports to show the presence or absence of an identifiable relationship between industry structure and industry performance.     

Pulse evolution and mode selection characteristics in a TEA-CO2 laser perturbed by injection of external radiation

A grating-tunable TEA-CO₂ laser with an unstable resonator cavity, modified to allow injection of cw CO₂ laser radiation at the resonant transition line by means of an intracavity NaCl window, has been used to study the coupling requirements for generation of single frequency pulses. The width and shape of the mode selection region, and the dependence of the gain-switched spike buildup time and the pulse shapes on the intensity and detuning frequency of the injected radiation are reported. Comparisons of the experimental results with previously reported mode selection behavior are discussed.     

Infrared laser enhancement of the reaction of sulfur hexafluoride with sodium vapor

The vibrational energy dependence of the rate of the gas phase reaction of Na with SF₆ has been determined in a diffusion cloud experiment using CO₂ laser excitation. The “conversion efficiency” of ca. 40% for vibrational energy suggests a preference for vibrational over the translational energy when compared with “prior” statistical expectation.