The mean chromatic number of paths and cycles

The mean chromatic number of a graph is defined. This is a measure of the expected performance of the greedy vertex-colouring algorithm when each ordering of the vertices is equally likely. In this note, we analyse the asymptotic behaviour of the mean chromatic number for the paths and even cycles, using generating function techniques.     

When to rush a ‘behind’ in Australian rules football: a dynamic programming approach

In Australian rules football, points are scored when the ball passes over the goal line. Six points are awarded for a goal when the ball passes between the two centre posts, and one point for a ‘behind’, when the ball passes between a centre post and an adjacent outer post. After a behind, the defending team has a free kick from the goal line. It may be worthwhile, particularly in the closing stages of a game, for a defending team voluntarily to concede a behind, by themselves passing the ball between the two outer posts, either to avert the possibility of an imminent goal or to increase the probability of scoring a goal themselves. A dynamic programming model is used to analyse this situation.     

Attogram detection limits using laser induced fluorescence

Laser induced fluorescence has produced outstanding detection limits in liquid phase analysis. This paper presents a general method for optimizing detection limits as a function of sample volume.     

Preparation,crystal and molecular structure of,and NMR parameters for,the exopolyhedral heterocyclic platinaundecaborane [μ-2,7-(SCSNEt2)-7-(PMe2Ph)-NIDO-7-PtB10H11]

The compound [μ-2,7-(SCSNEt₂)-7-(PMe₂Ph)-nido-7-PtB₁₀H₁₁] has been obtained in a yield of 52% from the reaction of [7,7-(PMe₂Ph)-nido-7-PtB₁₀H₁₂] and [AuBr₂(S₂CNEt₂)], and identified by single crystal X-ray diffraction analysis and multi-element single and double resonance NMR spectroscopy. The yellow-orange compound crystallizes in the monoclinic space group P2₁/n with a 1179.2(2), b = 1244.9(5), c = 1641.4(2) pm, β = 95.45(1)°, Z = 4, and the structure (R 0.0209, R_w = 0.0211 for 3719 observed reflections) is that of a nido-7-platinaundecaborane with an exopolyhedral N,N-diethyldithiocarbamate ligand bridging the Pt(7) and B(2) positions to give a $\text{-Pt-B-C-S-}$ five-membered ring. The tetrahapto platinum-to-borane bonding has a considerable twist distortion relative to other nido-7-platinaundecaboranes which do not possess this cyclic feature. The NMR parameters exhibit no anomalies and are consistent with the crystal molecular structure. A plot of δ(¹¹B) vs δ(¹H) for directly bound exo-terminal hydrogen atoms shows good correlation with the slope 16 : 1.     

Deformation theory of quantizable dynamical systems

Deformations of families of quantizable dynamical systems are introduced and the local extension problem for quantizable dynamical systems is solved. This research was supported in part by NSF GP-13375 and NSF GP-20856. Entrata in Redazione il 2 dicembre 1970.     

Examples of quantisable dynamical systems: The hydrogen atom and automorphism groups

The regularised energy surface of then-dimensional hydrogen atom is shown to be naturally the total space of a quantisable dynamical system. The automorphism groups of dynamical systems are studied; and the connected Riemannian dynamical systems with automorphism groups of maximal dimension are classified. Finally, the compact, connected and simply connected quantisable dynamical system with automorphism group of maximal dimension is shown to be the set of independent harmonic oscillators with equal periods.This research was supported in part by NSF GP-20856A, No. 1.