A general pattern theorem for weighted self-avoiding polygons (SAPs) and selfavoiding walks (SAWs) in Z 2 is obtained. The pattern theorem for SAPs fits into the general framework of the pattern theorem for lattice clusters introduced by Madras (1999 Ann. Comb. 3 357-84) . Note that, unlike other pattern theorems proved for SAPs, this pattern theorem does not rely on first establishing a relationship between SAPs and SAWs. These results are applied to obtain pattern theorems for self-interacting SAPs and self-interacting SAWs.
We consider a self-avoiding walk on the simple cubic lattice, as a model of localization of a random copolymer at an interface between two immiscible liquids. The vertices of the walk are coloured A or B randomly and independently. The two liquid phases are represented by the two half-spaces z > 0 and z < 0, and the plane z = 0 corresponds to the interface between the two liquids. The energy depends on the numbers of A-vertices with positive z-coordinate and B-vertices with negative z-coordinate. In addition there is a vertex-interface interaction, irrespective of the colour of the vertex. We use exact enumeration and series analysis techniques to investigate the form of the phase diagram and how it changes as the magnitude of the vertex-interface interaction changes.
The detonation velocity of pressed TNT has been determined as a function of charge diameter at each of a series of loading densities ρ. Current theories of the diameter effect are discussed and used to compute infinite diameter detonation velocities (D∞) and detonation reaction-zone lengths from the experimental data. The results for the velocity-density dependence may be summarized as follows: D∞ = 1872.7 + 3187.2 ρ, (0.9 ≤ ρ ≤ 0.5342 g/cc); D∞ = 6762.5 + 3187.2 (ρ − 1.5342) − 25 102 (ρ − 1.5342)2 + 115 056 (ρ − 1.5342)3, (1.5342 ≤ ρ ≤ 1.636 g/cc). The reaction-zone lengths computed from the data are a decreasing function of the charge density and are in good agreement with predictions based on the grain-burning model of the reaction zone.
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