The crystal structure of the Cu+ ion conductor, (C5H5NH)2Cu5Br7

(C₅H₅NH)₂Cu₅Br₇, in which (C₅H₅NH)⁺ is the pyridinium ion, belongs to space group P2₁2₁2₁ with Z = 4 and a = 13.09 ± 0.03 Å, b = 14.04 ± 0.03 Å, c = 11.78 ± 0.02Å. The 20 Cu⁺ ions are distributed non-uniformly over 52 tetrahedral sites. The bromide tetrahedra share faces in such a manner that undulating channels are formed in the [100] and [010] directions, and right- and left-handed helical channels are formed in the c-direction. The channels are interconnected, thereby forming a three-dimensional solid electrolyte. A detailed examination of the conduction pathways and of Cu⁺ ion site occupancies leads to the predictions that σ₂ (|b) should be somewhat greater than σ₁ (|a) but σ₃ (|c) should be substantially less than σ₁. (σ_i is the specific conductivity in the ith direction.) The ratio of available sites to current carriers and the percentage of unit cell volume attributable to the conduction pathways are both rather low and the average conductivity is rather high relative to AgI-based solid electrolytes.     

Graph minors. III. Planar tree-width

The “tree-width” of a graph is defined and it is proved that for any fixed planar graph H, every planar graph with sufficiently large tree-width has a minor isomorphic to H. This result has several applications which are described in other papers in this series.     

Multiparton interactions in photoproduction at HERA

The high energy photoproduction of jets is being observed at theep collider, HERA. It may be that the HERA centre-of-mass energy is sufficiently large that the production of more than one pair of jets perep collision becomes possible, owing to the large number density of the probed gluons. We construct a Monte Carlo model of such multiparton interactions and study their effects on a wide range of physical observables. The conclusion is that multiple interactions could have very significant effects upon the photoproduction final state and that this would for example make extractions of the gluon density in the photon rather difficult. Total rates for the production of many (i.e.≥3) jets could provide direct evidence for the presence of multiple interactions, although parton showering and hadronization significantly affect low transverse energy jets.     

A generalization of chordal graphs

In a 3-connected planar triangulation, every circuit of length ≥ 4 divides the rest of the edges into two nontrivial parts (inside and outside) which are “separated” by the circuit. Neil Robertson asked to what extent triangulations are characterized by this property, and conjectured an answer. In this paper we prove his conjecture, that if G is simple and 3-connected and every circuit of length ≥ 4 has at least two “bridges,” then G may be built up by “clique-sums” starting from complete graphs and planar triangulations. This is a generalization of Dirac's theorem about chordal graphs.     

Counting points in hypercubes and convolution measure algebras

It is shown that ifA andB are non-empty subsets of {0, 1} ⁿ (for somenεN) then |A+B|≧(|A||B|)^α where α=(1/2) log₂ 3 here and in what follows. In particular if |A|=2 ^n-1 then |A+A|≧3 ^n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2 ^n-1 then |A+A|=3 ^n-1 if and only if the points ofA lie on a hyperplane inn-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)^α. The above results imply the following improvement of a result of Talagrand [7]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))^α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system.