We investigate two and three-dimensional shell-structured-inflatable froths, which can be constructed by a recursion procedure adding successive layers of cells around a germ cell. We prove that any froth can be reduced into a system of concentric shells. There is only a restricted set of local configurations for which the recursive inflation transformation is not applicable. These configurations are inclusions between successive layers and can be treated as vertices and edges decorations of a shell-structureinflatable skeleton. The recursion procedure is described by a logistic map, which provides a natural classification into Euclidean, hyperbolic and elliptic froths. Froths tiling manifolds with different curvature can be classified simply by distinguishing between those with a bounded or unbounded number of elements per shell, without any a-priori knowledge on their curvature. A new result, associated with maximal orientational entropy, is obtained on topological properties of natural cellular systems. The topological characteristics of all experimentally known tetrahedrally close-packed structures are retrieved.
We present expressions for the energy-averages of the most general products of fluctuating S-matrix elements required to calculate the variance in the energy-averaged cross-section and related observables for compound-nucleus processes. The results, which are exact and hold from the regime of isolated resonances through to that of strongly-overlapping resonances (independently of the number of open channels), involve no more than straightforward three-dimensional integrals. In line with earlier general arguments, they are functions only of average S-matrix elements. Explicit (asymptotic) expansions which approximate these results in the domain of strongly-overlapping resonances are also determined and the leading order corrections to Ericson's treatment of fluctuations deduced. Contrary to previous studies, we find that the fluctuating S-matrix is not necessarily Gaussian distributed in this regime. In addition, we demonstrate how unitarity can be used to check our results both numerically and analytically. Other technical issues addressed include the casting of the generating function used into an "optimal" form, the treatment of complications due to its non-trivial dependence on the source matrix, and identities for the extraction of the maximal order term. These lay a foundation for other applications of the stochastic model for compound-nucleus processes. PACS: 24.60.Dr
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