Intersection theorems in permutation groups

The Hamming distance between two permutations of a finite setX is the number of elements ofX on which they differ. In the first part of this paper, we consider bounds for the cardinality of a subset (or subgroup) of a permutation groupP onX with prescribed distances between its elements. In the second part. We consider similar results for sets ofs-tuples of permutations; the role of Hamming distance is played by the number of elements ofX on which, for somei, the ith permutations of the two tuples differ.     

Colourful Simplicial Depth

Inspired by Barany’s Colourful Caratheodory Theorem, we introduce a colourful generalization of Liu's simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d² + 1 and that the maximum is d^d+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.     

Bouquets of matroids,d-injection geometries and diagrams

F-squashed geometries, one of the many recent generalizations of matroids, include a wide range of combinatorial structures but still admit a direct extension of many matroidal axiomatizations and also provide a good framework for studying the performance of the greedy algorithm in any independence system. Here, after giving all necessary preliminaries in section 1, we consider in section 2F-squashed geometries which are exactly the shadow structures coming from the Buekenhout diagram:, i.e. bouquets of matroids. We introduce d-injective planes: (generalizing the case of dual net for d=1) which provide a diagram representation for high rank d-injective geometries. In section 3, after a brief survey of known constructions for d-injective geometries, we give two new constructions using pointwise and setwise action of a class of mappings. The first one, using some features of permutation geometries (i.e. 2-injection geometries), produces bouquets of pairwise isomorphic matroids. The last section 4 presents briefly some related problems for squashed geometries.     

Enhancement of stochastic resonance in distributed systems due to a selective coupling

Recent massive numerical simulations have shown that the response of a "stochastic resonator" is enhanced as a consequence of spatial coupling. Similar results have been analytically obtained in a reaction-diffusion model, using nonequilibrium potential techniques. We now consider a field-dependent diffusivity and show that the selectivity of the coupling is more efficient for achieving stochastic-resonance enhancement than its overall value in the constant-diffusivity case.     

Allowed boundary sequences for fused polycyclic patches and related algorithmic problems

We consider sequences that encode boundary circuits of fused polycycles made up of polygonal faces with p sides, p < or = 6. We give a constructive algorithm for recognizing such sequences when p = 5 or 6. A simpler algorithm is given for planar hexagonal sequences. Hexagonal and pentagonal sequences of length at most 8 are tabulated, the former corresponding to planar benzenoid hydrocarbons CxHy with y up to 14.     

More Colourful Simplices

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in ℝ ^d is contained in at least ⌈(d+1)²/2⌉ simplices with one vertex from each set. This improves the known lower bounds for all d≥4.     

Space fullerenes: computer search for new Frank–Kasper structures II

A Frank?CKasper structure is a 3-periodic tiling of the Euclidean space E ³ by tetrahedra such that the vertex figure of any vertex belongs to four specified fullerenes with, respectively, 12, 14, 15, and 16 faces. Frank?CKasper structures occur in the crystallography of metallic alloys, clathrates, zeolites, and in geometrical optimization. 27 such physical structures are known. In Dutour et ai. (Acta Crystallogr A 66:637?C639, 2010) we obtained, by computer enumeration, all 84 such structures with up to 20 cells in a reduced fundamental domain; 13 among them were known physical structures. In the present follow-up study, we managed, by improving the computation, to get all 37 new structures with 21 cells in a reduced fundamental domain. Those structures are described, using six invariants: group, the size of fundamental domain, mean coordination number $ \bar{f} $ , fraction sequence, cell orbits and major skeleton. We found pairs of distinct structures having all six invariants equal. So, we devised a new invariant, zigzag vector, and computed it for all 135 structures known from now; all have this invariant different. New bounds for $ \bar{f} $ and new directions (computational perspectives, number of Kekule structures, space octahedrites, space cubites) are also discussed.