Abstract. A result due in its various parts to Hendrickson, Connelly, and Jackson and Jordán, provides a purely combinatorial characterisation of global rigidity for generic barjoint frameworks in R 2 . The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in R 3 when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson's necessary conditions for the global rigidity of generic frameworks on the cylinder, cone and ellipsoid.

Abstract. We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops.In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for as many combinations of elements as possible.ese systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.

Let G be a triangulation of the sphere with vertex set V , such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined A W to be the abelian group generated by the set V , with relations r + c + s = 0 for all white triangles with vertices r, c and s. The group A B can be defined similarly, using black triangles.The paper shows that A W and A B are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the structure of A W and the theory of asymmetric Laplacians of finite directed graphs, and weaker results for orientable surfaces of higher genus are given. The relevance of the group A W to the understanding of the embeddings of a partial latin square in an abelian group is also explained.

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In this paper we provide estimates for the coverage of parameter space when using Latin Hypercube Sampling, which forms the basis of building so-called populations of models. The estimates are obtained using combinatorial counting arguments to determine how many trials, k, are needed in order to obtain specified parameter space coverage for a given value of the discretisation size n. In the case of two dimensions, we show that if the ratio (θ) of trials to discretisation size is greater than 1, then as n becomes moderately large the fractional coverage behaves as 1 − e −θ. We compare these estimates with simulation results obtained from an implementation of Latin Hypercube Sampling using MATLAB.

Let G be a properly face 2-coloured (say black and white) piecewiselinear triangulation of the sphere with vertex set V . Consider the abelian group A W generated by the set V , with relations r + c + s = 0 for all white triangles with vertices r, c and s. The group A B can be defined similarly, using black triangles. These groups are related in the following mannerThe finite torsion subgroup C is referred to as the canonical group of the triangulation. Let m t be the maximal order of C over all properly face two-coloured spherical triangulations with t triangles of each colour. By relating such a triangulation to certain directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to the triangulation's canonical group we provide improved upper and lower bounds for lim sup t→∞ (m t ) 1/t .

We construct a face two‐colourable, blue and green say, embedding of the complete graph Kn in a nonorientable surface in which there are (n−1)/2 blue faces each of which have a hamilton cycle as their facial walk and n(n−1)/6 green faces each of which have a triangle as their facial walk; equivalently a biembedding of a Steiner triple system of order n with a hamilton cycle decomposition of Kn, for all n≡3(mod36) and n≠3. Using a variant of this construction, we establish the minimum genus of nonorientable embeddings of the graph K36k+3+Km¯, for m=18k+1+6s where k≥1 and 0≤s≤k−1.

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