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Convex Analysisの二,三の進展について

Sphere Packings, Lattices and Groups

Distance-Regular Graphs

Rational control of the self-assembly of large structures is one of the key challenges in chemistry, and is believed to become increasingly difficult and ultimately impossible as the number of components involved increases. So far, it has not been possible to design a self-assembled discrete molecule made up of more than 100 components. Such molecules-for example, spherical virus capsids-are prevalent in nature, which suggests that the difficulty in designing these very large self-assembled molecules is due to a lack of understanding of the underlying design principles. For example, the targeted assembly of a series of large spherical structures containing up to 30 palladium ions coordinated by up to 60 bent organic ligands was achieved by considering their topologies. Here we report the self-assembly of a spherical structure that also contains 30 palladium ions and 60 bent ligands, but belongs to a shape family that has not previously been observed experimentally. The new structure consists of a combination of 8 triangles and 24 squares, and has the symmetry of a tetravalent Goldberg polyhedron. Platonic and Archimedean solids have previously been prepared through self-assembly, as have trivalent Goldberg polyhedra, which occur naturally in the form of virus capsids and fullerenes. But tetravalent Goldberg polyhedra have not previously been reported at the molecular level, although their topologies have been predicted using graph theory. We use graph theory to predict the self-assembly of even larger tetravalent Goldberg polyhedra, which should be more stable, enabling another member of this polyhedron family to be assembled from 144 components: 48 palladium ions and 96 bent ligands.

Self-assembly of tetravalent Goldberg polyhedra from 144 small components

1 General Properties of Lattices.- 2 Geometric Inequalities.- 3 Perfection and Eutaxy.- 4 Root Lattices.- 5 Lattices Related to Root Lattices.- 6 Low-Dimensional Perfect Lattices.- 7 The Voronoi Algorithm.- 8 Hermitian Lattices.- 9 The Configurations of Minimal Vectors.- 10 Extremal Properties of Families of Lattices.- 11 Group Actions.- 12 Cross-Sections.- 13 Extensions of the Voronoi Algorithm.- 14 Numerical Data.- 15 Appendix 1: Semi-Simple Algebras and Quaternions.- 16 Appendix 2: Strongly Perfect Lattices.- References.- List of Symbols.

Perfect Lattices in Euclidean Spaces

A new approach to bathymetry smoothing in sigma-coordinate ocean models

In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a #P-hard problem. On the other hand we describe an algorithm for this problem which is especially suited for low dimensional (say dimensions at most 12) and for highly-symmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.

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Complexity and algorithms for computing Voronoi cells of lattices

A Frank–Kasper structure is a 3-periodic tiling of the Euclidean space {\bb E}^3 by tetrahedra such that the vertex figure of any vertex belongs to four specified patterns with, respectively, 20, 24, 26 and 28 faces. Frank–Kasper structures occur in the crystallography of metallic alloys and clathrates. A new computer enumeration method has been devised for obtaining Frank–Kasper structures of up to 20 cells in a reduced fundamental domain. Here, the 84 obtained structures have been compared with the known 27 physical structures and the known special constructions by Frank–Kasper–Sullivan, Shoemaker–Shoemaker, Sadoc–Mosseri and Deza–Shtogrin.

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

In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.

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Classification of eight-dimensional perfect forms

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Mathieu Dutour Sikirić

Papers

A new approach to bathymetry smoothing in sigma-coordinate ocean models

Complexity and algorithms for computing Voronoi cells of lattices

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

Classification of eight-dimensional perfect forms

Complexity and algorithms for computing Voronoi cells of lattices