There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Sphere Packings, Lattices and Groups

Distance-Regular Graphs

The magnet-like behavior can beobserved by slow relaxation of the magnetization below theblocking temperature. Since the discovery of SMMs in theearly 1990s, this assumption has formed the basis for theunderstanding of the origin of the anisotropic barrier.However, in recent years the development of novel lantha-nide-only SMMs that challenge and defy this theory pose anumber of questions:

A Polynuclear Lanthanide Single-Molecule Magnet with a Record

On the nature of

Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

Distance degree regular graphs

A type II matrix is a square matrixW with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix W, of a Bose-Mesner algebraN(W) , which is a commutative algebra of matrices containing the identity I, the all-one matrix J, closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, ifW is a spin model, it belongs to N(W). The transposition of matrices W corresponds to a classical notion of duality for the corresponding Bose-Mesner algebrasN(W) . Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute N(W) for a number of examples.

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Kazumasa Nomura

Papers

Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

Distance degree regular graphs

Bose-Mesner Algebras Related to Type II Matrices and Spin Models

Homogeneous graphs and regular near polygons

A refinement of the split decomposition of a tridiagonal pair