Local duality for equitable partitions of a Hamming space

AbstractThe Hamming space Qn is the set of binary words of length n. A partition (C1,C2,…,Cr) of Qn with quotient matrix B=[bij]r×r is equitable if for all i and j, any word in the cell Ci has exactly bij neighbors in the cell Cj. In this paper, we provide an explicit formula relating the local spectrum of cells in the face to that in the orthogonal face

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Non-unique topological sofic entropy and a von Neumann algebra multiplicative ergodic theorem

A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. In joint work with D. Airey and L.Bowen an explicit example is exhibited of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-coloringsof random hyper-graphs. Also, in joint work with L. Bowen and B. Hayes, the classical Multiplicative Ergodic Theorem (MET) of Oseledets is generalized to cocycles taking values in a type II von Neumann algebra. This appears to be the first MET involving operators with continuous spectrum.Mathematic