The A∞ T-system, also called the octahedron recurrence, is a dynamical recurrence relation. It can be realized as mutation in a coefficient-free cluster algebra (Kedem 2008, Di Francesco and Kedem 2009). We define T-systems with principal coefficients from cluster algebra aspect, and give combinatorial solutions with respect to any valid initial condition in terms of partition functions of perf...

We report herein the size-controlled preparation of monodispersed cuprous oxide octahedron nanocrystals smaller than 100 nm. The method is based on the reduction of copper nitrate in Triton X-100 water-in-oil (w/o) microemulsions by gamma-irradiation. The average edge length of the octahedron-shaped nanocrystals varies from 45 to 95 nm as a function of the dose rate. The quantum confinement eff...

In the code static analysis, the choice of an adequate abstract domain is an interesting issue. In this paper, we provide a new numerical abstract domain: 4-Octahedron. It is an Octahedra subclass that infers relations of the form : { x ∼ α, x−y ∼ β, (x−y)−(z−t) ∼ λ}, such that: x, y, z and t are real variables, α, β and λ are real constants and ∼ ∈ {≤,≥}. The 4-Octahedron extends the conjuncti...

The octahedron recurrence lives on a 3-dimensional lattice and is given by f (x, y, t + 1) = ( f (x + 1, y, t) f (x − 1, y, t) + f (x, y + 1, t) f (x, y − 1, t))/ f (x, y, t − 1). In this paper, we investigate a variant of this recurrence which lives in a lattice contained in [0, m] × [0, n] × R. Following Speyer, we give an explicit non-recursive formula for the values of this recurrence and u...

The extraordinarily rich chemistry of octahedral transition metal halide and chalcohalide clusters has begun to see use in a variety of applications, ranging from catalysis to chemical sensing. In facilitating the design of such cluster-based materials, new synthetic methods for adjusting the core electronic character by atom substitution and for exchanging the six outer terminal ligands are of...

Wemake the statement rigorous that the Robinson–Schensted–Knuth correspondence is a tropicalization of the Dodgson condensation rule.

This paper is one of a series of papers in which, for a family L of graphs, we describe the typical structure of graphs not containing any L ∈ L. In this paper, we prove sharp results about the case L = {O6}, where O6 is the graph with 6 vertices and 12 edges, given by the edges of an octahedron. Among others, we prove the following results. (a) The vertex set of almost every O6-free graph can ...