Incidence combinatorics of resolutions

We introduce notions of combinatorial blowups, building sets, and nested sets for arbitrary meet-semilattices. This gives a common abstract framework for the incidence combinatorics occurring in the context of De Concini-Procesi models of subspace arrangements and resolutions of singularities in toric varieties. Our main theorem states that a sequence of combinatorial blowups, prescribed by a building set in linear extension compatible order, gives the face poset of the corresponding simplicial complex of nested sets. As applications we trace the incidence combinatorics through every step of the De Concini-Procesi model construction, and we introduce the notions of building sets and nested sets to the context of toric varieties. There are several other instances, such as models of stratified manifolds and certain graded algebras associated with finite lattices, where our combinatorial framework has been put to work; we present an outline in the end of this paper.Comment: 20 pages; this is a revised version of our preprint dated Nov 2000 and May 2003; to appear in Selecta Mathematica (N.S.

The Tchebyshev transforms of the first and second kind

We give an in-depth study of the Tchebyshev transforms of the first and second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform (of the first kind) preserves desirable combinatorial properties, including Eulerianess (due to Hetyei) and EL-shellability. It is also a linear transformation on flag vectors. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg and Readdy omega map of oriented matroids. One consequence is that nonnegativity of the cd-index is maintained. The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on the space of quasisymmetric functions QSym. It coincides with Stembridge's peak enumerator for Eulerian posets, but differs for general posets. The complete spectrum is determined, generalizing work of Billera, Hsiao and van Willigenburg. The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's classical quasisymmetric function of a poset, this map is a comodule morphism with respect to the quasisymmetric functions QSym. Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps. One such occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and Sottile's result on the terminal object in the category of combinatorial Hopf algebras. In contrast, the chain map of the first kind is both an algebra map and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page

Non-local energetics of random heterogeneous lattices

In this paper, we study the mechanics of statistically non-uniform two-phase elastic discrete structures. In particular, following the methodology proposed in (Luciano and Willis, Journal of the Mechanics and Physics of Solids 53, 1505-1522, 2005), energetic bounds and estimates of the Hashin-Shtrikman-Willis type are developed for discrete systems with a heterogeneity distribution quantified by second-order spatial statistics. As illustrated by three numerical case studies, the resulting expressions for the ensemble average of the potential energy are fully explicit, computationally feasible and free of adjustable parameters. Moreover, the comparison with reference Monte-Carlo simulations confirms a notable improvement in accuracy with respect to approaches based solely on the first-order statistics.Comment: 32 pages, 8 figure