In this extended abstract, we exploit the combinatorics and geometry of triangulations of products of simplices to reinterpret and generalize a number of constructions in Catalan combinatorics. In our framework, the main role of “Catalan objects” is played by (I, J)-trees: bipartite trees associated to a pair (I, J) of finite index sets that stand in simple bijection with lattice paths weakly a...

This paper is a sequel of [8]. In algebraic geometry, when discussing the compactification of the moduli space of complex manifold X with ample canonical bundle KX , it is necessary to consider holomorphic degeneration family π : X → B, where Xt = π(t) are smooth for t 6= 0, X and X0 are QGorenstein, such that the canonical bundle of Xt for t 6= 0 and the dualizing sheaf of X0 are ample. We wil...

We begin this paper by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal’s very special Γ–spaces indeed this is roughly what Segal did and then to a more modern situation: K ⊗ R where K is a based space and R is...

Introduction 2 1. A Guide to Duality 3 1.1. Local Duality 3 1.2. Dualizing Complexes and Some Vanishing Theorems 10 1.3. Cohomological Annihilators 17 2. A Few Applications of Local Cohomology 21 2.1. On Ideal Topologies 21 2.2. On Ideal Transforms 25 2.3. Asymptotic Prime Divisors 28 2.4. The Lichtenbaum-Hartshorne Vanishing Theorem 35 2.5. Connectedness results 37 3. Local Cohomology and Syzy...

Dualizing the “extended bipartite double” construction for distance-regular graphs, we construct a new family of cometric (or Q-polynomial) association schemes with four associate classes based on linked systems of symmetric designs. The analysis of these new schemes naturally leads to structural questions concerning imprimitive cometric association schemes, some of which we answer with others ...

First, the dynamics of LC-circuits are formulated as a Hamiltonian system defined with respect to a Poisson bracket which may be degenerate, i.e., nonsymplectic. This Poisson bracket is deduced from the network graph of the circuit and captures the dynamic invariants due to KirchhoWs laws. Second, the antisymmetric relations defining the Poisson bracket are realized as a physical network using ...

Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group GS(Q)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that GS(Q)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension QS(p)/Q a...

This paper develops a duality theory for connected cochain DG algebras, with particular emphasis on the non-commutative aspects. One of the main items is a dualizing DG module which induces a duality between the derived categories of DG left-modules and DG right-modules with finitely generated cohomology. As an application, it is proved that if the canonical module A/A has a semi-free resolutio...