Exponential formulas for models of complex reflection groups

In this paper we find some exponential formulas for the Betti numbers of the De Concini-Procesi minimal wonderful models Y_{G(r,p,n)} associated to the complex reflection groups G(r,p,n). Our formulas are different from the ones already known in the literature: they are obtained by a new combinatorial encoding of the elements of a basis of the cohomology by means of set partitions with weights and exponents. We also point out that a similar combinatorial encoding can be used to describe the faces of the real spherical wonderful models of type A_{n-1}=G(1,1,n), B_n=G(2,1,n) and D_n=G(2,2,n). This provides exponential formulas for the f-vectors of the associated nestohedra: the Stasheff's associahedra (in this case closed formulas are well known) and the graph associahedra of type D_n.Comment: with respect to v.1: misprint corrected in Example 3.

Chow rings of toric varieties defined by atomic lattices

We study a graded algebra D=D(L,G) defined by a finite lattice L and a subset G in L, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi. Our main result is a representation of D, for an arbitrary atomic lattice L, as the Chow ring of a smooth toric variety that we construct from L and G. We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Groebner basis of the relation ideal of D and a monomial basis of D over Z.Comment: 23 pages, 7 figures, final revision with minor changes, to appear in Invent. Mat

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.

Der Kampf um Rohstoffe : wie das Recht Kosten und Nutzen der Rohstoffausbeutung verteilt

Große Rohstoffvorräte lagern in den Entwicklungsländern, doch ihre Ausbeutung führt in diesen Ländern oft weder zu steigendem Wirtschaftswachstum noch zu verbesserten Lebensverhältnissen der Bevölkerung. Von der Milliarde der ärmsten Menschen lebt fast ein Drittel in den rohstoffreichen Ländern. Kann das transnationale Rohstoffrecht dazu beitragen, dass die Verteilung gerechter abläuft und nicht nur die Investoren und Konsumenten der Nordhemisphäre und der Schwellenländer von den Rohstoffen der Welt profitieren? Die Juniorprofessorin Isabel Feichtner untersucht die Verteilungsgerechtigkeit im Rohstoffrecht

Tropical Discriminants

Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space of A and of the tropicalization of the kernel of A. This leads to an explicit positive formula for the extreme monomials of any A-discriminant, without any smoothness assumption.Comment: Major revisions, including several improvements and the correction of Section 5. To appear: Journal of the American Mathematical Societ

CFD modelling of ocean wave interaction with thin perforated structures represented by their macro-scale effects

Fluid interaction with thin perforated structures is of interest in a range of contexts. Applications in marine engineering include current and wave interaction with aquaculture containers, breakwaters and, as a new application, platforms for floating wind turbines with perforated outer shrouds. Another more general application is for tuned liquid dampers with baffles for motion attenuation. Thus, there is significant interest in the challenge of simulating the effect of these thin porous structures using Computational Fluid Dynamics (CFD). This thesis proposes and assesses the use of a macro-scale approach to CFD modelling of wave interaction with thin perforated structures. The structures are not resolved explicitly but represented by their spatially averaged effects on the flow by means of a homogeneous porous pressure-drop applied to the Navier-Stokes momentum equation. Two options are explored where the pressure-drop is either applied as a volumetric porous zone or as a jump-condition across a porous surface. The wave modelling capabilities and the basis of the macroscopic porosity implementations are readily available in the open-source code OpenFOAM®, which is used in this work. Minor code modifications were necessary to introduce orthotropic porosity for a cylindrically shaped structure. More significant code development was required to implement accurate motion of a floating porous structure as a new capability as part of a custom motion solver. The method is applied to fixed perforated sheets and cylinders as well as a floating tension leg platform (TLP), and the overall fluid flow behaviour and global forces and motions are assessed. The validation against experimental and potential-flow results demonstrates that a macro-scale porosity representation can accurately reproduce large-scale flow, force and motion effects of all conditions investigated. As the most representative case, the CFD results of the horizontal force on the perforated cylinder differ between 2 and 12% from the experimental results. As part of this work, it is shown that, firstly, the Volume-Averaged Reynolds-Averaged Navier-Stokes (VARANS) equations can not only be used for large volumetric granular material, but also for thin perforated structures, and secondly, that the effects of applying a RANS turbulence model on the results are of minor significance and that the full Navier-Stokes equations give good results. The presented macro-scale approach offers greater flexibility in the range of wave conditions that can be modelled compared to approaches based on linear potential-flow theory and requires a smaller computational effort compared to CFD approaches which resolve the micro-structural geometry of the openings and the fluid flow across it explicitly. This approach can therefore be an efficient alternative to assess large-scale effects for engineering problems