A sagbi basis for the quantum Grassmannian

The maximal minors of a p by (m + p) matrix of univariate polynomials of degree n with indeterminate coefficients are themselves polynomials of degree np. The subalgebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree np in the Grassmannian of p-planes in (m + p)-space. These subalgebra generators are shown to form a sagbi basis. The resulting flat deformation from the quantum Grassmannian to a toric variety gives a new `Gr\"obner basis style' proof of the Ravi-Rosenthal-Wang formulas in quantum Schubert calculus. The coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and Koszul, and the ideal of quantum Pl\"ucker relations has a quadratic Gr\"obner basis. This holds more generally for skew quantum Schubert varieties. These results are well-known for the classical Schubert varieties (n=0). We also show that the row-consecutive p by p-minors of a generic matrix form a sagbi basis and we give a quadratic Gr\"obner basis for their algebraic relations.Comment: 18 pages, 3 eps figure, uses epsf.sty. Dedicated to the memory of Gian-Carlo Rot

Climate protection through tradable permits: The EU proposal for a CO2 emissions trading system in Europe

On 23 October 2001 the European Commission adopted a proposal for a directive for trade in greenhouse gas emissions. Following the US experience of emissions trading systems, this marks the first large-scale attempt to deploy this instrument of environmental policy in Europe. The proposal places European climate protection policy on a completely new footing. The prospects of its implementation have increased since the climate change conference in Marrakech. This paper introduces the draft directive and gives an initial economic appraisal. It concludes that the directive deliberately and wisely - limits the scope of the first trading phase starting 2005. Consequently, there is still considerable scope for increasing its efficiency (resulting from gains from trading) in later phases, namely by extending both the number of participants and the gases included. The number of participants in the first phase and the institutional arrangements, however, appear sufficient to enable a liquid, functioning market. In this respect, also the politically difficult decision to start with a compulsory system is to be welcomed. Important issues not yet sufficiently clarified include the concrete rules for defining the total permit quantity issued to participants by each Member State, and whether other economic sectors, which are to be treated by other policies and measures, will bear a comparable burden. Furthermore, the essential questions of primary allocation and treatment of newcomers which are in principle left up to Member States as well as the linkage of emissions trading with existing policies affecting the participating sectors must be solved before legal implementation is recommended. --

Orbitopes

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, in particular convex geometry, optimization, and algebraic geometry. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n). These include Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese orbitopes and Grassmann orbitopes. We study their face lattices, their algebraic boundary hypersurfaces, and representations as spectrahedra or projected spectrahedra.Comment: 37 pages. minor revisions of origina

Numerical Schubert calculus

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gr\"obner basis for the Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.Comment: 24 pages, LaTeX 2e with 2 figures, used epsf.st

Modelling Correlations in Portfolio Credit Risk

The risk of a credit portfolio depends crucially on correlations between the probability of default (PD) in different economic sectors. Often, PD correlations have to be estimated from relatively short time series of default rates, and the resulting estimation error hinders the detection of a signal. We present statistical evidence that PD correlations are well described by a (one-)factorial model. We suggest a method of parameter estimation which avoids in a controlled way the underestimation of correlation risk. Empirical evidence is presented that, in the framework of the CreditRisk+ model with integrated correlations, this method leads to an increased reliability of the economic capital estimate.Comment: 5 pages, 4 figure

Supporting decisions on conflicting land-uses: An integrated ecological-economic approach

An integrated ecological-economic decision-making approach is developed to help local stakeholders decide on land use in rural areas where the conflict between natural resource protection and economic development is pressing. It consists of four methodological steps. In the first step the political options and alternatives for action regarding changes in the land-use pattern are specified in order to derive politically relevant land-use strategies (scenarios). In the second step economic, ecological and social indicators are derived. The third step includes economic modelling (economic input-output model), environmental modelling (modelling of landscape water balance) and the qualitative and quantitative estimation of ecological and environmental effects. These efforts result in the production of a multi-indicator matrix. Finally, the fourth step deals with a combined monetary and multi-criteria evaluation resulting in a ranking of the land-use strategies. The discussion of the decision-making approach concentrates on the necessity of preliminary decisions and the possibility and necessity of stakeholders participation in the decisionmaking process. --evaluation,decision-making,multi-criteria analysis,land-use management,scenarios,benefit-cost analysis

Multicriteria analysis under uncertainty with IANUS - method and empirical results

IANUS is a method for aiding public decision-making that supports efforts towards sustainable development and has a wide range of application. IANUS stands for Integrated Assessment of Decisions uNder Uncertainty for Sustainable Development. This paper introduces the main features of IANUS and illustrates the method using the results of a case study in the Torgau region (eastern Germany). IANUS structures the decision process into four steps: scenario derivation, criteria selection, modeling, evaluation. Its overall aim is to extract the information needed for a sound, responsible decision in a clear, transparent manner. The method is designed for use in conflict situations where environmental and socioeconomic effects need to be considered and so an interdisciplinary approach is required. Special emphasis is placed on a broad perception and consideration of uncertainty. Three types of uncertainty are explicitly taken into account by IANUS: development uncertainty (uncertainty about the social, economic and other developments that affect the consequences of decision), model uncertainty (uncertainty associated with the prediction of the effects of decisions), and weight uncertainty (uncertainty about the appropriate weighting of the criteria). The backbone of IANUS is a multicriteria method with the ability to process uncertain information. In the case study the multicriteria method PROMETHEE is used. Since PROMETHEE in its basic versions is not able to process uncertain information an extension of this method is developed here and described in detail. --

Assessing impacts of CAP reform in France and Germany

The 2003 CAP Reform left EU member states much room for national implementation. The farm group model EU-FARMIS is applied to quantify the effects of the reform and the impacts of the options for national implementation. The analysis is done for France and Germany because their implementation schemes adequately reflect the broad range of options. It is found that cereal and fodder maize production is reduced both in France and Germany. In contrast, the acreage of other arable fodder crops, of set-aside and of non-food crops is expanded. While bull fattening is substantially reduced in both countries, suckler cow production is extended in France due to partial decoupling, but reduced in Germany due to full decoupling. Sectoral income effects measured in Farm Net Value Added are similar. The regional implementation of decoupling in Germany induces a significant redistribution of direct payments and therefore causes differences in income effects depending on farm type, location and size.CAP Reform, decoupling, farm group model, FADN, Agricultural and Food Policy, Land Economics/Use,

Simulations of black-hole binaries with unequal masses or non-precessing spins: accuracy, physical properties, and comparison with post-Newtonian results

We present gravitational waveforms for the last orbits and merger of black-hole-binary (BBH) systems along two branches of the BBH parameter space: equal-mass binaries with equal non-precessing spins, and nonspinning unequal-mass binaries. The waveforms are calculated from numerical solutions of Einstein's equations for black-hole binaries that complete between six and ten orbits before merger. Along the equal-mass spinning branch, the spin parameter of each BH is $\chi_i = S_i/M_i^2 \in [-0.85,0.85]$, and along the unequal-mass branch the mass ratio is $q =M_2/M_1 \in [1,4]$. We discuss the construction of low-eccentricity puncture initial data for these cases, the properties of the final merged BH, and compare the last 8-10 GW cycles up to $M\omega = 0.1$ with the phase and amplitude predicted by standard post-Newtonian (PN) approximants. As in previous studies, we find that the phase from the 3.5PN TaylorT4 approximant is most accurate for nonspinning binaries. For equal-mass spinning binaries the 3.5PN TaylorT1 approximant (including spin terms up to only 2.5PN order) gives the most robust performance, but it is possible to treat TaylorT4 in such a way that it gives the best accuracy for spins $\chi_i > -0.75$. When high-order amplitude corrections are included, the PN amplitude of the $(\ell=2,m=\pm2)$ modes is larger than the NR amplitude by between 2-4%.Comment: 21 pages, 9 figures, 6 tables. Version accepted by PR