Multifraction reduction III: The case of interval monoids

We investigate gcd-monoids, which are cancellative monoids in which any two elements admit a left and a right gcd, and the associated reduction of multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the word problem for the enveloping group. Here we consider the particular case of interval monoids associated with finite posets. In this way, we construct gcd-monoids, in which reduction of multifractions has prescribed properties not yet known to be compatible: semi-convergence of reduction without convergence, semi-convergence up to some level but not beyond, non-embeddability into the enveloping group (a strong negation of semi-convergence).Comment: 23 pages ; v2 : cross-references updated ; v3 : one example added, typos corrected; final version due to appear in Journal of Combinatorial Algebr

Modelling the CAP Reform at the Regional Level with ProLand

The reform of the European Common Agricultural Policy (CAP) will fundamentally affect the decision behaviour of land users. So far transfer payments were coupled to specific forms of land use. The reform encourages land users to make decisions concerning production based solely on market aspects. The effects of the CAP reform on the Lahn Dill region in Germany are simulated with the spatially explicit land use model ProLand. The results show that land use decisions will be based stronger on site specific natural conditions than was the case in the Agenda 2000 scenario. The transfer payment volume directed into the region increases considerably.modelling, decision support, land use, spatially explicit, Agricultural and Food Policy, Q01,

A critical analysis of Zimbabwe’s codified business judgment rule and its place in the corporate governance landscape

The business judgment rule (BJR or the Rule) is an American legal export which has become a key corporate governance tool in most leading common law jurisdictions, such as, Australia, Canada and South Africa. However, the Rule has not been formally embraced in the United Kingdom. In Zimbabwe, the Rule has traditionally been treated as a common law feature. However, section 54 of Zimbabwe’s new Companies and Other Business Entities Act represents one of the significant advances in strengthening the jurisdiction’s corporate governance principles by codifying the Rule. The BJR originated together with the directors’ duty of care and skill. There are two main formulations of the BJR. The first one is by the Delaware Chancery Court and the second one derives from the American Law Institute’s Principles of Corporate Governance. The Rule mostly applies in determining the procedural aspects of the directors’ decision or the decision-making process and only in exceptional cases is it invoked to review the merits of their decision. This article seeks to critically analyse the major elements of Zimbabwe’s codified BJR and to ascertain its place in the corporate governance framework. As will become clear, it will also be argued that the statutory BJR is intended for the enhancement of directorial accountability

A Typology to Explore the Mitigation of Shortcut Behavior

As machine learning models become increasingly larger, trained weakly supervised on large, possibly uncurated data sets, it becomes increasingly important to establish mechanisms for inspecting, interacting, and revising models to mitigate learning shortcuts and guarantee their learned knowledge is aligned with human knowledge. The recently proposed XIL framework was developed for this purpose, and several such methods have been introduced, each with individual motivations and methodological details. In this work, we provide a unification of various XIL methods into a single typology by establishing a common set of basic modules. In doing so, we pave the way for a principled comparison of existing, but, importantly, also future XIL approaches. In addition, we discuss existing and introduce novel measures and benchmarks for evaluating the overall abilities of a XIL method. Given this extensive toolbox, including our typology, measures, and benchmarks, we finally compare several recent XIL methods methodologically and quantitatively. In our evaluations, all methods prove to revise a model successfully. However, we found remarkable differences in individual benchmark tasks, revealing valuable application-relevant aspects for integrating these benchmarks in developing future methods

Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms

Given a unimodular real spherical space $Z=G/H$ we construct for each boundary degeneration $Z_I=G/H_I$ of $Z$ a Bernstein morphism $B_I: L^2(Z_I)_{\rm disc }\to L^2(Z)$. We show that $B:=\bigoplus_I B_I$ provides an isospectral $G$-equivariant morphism onto $L^2(Z)$. Further, the maps $B_I$ are finite linear combinations of orthogonal projections which translates in the known cases where $Z$ is a group or a symmetric space into the familiar Maass-Selberg relations. As a corollary we obtain that $L^2(Z)_{\rm disc }\neq \emptyset$ provided that ${\mathfrak h}^\perp$ contains elliptic elements in its interior.Comment: Extended revised version with 4 additional sections. In particular, we derive the Plancherel Theorem of Harish-Chandra (Section 14) for a real reductive group as well as the Plancherel formula of Delorme and van den Ban-Schlichtkrull for symmetric spaces (Section 15). Further additional Section 13 on elliptic elements and moment maps. 98 pages. To be submitte