Labelled seeds and the mutation group

We study the set S of labelled seeds of a cluster algebra of rank n inside a field F as a homogeneous space for the group M_n of (globally defined) mutations and relabellings. Regular equivalence relations on S are associated to subgroups W of Aut_{M_n}(S), and we thus obtain groupoids W \ S. We show that for two natural choices of equivalence relation, the corresponding groups W^c and W^+ act on F, and the groupoids W^c \ S and W^+ \ S on the model field K=Q(x_1,...,x_n). The groupoid W^+ \ S is equivalent to Fock-Goncharov's cluster modular groupoid. Moreover, W^c is isomorphic to the group of cluster automorphisms, and W^+ to the subgroup of direct cluster automorphisms, in the sense of Assem-Schiffler-Shramchenko. We also prove that, for mutation classes whose seeds have mutation finite quivers, the stabilizer of a labelled seed under M_n determines the quiver of the seed up to 'similarity', meaning up to taking opposites of some of the connected components. Consequently, the subgroup W^c is the entire automorphism group of S in these cases.Comment: 29 pages. v4; accepted manuscript, to appear, subject to editorial input, in the Mathematical Proceedings of the Cambridge Philosophical Societ

Has Salary Discrimination Really Disappeared From Major League Baseball?

Analysis of a detailed data set for the 2000-2001 period shows no evidence of overall racial or ethnic salary discrimination for baseball players. Given prior research, that finding is not unusual. However, when the data set is divided into low, middle, and high salary ranges, a RESST test shows that minorities in the lowest salary group receive significantly lower returns to their skills than do whites. A decomposition of the wage differences for the lowest salary group shows that as much as 86.3% of the black/white and 91.5% of the Hispanic/white salary gap may be due to discrimination.

Reclaiming the differences : Three neglected marxian theories of fascism in Lukács, Marcuse, and Bloch

No abstract available

An investigation into the potential benefit/value of Ipswich CBD residential revitalisation

CBD revitalisation strategies have been successfully implemented on a worldwide scale for many decades. Common CBD revitalisation techniques often include, maintaining high density levels, emphasizing the importance of historic preservation, being sure not to suburbanise the CBD by creating strict design control measures, and highlighting the importance of street activity by creating a pedestrian friendly urban environment (Ferguson, 2004). However, revitalisation projects are often undertaken by public developers or given to private developers with hefty government funded incentives. Government funded development generally occurs only in public domains, for example the Icon project in Ipswich CBD. This raises the question: 'Can residential revitalisation in decaying city centres be just as beneficial/valuable as large government funded revitalisation projects?' 'Will small scale residential revitalisation be an effective strategy in contributing to CBD sustainability?' The aim of this dissertation is to investigate the potential benefit/value of a revitalisation to residential properties in Ipswich CBD. By undertaking an Ipswich community survey, public/private developer survey and grading appraisal of 10 potential development sites, the research hopes to uncover the potential for residential development in Ipswich CBD

Socrates\u27s Great Escape: Philosophy and Politics in the Crito

Many contemporary students of Plato hold that the arguments Socrates gives the personified Laws in the Crito do not represent Socrates’s own views, but rather work on assumptions to which Crito adheres, but Socrates does not. But if the Laws’ arguments are not Socrates’s own, then we seem to be left with a bewildering problem: why would Plato provide us with arguments that Socrates does not believe in, for a conclusion which Socrates evidently does believe in? After all, Socrates does remain in prison to face his execution; evidently, he believes that that is what he ought to do. This problem has been posed by Leo Strauss, and the solution Strauss proposes has gained some currency, namely, that Crito is not sufficiently sophisticated to understand why Socrates really thinks he should accept his execution. But still, even if that explains why Socrates can’t tell Crito what he really thinks, why doesn’t Plato have his Socrates tell us what he really thinks? And just what does Plato’s Socrates really think, anyway? I propose that Plato does tell us what his Socrates really thinks, but implicitly, through what is shown up in certain inconsistencies between what the Laws argue and certain elements elsewhere in the Crito and other dialogues. In this paper I discuss four such inconsistencies, each of which is both a reason not to believe that the Laws’ arguments represent Socrates’s own view and an indication of what Socrates’s own view actually is