The role of response mechanisms in determining reaction time performance: Piéron’s Law revisited

A response mechanism takes evaluations of the importance of potential actions and selects the most suitable. Response mechanism function is a nontrivial problem that has not received the attention it deserves within cognitive psychology. In this article, we make a case for the importance of considering response mechanism function as a constraint on cognitive processes and emphasized links with the wider problem of behavioral action selection. First, we show that, contrary to previous suggestions, a well–known model of the Stroop task (Cohen, Dunbar, & McClelland, 1990) relies on the response mechanism for a key feature of its results—the interference–facilitation asymmetry. Second, we examine a variety of response mechanisms (including that in the model of Cohen et al., 1990) and show that they all follow a law analogous to Piéron's law in relating their input to reaction time. In particular, this is true of a decision mechanism not designed to explain RT data but based on a proposed solution to the general problem of action selection and grounded in the neurobiology of the vertebrate basal ganglia. Finally, we show that the dynamics of simple artificial neurons also support a Piéron–like law

The Auslander-Gorenstein property for Z-algebras

We provide a framework for part of the homological theory of Z-algebras and their generalizations, directed towards analogues of the Auslander-Gorenstein condition and the associated double Ext spectral sequence that are useful for enveloping algebras of Lie algebras and related rings. As an application, we prove the equidimensionality of the characteristic variety of an irreducible representation of the Z-algebra, and for related representations over quantum symplectic resolutions. In the special case of Cherednik algebras of type A, this answers a question raised by the authors.Comment: 31 page

Noncommutative curves and noncommutative surfaces

In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.Comment: Suggestions by many people (in particular Haynes Miller and Dennis Keeler) have been incorporated. The formulation of some results has been improve

Differential operators and Cherednik algebras

We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction, used in [GS]; the other involving quantum hamiltonian reduction of an algebra of differential operators, used in [GG]. In the present paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result, [GS, Theorem 1.4] without recourse to Haiman's deep results on the n! theorem. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG].Comment: 37 p