An index theory for -actions

This paper concerns an index theory for -actions induced by a homeomorphism of a compact space. We give a definition of a genus for uniform spaces and prove that the genus for compact spaces is an index. To this end we show a -version of the Borsuk-Ulam theorem and the existence of a continuous equivariant extension for these -actions.

     

Group Actions on the Cubic Tree

It is known that every group which acts transitively on the ordered edges of the cubic tree ₃, with finite vertex stabilizer, is isomorphic to one of seven finitely presented subgroups of the full automorphism group of ₃–one of which is the modular group. In this paper a complete answer is given for the question (raised by Djokovi and Miller) as to whether two such subgroups which intersect in the modular group generate their free product with the modular group amalgamated.     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.     

Meson-Meson Scattering on Anisotropic Lattices

Using the tadpole improved Wilson quark action on small, coarse, and anisotropic lattices, meson-meson scattering lengths are calculated within quenched approximation. The study covers pion-pion scattering in the I=2 channel and kaon-pion scattering in the I=3/2 channel. The results are extrapolated towards the chiral limit. Finite volume and finite lattice spacing errors are also analyzed and results in the infinite volume and continuum limit are obtained. Our results are compared with the results obtained using Roy equations, chiral perturbation theory, dispersion relations, and the experimental data. We also compare our results with other lattice results on the scattering lengths.     

On the Construction of Geometric Integrators in the RKMK Class

We consider the construction of geometric integrators in the class of RKMK methods. Any differential equation in the form of an infinitesimal generator on a homogeneous space is shown to be locally equivalent to a differential equation on the Lie algebra corresponding to the Lie group acting on the homogeneous space. This way we obtain a distinction between the coordinate-free phrasing of the differential equation and the local coordinates used. In this paper we study methods based on arbitrary local coordinates on the Lie group manifold. By choosing the coordinates to be canonical coordinates of the first kind we obtain the original method of Munthe-Kaas [16]. Methods similar to the RKMK method are developed based on the different coordinatizations of the Lie group manifold, given by the Cayley transform, diagonal Padé approximants of the exponential map, canonical coordinates of the second kind, etc. Some numerical experiments are also given.     

Flexibility of disagreement actions in negotiations

Negotiation games in which two players engage in disagreement games to determine their payoffs during the bargaining generally admit multiple perfect equilibrium outcomes, including inefficient ones. This paper shows that if the two players cannot change their disagreement actions as frequent as they bargain, then the set of perfect equilibrium payoffs shrinks. As disagreement actions become completely inflexible relative to the bargaining frequency, the set of equilibrium payoffs shrinks to Nash's (1953) bargaining solution with strategic threat. Received January 1999/Final version March 2001     

Type III1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers

We complete the analysis of KMS-states of the Toeplitz algebra T(N?N^×) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1?β?2, the unique KMSβ-state is of type III₁. We prove this by reducing the type classification from T(N?N^×) to that of the symmetric part of the Bost-Connes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of N?N^× in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of N?N^× on the Nica spectrum, we can also recover all the KMS-states of T(N?N^×) originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMSβ-states for β>2 that does not ostensibly come from an action of T on the C^?-algebra.     

Finding optimal memoryless policies of POMDPs under the expected average reward criterion

In this paper, partially observable Markov decision processes (POMDPs) with discrete state and action space under the average reward criterion are considered from a recent-developed sensitivity point of view. By analyzing the average-reward performance difference formula, we propose a policy iteration algorithm with step sizes to obtain an optimal or local optimal memoryless policy. This algorithm improves the policy along the same direction as the policy iteration does and suitable step sizes guarantee the convergence of the algorithm. Moreover, the algorithm can be used in Markov decision processes (MDPs) with correlated actions. Two numerical examples are provided to illustrate the applicability of the algorithm.     

Modeling action-interdependence in multiple criteria decision making

A problem of subset selection when actions are interdependent is formulated within a multiple criteria framework. More specifically, a novel definition and characterization of interdependence of actions applicable to Multiple Criteria Decision Making (MCDM) are presented. The effects of interdependence of actions on the modeling and resolution of a subset choice problem are shown, and the importance of taking interdependence of actions into account is discussed. Most of the discussion is generalized to independence and interdependence of sets of actions, which are then compared to the case of individual actions. A general approach to evaluate a combination of interdependent actions is proposed and the use of the multiple criteria structure to eliminate some difficulties in evaluating a set of interdependent actions is explained.