Topologies on Types: Connections

For different purposes, economists may use different topologies on types. We char- acterize the relationship among these various topologies. First, we show that for any general types, convergence in the uniform-weak topology implies convergence in both the strategic topology and the uniform strategic topology. Second, we explicitly con- struct a type which is not the limit of any …finite types under the uniform strategic topology, showing that the uniform strategic topology is strictly fi…ner than the strategic topology. With these results, we can linearly rank various topologies on the universal type space, which gives a clear picture of the relationship between the implication of types for beliefs and their implication for behaviors.the universal type space, the strategic topology; the uniform strategic topology; the uniform-weak topology; interim correlated rationalizable actions

Casimir forces for inhomogeneous planar media

Casimir forces arise from vacuum uctuations. They are fully understood only for simple models, and are important in nano- and microtechnologies. We report our experience of computer algebra calculations towards the Casimir force for models involving inhomogeneous dielectrics. We describe a methodology that greatly increases condence in any results obtained, and use this methodology to demonstrate that the analytic derivation of scalar Green's functions is at the boundary of current computer algebra technology. We further demonstrate that Lifshitz theory of electromagnetic vacuum energy can not be directly applied to calculate the Casimir stress for models of this type, and produce results that have led to alternative regularisations. Using a combination of our new computational framework and the new theory based on our results, we provide specic calculations of Casimir forces for planar dielectrics having permittivity that declines exponentially. We discuss the relative strengths and weaknesses of computer algebra systems when applied to this type of problem, and describe a combined numerical and symbolic computational framework for calculating Casimir forces for arbitrary planar models.Publisher PD

Magnetic Coupling of a Rotating Black Hole with Advection-Dominated Accretion Flows

A model of magnetic coupling (MC) of a rotating black hole (BH) with advection- dominated accretion flow (MCADAF) is proposed. It turns out that MCADAF providers a natural explanation for the transition radius between ADAF and SSD, and could be used to interpret the highest luminosity of GX 339-4 in hard-state. A very steep emissivity index can be produced in the innermost part of the MCADAF,which is consistent with the recent XMM-Newton observations of the nearby bright Seyfert 1 galaxy MCG-6-30-15 and with two X-ray binaries (XRBs): XTE J1655-500 and GX 339-4. In addition, we estimate the BH spins in Seyfert 1 galaxy MCG-6-30-15 and in the two XRBs based on this model.Comment: 18 pages, 9 figures,acceptted by New Astronom

Uniform Topologies on Types

We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, called uniform weak topology, under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. This topology generalizes the now classic notion of proximity to common knowledge based on common p-beliefs (Monderer and Samet (1989)). We show that convergence in the uniform weak topology implies convergence in the uniform strategic topology (Dekel, Fudenberg, and Morris (2006)). Moreover, when the limit is a finite type, uniform-weak convergence is also a necessary condition for convergence in the strategic topology. Finally, we show that the set of finite types is nowhere dense under the uniform strategic topology. Thus, our results shed light on the connection between similarity of beliefs and similarity of behaviors in games.Rationalizability, Incomplete information, Higher-order beliefs, Strategic topology, Electronic mail game