44,291 research outputs found
Metric characterization of connectedness for topological spaces
Connectedness, path connectedness, and uniform connectedness are well-known
concepts. In the traditional presentation of these concepts there is a
substantial difference between connectedness and the other two notions, namely
connectedness is defined as the absence of disconnectedness, while path
connectedness and uniform connectedness are defined in terms of connecting
paths and connecting chains, respectively. In compact metric spaces uniform
connectedness and connectedness are well-known to coincide, thus the apparent
conceptual difference between the two notions disappears. Connectedness in
topological spaces can also be defined in terms of chains governed by open
coverings in a manner that is more reminiscent of path connectedness. We
present a unifying metric formalism for connectedness, which encompasses both
connectedness of topological spaces and uniform connectedness of uniform
spaces, and which further extends to a hierarchy of notions of connectedness
Some Remarks on the Model Theory of Epistemic Plausibility Models
Classical logics of knowledge and belief are usually interpreted on Kripke
models, for which a mathematically well-developed model theory is available.
However, such models are inadequate to capture dynamic phenomena. Therefore,
epistemic plausibility models have been introduced. Because these are much
richer structures than Kripke models, they do not straightforwardly inherit the
model-theoretical results of modal logic. Therefore, while epistemic
plausibility structures are well-suited for modeling purposes, an extensive
investigation of their model theory has been lacking so far. The aim of the
present paper is to fill exactly this gap, by initiating a systematic
exploration of the model theory of epistemic plausibility models. Like in
'ordinary' modal logic, the focus will be on the notion of bisimulation. We
define various notions of bisimulations (parametrized by a language L) and show
that L-bisimilarity implies L-equivalence. We prove a Hennesy-Milner type
result, and also two undefinability results. However, our main point is a
negative one, viz. that bisimulations cannot straightforwardly be generalized
to epistemic plausibility models if conditional belief is taken into account.
We present two ways of coping with this issue: (i) adding a modality to the
language, and (ii) putting extra constraints on the models. Finally, we make
some remarks about the interaction between bisimulation and dynamic model
changes.Comment: 19 pages, 3 figure
Under-connected and over-connected networks
Since the seminal contribution of Jackson & Wolinsky 1996 [A Strategic Model of Social and Economic Networks, JET 71, 44-74] it has been widely acknowledged that the formation of social networks exhibits a general conflict between individual strategic behavior and collective outcome. What has not been studied systematically are the sources of inefficiency. We approach this gap by analyzing the role of positive and negative externalities of link formation. We find general results that relate situations of positive externalities with stable networks that cannot be "too dense" in a well-defined sense, while situations with negative externalities, tend to induce "too dense" networks.networks, network formation, connections, game theory, externalities, spillovers, stability, efficiency
On asymptotically hereditarily aspherical groups
We undertake a systematic study of asymptotically hereditarily aspherical
(AHA) groups - the class of groups introduced by Tadeusz Januszkiewicz and the
second author as a tool for exhibiting exotic properties of systolic groups. We
provide many new examples of AHA groups, also in high dimensions. We relate AHA
property with the topology at infinity of a group, and deduce in this way some
new properties of (weakly) systolic groups. We also exhibit an interesting
property of boundary at infinity for few classes of AHA groups.Comment: 36 pages, minor modifications to v
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