9,574 research outputs found
Entropies from coarse-graining: convex polytopes vs. ellipsoids
We examine the Boltzmann/Gibbs/Shannon and the
non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \
\ and the Kaniadakis -entropy \ \
from the viewpoint of coarse-graining, symplectic capacities and convexity. We
argue that the functional form of such entropies can be ascribed to a
discordance in phase-space coarse-graining between two generally different
approaches: the Euclidean/Riemannian metric one that reflects independence and
picks cubes as the fundamental cells and the symplectic/canonical one that
picks spheres/ellipsoids for this role. Our discussion is motivated by and
confined to the behaviour of Hamiltonian systems of many degrees of freedom. We
see that Dvoretzky's theorem provides asymptotic estimates for the minimal
dimension beyond which these two approaches are close to each other. We state
and speculate about the role that dualities may play in this viewpoint.Comment: 63 pages. No figures. Standard LaTe
Parovicenko spaces with structures
We study an analogue of the Parovicenko property in categories of compact
spaces with additional structures. In particular, we present an internal
characterization of this property in the class of compact median spaces.Comment: Minor revision, final version, 19 page
Convexity of Momentum Maps: A Topological Analysis
The Local-to-Global-Principle used in the proof of convexity theorems for
momentum maps has been extracted as a statement of pure topology enriched with
a structure of convexity. We extend this principle to not necessarily closed
maps f\colon X\ra Y where the convexity structure of the target space
need not be based on a metric. Using a new factorization of , convexity of
the image is proved without local fiber connectedness, and for arbitrary
connected spaces .Comment: 21 pages LaTeX2e; minor revisions, to appear in Topology and its
Applications; Dedicated to Alan D. Weinstein, Dennis P. Sullivan, and in
memory of Jerrold E. Marsden. arXiv admin note: substantial text overlap with
arXiv:1009.252
Perfect Competition
In his 1987 entry on ‘Perfect Competition’ in The New Palgrave, the author reviewed the question of the perfectness of perfect competition, and gave four alternative formalisations rooted in the so-called Arrow-Debreu-Mckenzie model. That entry is now updated for the second edition to include work done on the subject during the last twenty years. A fresh assessment of this literature is offered, one that emphasises the independence assumption whereby individual agents are not related except through the price system. And it highlights a ‘linguistic turn’ whereby Hayek’s two fundamental papers on ‘division of knowledge’ are seen to have devastating consequences for this research programmeAllocation of Resources, Perfect Competition, Exchange Economy
Perfect Competition
In his 1987 entry on ‘Perfect Competition’ in The New Palgrave, the author reviewed the question of the perfectness of perfect competition, and gave four alternative formalisations rooted in the so-called Arrow-Debreu-Mckenzie model. That entry is now updated for the second edition to include work done on the subject during the last twenty years. A fresh assessment of this literature is offered, one that emphasises the independence assumption whereby individual agents are not related except through the price system. And it highlights a ‘linguistic turn’ whereby Hayek’s two fundamental papers on ‘division of knowledge’ are seen to have devastating consequences for this research programme.Allocation of Resources; Perfect Competition; Exchange Economy
Computational universes
Suspicions that the world might be some sort of a machine or algorithm
existing ``in the mind'' of some symbolic number cruncher have lingered from
antiquity. Although popular at times, the most radical forms of this idea never
reached mainstream. Modern developments in physics and computer science have
lent support to the thesis, but empirical evidence is needed before it can
begin to replace our contemporary world view.Comment: Several corrections of typos and smaller revisions, final versio
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