795 research outputs found
Le Chatelier principle in replicator dynamics
The Le Chatelier principle states that physical equilibria are not only
stable, but they also resist external perturbations via short-time
negative-feedback mechanisms: a perturbation induces processes tending to
diminish its results. The principle has deep roots, e.g., in thermodynamics it
is closely related to the second law and the positivity of the entropy
production. Here we study the applicability of the Le Chatelier principle to
evolutionary game theory, i.e., to perturbations of a Nash equilibrium within
the replicator dynamics. We show that the principle can be reformulated as a
majorization relation. This defines a stability notion that generalizes the
concept of evolutionary stability. We determine criteria for a Nash equilibrium
to satisfy the Le Chatelier principle and relate them to mutualistic
interactions (game-theoretical anticoordination) showing in which sense
mutualistic replicators can be more stable than (say) competing ones. There are
globally stable Nash equilibria, where the Le Chatelier principle is violated
even locally: in contrast to the thermodynamic equilibrium a Nash equilibrium
can amplify small perturbations, though both this type of equilibria satisfy
the detailed balance condition.Comment: 12 pages, 3 figure
The baker's map with a convex hole
We consider the baker's map on the unit square and an open convex set
which we regard as a hole. The survivor set is
defined as the set of all points in whose -trajectories are disjoint
from . The main purpose of this paper is to study holes for which
(dimension traps) as well as those for which any
periodic trajectory of intersects (cycle traps).
We show that any which lies in the interior of is not a dimension
trap. This means that, unlike the doubling map and other one-dimensional
examples, we can have for whose Lebesgue measure
is arbitrarily close to one. Also, we describe holes which are dimension or
cycle traps, critical in the sense that if we consider a strictly convex
subset, then the corresponding property in question no longer holds.
We also determine such that for all
convex whose Lebesgue measure is less than .
This paper may be seen as a first extension of our work begun in [3, 4, 6, 7,
13] to higher dimensions.Comment: 31 pages, 10 figure
Interference, Cooperation and Connectivity - A Degrees of Freedom Perspective
We explore the interplay between interference, cooperation and connectivity
in heterogeneous wireless interference networks. Specifically, we consider a
4-user locally-connected interference network with pairwise clustered decoding
and show that its degrees of freedom (DoF) are bounded above by 12/5.
Interestingly, when compared to the corresponding fully connected setting which
is known to have 8/3 DoF, the locally connected network is only missing
interference-carrying links, but still has lower DoF, i.e., eliminating these
interference-carrying links reduces the DoF. The 12/5 DoF outer bound is
obtained through a novel approach that translates insights from interference
alignment over linear vector spaces into corresponding sub-modularity
relationships between entropy functions.Comment: Submitted to 2011 IEEE International Symposium on Information Theory
(ISIT
Entropy of Operator-valued Random Variables: A Variational Principle for Large N Matrix Models
We show that, in 't Hooft's large N limit, matrix models can be formulated as
a classical theory whose equations of motion are the factorized
Schwinger--Dyson equations. We discover an action principle for this classical
theory. This action contains a universal term describing the entropy of the
non-commutative probability distributions. We show that this entropy is a
nontrivial 1-cocycle of the non-commutative analogue of the diffeomorphism
group and derive an explicit formula for it. The action principle allows us to
solve matrix models using novel variational approximation methods; in the
simple cases where comparisons with other methods are possible, we get
reasonable agreement.Comment: 45 pages with 1 figure, added reference
Heteroclinic Chaos, Chaotic Itinerancy and Neutral Attractors in Symmetrical Replicator Equations with Mutations
A replicator equation with mutation processes is numerically studied.
Without any mutations, two characteristics of the replicator dynamics are
known: an exponential divergence of the dominance period, and hierarchical
orderings of the attractors. A mutation introduces some new aspects: the
emergence of structurally stable attractors, and chaotic itinerant behavior. In
addition, it is reported that a neutral attractor can exist in the mutataion
rate -> +0 region.Comment: 4 pages, 9 figure
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