1,170 research outputs found
From winning strategy to Nash equilibrium
Game theory is usually considered applied mathematics, but a few
game-theoretic results, such as Borel determinacy, were developed by
mathematicians for mathematics in a broad sense. These results usually state
determinacy, i.e. the existence of a winning strategy in games that involve two
players and two outcomes saying who wins. In a multi-outcome setting, the
notion of winning strategy is irrelevant yet usually replaced faithfully with
the notion of (pure) Nash equilibrium. This article shows that every
determinacy result over an arbitrary game structure, e.g. a tree, is
transferable into existence of multi-outcome (pure) Nash equilibrium over the
same game structure. The equilibrium-transfer theorem requires cardinal or
order-theoretic conditions on the strategy sets and the preferences,
respectively, whereas counter-examples show that every requirement is relevant,
albeit possibly improvable. When the outcomes are finitely many, the proof
provides an algorithm computing a Nash equilibrium without significant
complexity loss compared to the two-outcome case. As examples of application,
this article generalises Borel determinacy, positional determinacy of parity
games, and finite-memory determinacy of Muller games
Mechanical noise dependent Aging and Shear Banding behavior of a mesoscopic model of amorphous plasticity
We discuss aging and localization in a simple "Eshelby" mesoscopic model of
amorphous plasticity. Plastic deformation is assumed to occur through a series
of local reorganizations. Using a discretization of the mechanical fields on a
discrete lattice, local reorganizations are modeled as local slip events. Local
yield stresses are randomly distributed in space and invariant in time. Each
plastic slip event induces a long-ranged elastic stress redistribution.
Mimicking the effect of aging, we focus on the behavior of the model when the
initial state is characterized by a distribution of high local yield stress
values. A dramatic effect on the localization behavior is obtained: the system
first spontaneously self-traps to form a shear band which then only slowly
widens. The higher the "age" parameter the more localized the plastic strain
field. Two-time correlation computed on the stress field show a divergent
correlation time with the age parameter. The amplitude of a local slip event
(the prefactor of the Eshelby singularity) as compared to the yield stress
distribution width acts here as an effective temperature-like parameter: the
lower the slip increment, the higher the localization and the decorrelation
time
Quantitative prediction of effective toughness at random heterogeneous interfaces
The propagation of an adhesive crack through an anisotropic heterogeneous
interface is considered. Tuning the local toughness distribution function and
spatial correlation is numerically shown to induce a transition between weak to
strong pinning conditions. While the macroscopic effective toughness is given
by the mean local toughness in case of weak pinning, a systematic toughness
enhancement is observed for strong pinning (the critical point of the depinning
transition). A self-consistent approximation is shown to account very
accurately for this evolution, without any free parameter
Tracer Dispersion in Rough Open Cracks
Tracer dispersion is studied in an open crack where the two rough crack faces have been translated with respect to each other. The different dispersion regimes encountered in rough-wall Hele-Shaw cell are first introduced, and the geometric dispersion regime in the case of self-affine crack surfaces is treated in detail through perturbation analysis. It is shown that a line of tracer is progressively wrinkled into a self-affine curve with an exponent equal to that of the crack surface.This leads to a global dispersion coefficient which depends on the distance from the tracer inlet, but which is still proportional to the mean advection velocity. Besides, the tracer front is subjected to a local dispersion (as could be revealed by point measurements or echo experiments) very different from the global one. The expression of this anomalous local dispersion coefficient is also obtained
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