3,955 research outputs found
Varying the direction of propagation in reaction-diffusion equations in periodic media
We consider a multidimensional reaction-diffusion equation of either ignition
or monostable type, involving periodic heterogeneity, and analyze the
dependence of the propagation phenomena on the direction. We prove that the
(minimal) speed of the underlying pulsating fronts depends continuously on the
direction of propagation, and so does its associated profile provided it is
unique up to time shifts. We also prove that the spreading properties
\cite{Wein02} are actually uniform with respect to the direction
Collet, Eckmann and the bifurcation measure
The moduli space of degree rational maps can
naturally be endowed with a measure detecting maximal
bifurcations, called the bifurcation measure. We prove that the support of the
bifurcation measure has positive Lebesgue measure. To do so,
we establish a general sufficient condition for the conjugacy class of a
rational map to belong to the support of and we exhibit a
large set of Collet-Eckmann rational maps which satisfy this condition. As a
consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue
measure which are approximated by hyperbolic rational maps
Criticality in the approach to failure in amorphous solids
Failure of amorphous solids is fundamental to various phenomena, including
landslides and earthquakes. Recent experiments indicate that highly plastic
regions form elongated structures that are especially apparent near the maximal
shear stress where failure occurs. This observation suggested
that acts as a critical point where the length scale of those
structures diverges, possibly causing macroscopic transient shear bands. Here
we argue instead that the entire solid phase () is
critical, that plasticity always involves system-spanning events, and that
their magnitude diverges at independently of the presence of
shear bands. We relate the statistics and fractal properties of these
rearrangements to an exponent that captures the stability of the
material, which is observed to vary continuously with stress, and we confirm
our predictions in elastoplastic models.Comment: 6 pages, 5 figure
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