432 research outputs found
A measure of non-convexity in the plane and the Minkowski sum
In this paper a measure of non-convexity for a simple polygonal region in the
plane is introduced. It is proved that for "not far from convex" regions this
measure does not decrease under the Minkowski sum operation, and guarantees
that the Minkowski sum has no "holes".Comment: 5 figures; Discrete and Computational Geometry, 201
L1-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups
In this paper, we prove interior Poincar\ue9 and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L1 norm. Unlike for Lp, p>1, the estimates are doomed to fail in top degree. The singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis in Euclidean spaces, and to Chanillo-Van Schaftingen in Heisenberg groups
ORLICZ SPACES AND ENDPOINT SOBOLEV-POINCAR \u301EINEQUALITIES FOR DIFFERENTIAL FORMS INHEISENBERG GROUPS
In this paper we prove Poincar \u301e and Sobolev inequalities for differ-ential forms in the Rumin\u2019s contact complex on Heisenberg groups. Inparticular, we deal with endpoint values of the exponents, obtaining fi-nally estimates akin to exponential Trudinger inequalities for scalar func-tion. These results complete previous results obtained by the authors awayfrom the exponential case. From the geometric point of view, Poincar \u301eand Sobolev inequalities for differential forms provide a quantitative for-mulation of the vanishing of the cohomology. They have also applicationsto regularity issues for partial differential equations
Modelling microbial exchanges between forms of soil nitrogen in contrasting ecosystems
Although nitrogen (N) is often combined with carbon (C) in organic
molecules, C passes from the air to the soil through plant photosynthesis,
whereas N passes from the soil to plants through a chain of microbial
conversions. However, dynamic models do not fully consider the
microorganisms at the centre of exchange processes between organic and
mineral forms of N. This study monitored the transfer of <sup>14</sup>C and
<sup>15</sup>N between plant materials, microorganisms, humified compartments, and
inorganic forms in six very different ecosystems along an altitudinal
transect. The microbial conversions of the <sup>15</sup>N forms appear to be
strongly linked to the previously modelled C cycle, and the same equations
and parameters can be used to model both C and N cycles. The only
difference is in the modelling of the flows between microbial and inorganic
forms. The processes of mineralization and immobilization of N appear to be
regulated by a two-way microbial exchange depending on the C : N ratios of
microorganisms and available substrates. The MOMOS (Modelling of Organic Matter of Soils) model has already been
validated for the C cycle and also appears to be valid for the prediction of
microbial transformations of N forms. This study shows that the hypothesis
of microbial homeostasis can give robust predictions at global scale.
However, the microbial populations did not appear to always be independent
of the external constraints. At some altitudes their C : N ratio could be
better modelled as decreasing during incubation and increasing with
increasing C storage in cold conditions. The ratio of potentially
mineralizable-<sup>15</sup>N/inorganic-<sup>15</sup>N and the <sup>15</sup>N stock in the
plant debris and the microorganisms was modelled as increasing with altitude,
whereas the <sup>15</sup>N storage in stable humus was modelled as decreasing with
altitude. This predicts that there is a risk that mineralization of organic
reserves in cold areas may increase global warming
Smectic blue phases: layered systems with high intrinsic curvature
We report on a construction for smectic blue phases, which have quasi-long
range smectic translational order as well as three dimensional crystalline
order. Our proposed structures fill space by adding layers on top of a minimal
surface, introducing either curvature or edge defects as necessary. We find
that for the right range of material parameters, the favorable saddle-splay
energy of these structures can stabilize them against uniform layered
structures. We also consider the nature of curvature frustration between mean
curvature and saddle-splay.Comment: 15 pages, 11 figure
The dynamics of thin vibrated granular layers
We describe a series of experiments and computer simulations on vibrated
granular media in a geometry chosen to eliminate gravitationally induced
settling. The system consists of a collection of identical spherical particles
on a horizontal plate vibrating vertically, with or without a confining lid.
Previously reported results are reviewed, including the observation of
homogeneous, disordered liquid-like states, an instability to a `collapse' of
motionless spheres on a perfect hexagonal lattice, and a fluctuating,
hexagonally ordered state. In the presence of a confining lid we see a variety
of solid phases at high densities and relatively high vibration amplitudes,
several of which are reported for the first time in this article. The phase
behavior of the system is closely related to that observed in confined
hard-sphere colloidal suspensions in equilibrium, but with modifications due to
the effects of the forcing and dissipation. We also review measurements of
velocity distributions, which range from Maxwellian to strongly non-Maxwellian
depending on the experimental parameter values. We describe measurements of
spatial velocity correlations that show a clear dependence on the mechanism of
energy injection. We also report new measurements of the velocity
autocorrelation function in the granular layer and show that increased
inelasticity leads to enhanced particle self-diffusion.Comment: 11 pages, 7 figure
Symbolic approach and induction in the Heisenberg group
We associate a homomorphism in the Heisenberg group to each hyperbolic
unimodular automorphism of the free group on two generators. We show that the
first return-time of some flows in "good" sections, are conjugate to
niltranslations, which have the property of being self-induced.Comment: 18 page
Buckling Instabilities of a Confined Colloid Crystal Layer
A model predicting the structure of repulsive, spherically symmetric,
monodisperse particles confined between two walls is presented. We study the
buckling transition of a single flat layer as the double layer state develops.
Experimental realizations of this model are suspensions of stabilized colloidal
particles squeezed between glass plates. By expanding the thermodynamic
potential about a flat state of confined colloidal particles, we derive
a free energy as a functional of in-plane and out-of-plane displacements. The
wavevectors of these first buckling instabilities correspond to three different
ordered structures. Landau theory predicts that the symmetry of these phases
allows for second order phase transitions. This possibility exists even in the
presence of gravity or plate asymmetry. These transitions lead to critical
behavior and phases with the symmetry of the three-state and four-state Potts
models, the X-Y model with 6-fold anisotropy, and the Heisenberg model with
cubic interactions. Experimental detection of these structures is discussed.Comment: 24 pages, 8 figures on request. EF508
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