156,434 research outputs found
Extending Whitney's extension theorem: nonlinear function spaces
We consider a global, nonlinear version of the Whitney extension problem for
manifold-valued smooth functions on closed domains , with non-smooth
boundary, in possibly non-compact manifolds. Assuming is a submanifold with
corners, or is compact and locally convex with rough boundary, we prove that
the restriction map from everywhere-defined functions is a submersion of
locally convex manifolds and so admits local linear splittings on charts. This
is achieved by considering the corresponding restriction map for locally convex
spaces of compactly-supported sections of vector bundles, allowing the even
more general case where only has mild restrictions on inward and outward
cusps, and proving the existence of an extension operator.Comment: 37 pages, 1 colour figure. v2 small edits, correction to Definition
A.3, which makes no impact on proofs or results. Version submitted for
publication. v3 small changes in response to referee comments, title
extended. v4 crucial gap filled, results not affected. v5 final version to
appear in Annales de l'Institut Fourie
Stick-slip statistics for two fractal surfaces: A model for earthquakes
Following the observations of the self-similarity in various length scales in
the roughness of the fractured solid surfaces, we propose here a new model for
the earthquake. We demonstrate rigorously that the contact area distribution
between two fractal surfaces follows an unique power law. This is then utilised
to show that the elastic energy releases for slips between two rough fractal
surfaces indeed follow a Guttenberg-Richter like power law.Comment: 9 pages (Latex), 4 figures (postscript
The enhanced Sanov theorem and propagation of chaos
We establish a Sanov type large deviation principle for an ensemble of
interacting Brownian rough paths. As application a large deviations for the
(-layer, enhanced) empirical measure of weakly interacting diffusions is
obtained. This in turn implies a propagation of chaos result in rough path
spaces and allows for a robust subsequent analysis of the particle system and
its McKean-Vlasov type limit, as shown in two corollaries.Comment: 42 page
Multi-scale analysis of the roughness effect on lubricated rough contact
Determining friction is as equally essential as determining the film thickness in the lubricated contact, and is an important research subject. Indeed, reduction of friction in the automotive industry is important for both the minimization of fuel consumption as well as the decrease in the emissions of greenhouse gases. However, the progress in friction reduction has been limited by the difficulty in understanding the mechanism of roughness effects on friction. It was observed that micro-surface geometry or roughness was one of the major factors that affected the friction coefficient. In the present study, a new methodology coupling the multi-scale decomposition of the surface and the prediction of the friction coefficient by numerical simulation was developed to understand the influence of the scale of roughness in the friction coefficient. In particular, the real surface decomposed in different roughness scale by multi-scale decomposition, based on ridgelets transform was used as input into the model. This model predicts the effect of scale on mixed elastohydroynamic point contact. The results indicate a good influence of the fine scale of surface roughness on the friction coefficient for full-film lubrication as well as a beginning of improvement for mixed lubrication
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