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Briane M.,INSA Rennes | Capdeboscq Y.,Mathematical Institute | Nguyen L.,Fine Hall
Archive for Rational Mechanics and Analysis | Year: 2013

In this paper, uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by an ε-periodic lattice of highly conducting thin rods. The estimates are derived only at a distance ε1+τ (for some τ > 0) away from the fibres. This distance constraint is rather sharp since the gradients of the solutions are shown to be unbounded locally in Lp as soon as p > 2. One key ingredient is the derivation in dimension two of regularity estimates to the solutions of the equations deduced from a Fourier series expansion with respect to the fibres' direction, and weighted by the high-contrast conductivity. The dependence on powers of ε of these two-dimensional estimates is shown to be sharp. The initial motivation for this work comes from imaging, and enhanced resolution phenomena observed experimentally in the presence of micro-structures (Lerosey et al., Science 315:1120-1124, 2007). We use these regularity estimates to characterize the signature of low volume fraction heterogeneities in the fibred reinforced medium, assuming that the heterogeneities stay at a distance ε1+τ away from the fibres. © 2012 Springer-Verlag.

Fefferman C.L.,Fine Hall | Rodrigo J.L.,University of Warwick
Archive for Rational Mechanics and Analysis | Year: 2015

In this paper we construct families of real analytic solutions of the Surface Quasi-Geostrophic equation (SQG) that are locally constant outside a thin neighborhood of a curve of arbitrarily small thickness. Despite the fact that only local existence results are known for SQG, and that our initial conditions have a arbitrarily large gradient we show that solutions exist for a time independent of the thickness of the neighborhood. © 2015, Springer-Verlag Berlin Heidelberg.

Dvorak Z.,Charles University | Norine S.,Fine Hall
Journal of Combinatorial Theory. Series B | Year: 2010

A class of simple undirected graphs is small if it contains at most n ! αn labeled graphs with n vertices, for some constant α. We prove that for any constants c, ε > 0, the class of graphs with expansion bounded by the function f (r) = cr1 / 3 - ε is small. Also, we show that the class of graphs with expansion bounded by 6 ṡ 3sqrt(r log (r + e)) is not small. © 2009 Elsevier Inc. All rights reserved.

Bukh B.,Fine Hall | Matousek J.,Charles University | Nivasch G.,Tel Aviv University
Discrete and Computational Geometry | Year: 2010

The following result was proved by Bárány in 1982: For every d≥1, there exists c d>0 such that for every n-point set S in ℝ d, there is a point p∈ℝ d contained in at least c dn d+1-O(n d) of the d-dimensional simplices spanned by S. We investigate the largest possible value of c d. It was known that c d≤1/(2 d(d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c d≤(d+1) -(d+1), and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, is c d≥γ d:=(d 2+1)/((d+1)!(d+1) d+1); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than γ dn d+1+O(n d) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved. We also prove that for every n-point set S⊂ℝ d, there exists a (d-2)-flat that stabs at least c d,d-2n 3-O(n 2) of the triangles spanned by S, with c d,d-2≥1/24(1-1/(2d-1) 2). To this end, we establish an equipartition result of independent interest (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in ℝ d can be divided into 4d-2 equal parts by 2d-1 hyperplanes intersecting in a common (d-2)-flat. © 2008 Springer Science+Business Media, LLC.

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