We prove a conjecture of Il'yashenko, that for a C 1 map in R n which locally contracts k-dimensional volumes, the box dimension of any compact invariant set is less than k. This result was proved independently by Douady and Oesterlé and by Il'yashenko for Hausdorff dimension. An upper bound on the box dimension of an attractor is valuable because, unlike a bound on the Hausdorff dimension, it implies an upper bound on the dimension needed to embed the attractor. We also get the same bound for the fractional part of the box dimension as is obtained by Douady and Oesterlé for Hausdorff dimension. This upper bound can be characterized in terms of a local version of the Lyapunov dimension defined by Kaplan and Yorke.
Key words Laser shock-wave, laser-driven foils, spallation, ultimate strength. PACS 41.75.Jv Direct laser interaction and laser-driven thin foils were used for investigation spallation phenomena in polymetyl metacrylate (PMMA) targets in case of high strain rate. The aluminium foils with thickness 8 and 15 mkm were used as impactors. Mass and velocity of the laser-driven foils after laser ablation and acceleration were determined by the method of foil deceleration in a gas atmosphere. Basing on experimental data, we determined the spallation plane position and the moment for the spall layer to arrive at an additional electrocontact sensor located beside the rear side of the target. Then the spall strength and strain rate have been calculated numerically using a hydrodynamic code based on a wide-range semi-empirical equation of state for PMMA. As a result of these experiments, we have for the first time shown that the ultimate spall strength of PMMA (about 10 kbar) is achieved in case of strain rate ranging 1.5×10 6 to 6×10 6
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.