On the zone of the boundary of a convex body

We consider an arrangement \A of $n$ hyperplanes in $\R^d$ and the zone $\Z$ in \A of the boundary of an arbitrary convex set in $\R^d$ in such an arrangement. We show that, whereas the combinatorial complexity of $\Z$ is known only to be $O$ \cite{APS}, the outer part of the zone has complexity $O$ (without the logarithmic factor). Whether this bound also holds for the complexity of the inner part of the zone is still an open question (even for $d=2$)

The Erd\H{o}s-Szekeres problem for non-crossing convex sets

We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and T\'{o}th on the Erd\H{o}s-Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk, on the Erd\H{o}s-Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erd\H{os}-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erd\H{o}s-Szekeres theorem of P\'{o}r and Valtr to arrangements of non-crossing convex bodies

Integration algorithms of elastoplasticity for ceramic powder compaction

Inelastic deformation of ceramic powders (and of a broad class of rock-like and granular materials), can be described with the yield function proposed by Bigoni and Piccolroaz (2004, Yield criteria for quasibrittle and frictional materials. Int. J. Solids and Structures, 41, 2855-2878). This yield function is not defined outside the yield locus, so that 'gradient-based' integration algorithms of elastoplasticity cannot be directly employed. Therefore, we propose two ad hoc algorithms: (i.) an explicit integration scheme based on a forward Euler technique with a 'centre-of-mass' return correction and (ii.) an implicit integration scheme based on a 'cutoff-substepping' return algorithm. Iso-error maps and comparisons of the results provided by the two algorithms with two exact solutions (the compaction of a ceramic powder against a rigid spherical cup and the expansion of a thick spherical shell made up of a green body), show that both the proposed algorithms perform correctly and accurately.Comment: 21 pages. Journal of the European Ceramic Society, 201

Approximating the Maximum Overlap of Polygons under Translation

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

A matrix stability analysis of the carbuncle phenomenon

The carbuncle phenomenon is a shock instability mechanism which ruins all efforts to compute grid-aligned shock waves using low-dissipative upwind schemes. The present study develops a stability analysis for two-dimensional steady shocks on structured meshes based on the matrix method. The numerical resolution of the corresponding eigenvalue problem confirms the typical odd–even form of the unstable mode and displays a Mach number threshold effect currently observed in computations. Furthermore, the present method indicates that the instability of steady shocks is not only governed by the upstream Mach number but also by the numerical shock structure. Finally, the source of the instability is localized in the upstream region, providing some clues to better understand and control the onset of the carbuncle

The George C. Davis Site, Cherokee County, Texas: Spring 1980 Archeological Investigations

The results of archeological investigations in portions of the George C. Davis Site are presented in this report prepared by Ross Fields and J. Peter Thurmond . Stringent contract requirements and the provisions of Texas Antiquities Permit No. 237 dictate that this report be brief and descriptive in nature. Requirements such as inclusion of a detailed site investigation history and environmental data were excluded and extensive excavations were stressed in preference to detailed analyses and comparisons. Field investigations were limited by contract to 20 working days and the analysis/write-up to 10 working days. The governing research design was prepared by the Texas Antiquities Committee staff to reflect agreements between the Committee and the Texas Forest Service. Sighificant findings include the general delineation of extensive, and in places intensive, Archaic and Late Prehistoric occupations. Of primary importance is the lack of an identifiable Alto Focus occupation within the northern portion of the Davis Site. This suggests that the small left bank tributary to Bowles Creek existed during Alto Focus times and served as a natural boundary for the village area during that period of site usage. There are hints that significant variations in the selection and use of lithic resources through time are identifiable. It is indeed ! unfortunate that time limitations precluded pursuit of this line of investigation. Ross and Pete have successfully accomplished a most challenging task under less than ideal conditions. They and their field crew are commended for their outstanding work at the Davis Site