On f-vectors of Minkowski additions of convex polytopes

The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytopes.Comment: 13 pages, submitted to Discrete & Computational Geometr

Congestion in planar graphs with demands on faces

We give an algorithm to route a multicommodity flow in a planar graph $G$ with congestion $O(\log k)$, where $k$ is the maximum number of terminals on the boundary of a face, when each demand edge lie on a face of $G$. We also show that our specific method cannot achieve a substantially better congestion

When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks

Let $G=(V,E)$ be a supply graph and $H=(V,F)$ a demand graph defined on the same set of vertices. An assignment of capacities to the edges of $G$ and demands to the edges of $H$ is said to satisfy the \emph{cut condition} if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair $(G,H)$ is called \emph{cut-sufficient} if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on $G$. We prove a previous conjecture, which states that when the supply graph $G$ is series-parallel, the pair $(G,H)$ is cut-sufficient if and only if $(G,H)$ does not contain an \emph{odd spindle} as a minor; that is, if it is impossible to contract edges of $G$ and delete edges of $G$ and $H$ so that $G$ becomes the complete bipartite graph $K_{2,p}$, with $p\geq 3$ odd, and $H$ is composed of a cycle connecting the $p$ vertices of degree 2, and an edge connecting the two vertices of degree $p$. We further prove that if the instance is \emph{Eulerian} --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.Comment: An extended abstract of this paper will be published at the 44th Symposium on Theory of Computing (STOC 2012

The preparation of double-walled carbon nanotube/Cu composites by spark plasma sintering, and their hardness and friction properties

Double-walled carbon nanotube (DWCNT)/copper composite powders were prepared by a rapid route involving freeze-drying without oxidative acidic treatment or ball-milling. The DWCNTs are not damaged and are homogeneously dispersed in the matrix. Dense specimens were prepared by spark plasma sintering. The Vickers microhardness is doubled, the wear against a steel or an alumina ball seems very low and the average friction coefficient is decreased by a factor of about 4 compared to pure copper. The best results are obtained for a carbon loading (5 vol%) significantly lower than those reported when using multi-walled carbon nanotubes (10–20 vol%). Maximum Hertzian contact pressure data could indicate that the surface DWCNTs and bundles of them are deformed and broken, possibly resulting in the formation of a graphitized lubricating tribofilm in the contac

f-Vectors of Minkowski Additions of Convex Polytopes

The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of the face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytope

Preparation-microstructure-property relationships in double-walled carbon nanotubes/alumina composites

Double-walled carbon nanotube/alumina composite powders with low carbon contents (2– 3 wt.%) are prepared using three different methods and densified by spark plasma sintering. The mechanical properties and electrical conductivity are investigated and correlated with the microstructure of the dense materials. Samples prepared by in situ synthesis of carbon nanotubes (CNTs) in impregnated submicronic alumina are highly homogeneous and present the higher electrical conductivity (2.2–3.5 Scm-1) but carbon films at grain boundaries induce a poor cohesion of the materials. Composites prepared by mixing using moderate sonication of as-prepared double-walled CNTs and lyophilisation, with little damage to the CNTs, have a fracture strength higher (+30%) and a fracture toughness similar (5.6 vs 5.4 MPa m1/2) to alumina with a similar submicronic grain size. This is correlated with crack-bridging by CNTs on a large scale, despite a lack of homogeneity of the CNT distribution

Toughening and hardening in double-walled carbon nanotube/nanostructured magnesia composites

Dense double-walled carbon nanotube (DWCNT)/nanostructured MgO composites were prepared using an in situ route obviating any milling step for the synthesis of powders and consolidation by spark-plasma-sintering. An unambiguous increase in both toughness and microhardness is reported. The mechanisms of crack-bridging on an unprecedented scale, crack-deflection and DWCNT pullout have been evidenced. The very long DWCNTs, which appear to be mostly undamaged, are very homogeneously dispersed at the grain boundaries of the matrix, greatly inhibiting the grain growth during sintering. These results arise because the unique microstructure (low content of long DWCNTs, nanometric matrix grains and grain boundary cohesion) provides the appropriate scale of the reinforcement to make the material tough