TY - CHAP
AU - Bang-Jensen, Jørgen
AU - Reed, Bruce
AU - Schacht, Bruce
AU - Šámal, Robert
AU - Toft, Bjarne
AU - Uli Wagner
ID - 2416
T2 - Topics in Discrete Mathematics
TI - On six problems posed by Jarik Nešetřil
VL - 26
ER -
TY - JOUR
AB - We show, with an elementary proof, that the number of halving simplices in a set of n points in 4 in general position is O(n4-2/45). This improves the previous bound of O(n4-1/134). Our main new ingredient is a bound on the maximum number of halving simplices intersecting a fixed 2-plane.
AU - Matoušek, Jiří
AU - Sharir, Micha
AU - Smorodinsky, Shakhar
AU - Uli Wagner
ID - 2429
IS - 2
JF - Discrete & Computational Geometry
TI - K-sets in four dimensions
VL - 35
ER -
TY - JOUR
AB - We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every interval I there is a color that appears exactly once in I. We present deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω(√̃) colors in the worst case. We then derive two efficient variants of this simple algorithm. The first is deterministic and uses O(log 2 n) colors, and the second is randomized and uses O(log n) colors with high probability. We also show that the O(log 2 n) bound on the number of colors used by our deterministic algorithm is tight on the worst case. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order and present an incomplete analysis that indicates that, with high probability, it uses only O(log n) colors. Finally, we show that in the extension of this problem to two dimensions, where the relevant ranges are disks, n colors may be required in the worst case.
AU - Chent, Ke
AU - Fiat, Amos
AU - Kaplan, Haim
AU - Levy, Meital B
AU - Matoušek, Jiří
AU - Mossel, Elchanan
AU - Pach, János
AU - Sharir, Micha
AU - Smorodinsky, Shakhar
AU - Uli Wagner
AU - Welzl, Emo
ID - 2430
IS - 5
JF - SIAM Journal on Computing
TI - Online conflict-free coloring for intervals
VL - 36
ER -
TY - CONF
AB - We prove an upper bound, tight up to a factor of 2, for the number of vertices of level at most t in an arrangement of n halfspaces in R , for arbitrary n and d (in particular, the dimension d is not considered constant). This partially settles a conjecture of Eckhoff, Linhart, and Welzl. Up to the factor of 2, the result generalizes McMullen's Upper Bound Theorem for convex polytopes (the case ℓ = O) and extends a theorem of Linhart for the case d ≤ 4. Moreover, the bound sharpens asymptotic estimates obtained by Clarkson and Shor. The proof is based on the h-matrix of the arrangement (a generalization, introduced by Mulmuley, of the h-vector of a convex polytope). We show that bounding appropriate sums of entries of this matrix reduces to a lemma about quadrupels of sets with certain intersection properties, and we prove this lemma, up to a factor of 2, using tools from multilinear algebra. This extends an approach of Alon and Kalai, who used linear algebra methods for an alternative proof of the classical Upper Bound Theorem. The bounds for the entries of the h-matrix also imply bounds for the number of i-dimensional faces, i > 0, at level at most ℓ. Furthermore, we discuss a connection with crossing numbers of graphs that was one of the main motivations for investigating exact bounds that are valid for arbitrary dimensions.
AU - Uli Wagner
ID - 2431
TI - On a geometric generalization of the Upper Bound Theorem
ER -
TY - CONF
AB - Often the properties of a single cell are considered as representative for a complete polymer electrolyte fuel cell stack or even a fuel cell system. In some cases this comes close, however, in many real cases differences on several scales become important. Cell interaction phenomena in fuel cell stacks that arise from inequalities between adjacent cells are investigated in detail experimentally. For that, a specialized 2-cell stack with advanced localized diagnostics was developed. The results show that inequalities propagate by electrical coupling, inhomogeneous cell polarization and inducing in-plane current in the common bipolar plate. The effects of the different loss-mechanisms are analyzed and quantified.
AU - Büchi, Felix N.
AU - Freunberger, Stefan Alexander
AU - Santis, Marco
ID - 7326
IS - 1
T2 - ECS Transactions
TI - What is learned beyond the scale of single cells?
VL - 3
ER -