Braided racks, Hurwitz actions and Nichols algebras with many cubic relations

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.Comment: v2: 35 pages, 6 tables, 14 figure

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Sampling-based Algorithms for Optimal Motion Planning

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.Comment: 76 pages, 26 figures, to appear in International Journal of Robotics Researc

Noise Sensitivity in Continuum Percolation

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c \ne 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p_c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.Comment: 42 page

Counting regions with bounded surface area

Define a cubical complex to be a collection of integer-aligned unit cubes in d dimensions. Lebowitz and Mazel (1998) proved that there are between (C_1d)n/2d and (C_2d)64n/d complexes containing a fixed cube with connected boundary of (d - 1)-volume n. In this paper we narrow these bounds to between (C_3d)n/d and (C_4d)2n/d . We also show that there are n{n/(2d(d-1))+o(1) connected complexes containing a fixed cube with (not necessarily connected) boundary of volume n. © Springer-Verlag 2007

Vertex distinguishing colorings of graphs with Δ(G) = 2

In a paper by Burris and Schelp (J. Graph Theory 26 (2) (1997) 70), a conjecture was made concerning the number of colors χ′s(G) required to proper edge-color G so that each vertex has a distinct set of colors incident to it. We consider the case when Δ(G) = 2, so that G is a union of paths and cycles. In particular we find the exact values of χ′(G) and hence verify the conjecture when G consists of just paths or just cycles. We also give good bounds on χ′s(G) when G contains both paths and cycles. © 2002 by the American College of Cardiology