Dilation-free graphs in the l1 metric

The dilation-free graph of a planar point set S is a graph that spans S in such a way that the distance between two points in the graph is no longer than their planar distance. Metrically speaking, those graphs are equivalent to complete graphs; however they have far fewer edges when considering the Manhattan distance (we give here an upper bound on the number of saved edges). This article provides several theoretical, algorithmic, and complexity features of dilation-free graphs in the l1-metric, giving several construction algorithms and proving some of their properties. Moreover, special attention is paid to the planar case due to its applications in the design of printed circuit boards

Cheeger's differentiation theorem via the multilinear Kakeya inequality

Suppose that $(X,d,\mu)$ is a metric measure space of finite Hausdorff dimension and that, for every Lipschitz $f \colon X \to \mathbb R$, $\operatorname{Lip}(f,\cdot)$ is dominated by every upper gradient of $f$. We show that $X$ is a Lipschitz differentiability space, and the differentiable structure of $X$ has dimension at most $\dim_{\mathrm{H}} X$. Since our assumptions are satisfied whenever $X$ is doubling and satisfies a Poincar\'e inequality, we thus obtain a new proof of Cheeger's generalisation of Rademacher's theorem. Our approach uses Guth's multilinear Kakeya inequality for neighbourhoods of Lipschitz graphs to show that any non-trivial measure with $n$ independent Alberti representations has Hausdorff dimension at least $n$.Comment: 14 page

Quantum graphs as holonomic constraints

We consider the dynamics on a quantum graph as the limit of the dynamics generated by a one-particle Hamiltonian in R^2 with a potential having a deep strict minimum on the graph, when the width of the well shrinks to zero. For a generic graph we prove convergence outside the vertices to the free dynamics on the edges. For a simple model of a graph with two edges and one vertex, we prove convergence of the dynamics to the one generated by the Laplacian with Dirichlet boundary conditions in the vertex.Comment: 28 pages, 3 figure

Finitely Correlated Representations of Product Systems of C∗C^*-Correspondences over Nk\mathbb{N}^k

We study isometric representations of product systems of correspondences over the semigroup $\mathbb{N}^k$ which are minimal dilations of finite dimensional, fully coisometric representations. We show the existence of a unique minimal cyclic coinvariant subspace for all such representations. The compression of the representation to this subspace is shown to be complete unitary invariant. For a certain class of graph algebras the nonself-adjoint \textsc{wot}-closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace. This class includes free semigroup algebras. This result extends to a class of higher-rank graph algebras which includes higher-rank graphs with a single vertex.Comment: 34 pages; Introduction extended; to appear in the Journal of Functional Analysi

09451 Abstracts Collection -- Geometric Networks, Metric Space Embeddings and Spatial Data Mining

From November 1 to 6, 2009, the Dagstuhl Seminar 09451 ``Geometric Networks, Metric Space Embeddings and Spatial Data Mining\u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available