2,039 research outputs found
Planar Manhattan Local Minimal and Critical Networks
AbstractWe investigate extremal networks of the Manhattan length functional. We give a criterion of that a local minimal network in the sense of the Manhattan length is a critical network with respect to the Manhattan length functional. It turns out that the critical condition is non-local
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
represent hyperconvex metric spaces that have the same topology (although not
necessarily the same distance function) as subsets of the plane. We show how to
use this data structure to construct the tight span of a metric space given as
an n x n distance matrix, when the tight span is homeomorphic to a subset of
the plane, in time O(n^2), and to add a single point to a planar tight span in
time O(n). As an application of this construction, we show how to test whether
a given finite metric space embeds isometrically into the Manhattan plane in
time O(n^2), and add a single point to the space and re-test whether it has
such an embedding in time O(n).Comment: 39 pages, 15 figure
Random Walks Along the Streets and Canals in Compact Cities: Spectral analysis, Dynamical Modularity, Information, and Statistical Mechanics
Different models of random walks on the dual graphs of compact urban
structures are considered. Analysis of access times between streets helps to
detect the city modularity. The statistical mechanics approach to the ensembles
of lazy random walkers is developed. The complexity of city modularity can be
measured by an information-like parameter which plays the role of an individual
fingerprint of {\it Genius loci}.
Global structural properties of a city can be characterized by the
thermodynamical parameters calculated in the random walks problem.Comment: 44 pages, 22 figures, 2 table
Ghetto of Venice: Access to the Target Node and the Random Target Access Time
Random walks defined on undirected graphs assign the absolute scores to all
nodes based on the quality of path they provide for random walkers. In city
space syntax, the notion of segregation acquires a statistical interpretation
with respect to random walks. We analyze the spatial network of Venetian canals
and detect its most segregated part which can be identified with canals
adjacent to the Ghetto of Venice.Comment: 14 pages, 3 figure
Variational approximation of functionals defined on 1-dimensional connected sets: the planar case
In this paper we consider variational problems involving 1-dimensional
connected sets in the Euclidean plane, such as the classical Steiner tree
problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal
partition problems and provide a variational approximation through
Modica-Mortola type energies proving a -convergence result. We also
introduce a suitable convex relaxation and develop the corresponding numerical
implementations. The proposed methods are quite general and the results we
obtain can be extended to -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
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