295 research outputs found
Minkowski Tensors of Anisotropic Spatial Structure
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences
Ising Model on Random Triangulations of the Disk : Phase Transition
In Chen and Turunen (Commun Math Phys 374(3):1577-1643, 2020), we have studied the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the infinite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform infinite half-planar triangulation decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of finite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in Chen and Turunen (2020) to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a. Liouville quantum gravity).Peer reviewe
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Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
Proactive-reactive, robust scheduling and capacity planning of deconstruction projects under uncertainty
A project planning and decision support model is developed and applied to identify and reduce risk and uncertainty in deconstruction project planning. It allows calculating building inventories based on sensor information and construction standards and it computes robust project plans for different scenarios with multiple modes, constrained renewable resources and locations. A reactive and flexible planning element is proposed in the case of schedule infeasibility during project execution
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