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We study a particle system in which particles reproduce, move randomly in space, and compete with each other. We prove global survival as well as a shape theorem describing the asymptotic spread of the population, when the population density is sufficiently large. In contrast to most previous studies, we allow the competition kernel to have an arbitrary, or even infinite range, whence the term 'non-local competition'. This makes the particle system non-monotone and of infinite-range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands-on approach. Based on joint work with Pascal Maillard.
In our manufacturing plants, many tens of thousands of components for the automotive industry, like cameras or brake boosters, are produced each day. For many of our products, thousands of quality measurements are collected and checked during their assembly process individually. Understanding the relations and interconnections between those measurements is key to obtain a high production uptime and keep scrap at a minimum. Graphical models, like Bayesian networks, provide a rich statistical framework to investigate these relationships, not alone because they represent them as a graph. However, learning their graph structure is an NP-hard problem and most existing algorithms designed to either deal with a small number of variables or a small number of observations. On our datasets, with many thousands of variables and many hundreds of thousands of observations, classic learning algorithms don’t converge. In this talk, we show how we use an adapted version of the NOTEARs algorithm that uses mixture density neural networks to learn the structure of Bayesian networks even for very high-dimensional manufacturing data.
We discuss recent developments in the use of interacting particle systems for solving high-dimensional optimization problems, and provide insights into analytical guarantees for convergence in the particle mean-field limit.
Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816
This talk will comprise two parts. Part one will -- in the style of a colloquium -- give a non-technical introduction into the mathematical problems of describing relativistic interaction between quantum particles, in particular, the well-known ultraviolet divergence. Part two will give a technical introduction into a method, developed jointly with A. Pizzo, which allows to construct the mass shell of a persistent charge model of quantum field theory in the problematic ultraviolet regime.
The semiclassical structure of resonance states of classically chaotic scattering systems with partial escape is investigated. We introduce a local randomization on phase space for the baker map with escape, which separates the smallest multifractal scale from the scale of the Planck cell [1]. This allows for deriving a semiclassical description of resonance states based on a local random wave model and conditional invariance. We numerically demonstrate that the resulting classical measures perfectly describe resonance states of all decay rates $\gamma$ for the randomized baker map. Comparison to the baker map without randomization shows very good agreement of the multifractal structures. Quantitative differences indicate that a semiclassical description for systems without randomization must take into account that the multifractal structures persist down to the Planck scale.
We consider a large class of random graphs on the points of a Poisson point process in d dimensions, the weight-dependent random connection model. To each Poisson point we associate a random weight and we independently connect two points to each other with a probability depending on the distance and the weights of the two points. We investigate the number of infinite clusters and prove that the infinite cluster is almost surely unique. The general concept of the proof stems from the paper 'Uniqueness of the infinite component in a random graph with application to percolation and spin glasses' by Gandolfi et al. We extend their methods from the lattice to the continuum setup required for the weight-dependent random connection model.