1,011 research outputs found
Optimal bounds for self-similar solutions to coagulation equations with product kernel
We consider mass-conserving self-similar solutions of Smoluchowski's
coagulation equation with multiplicative kernel of homogeneity . We establish rigorously that such solutions exhibit a singular behavior
of the form as . This property had been
conjectured, but only weaker results had been available up to now
Self-similar solutions with fat tails for a coagulation equation with diagonal kernel
We consider self-similar solutions of Smoluchowski's coagulation equation
with a diagonal kernel of homogeneity . We show that there exists a
family of second-kind self-similar solutions with power-law behavior
as with . To our knowledge
this is the first example of a non-solvable kernel for which the existence of
such a family has been established
Compressing networks with super nodes
Community detection is a commonly used technique for identifying groups in a
network based on similarities in connectivity patterns. To facilitate community
detection in large networks, we recast the network to be partitioned into a
smaller network of 'super nodes', each super node comprising one or more nodes
in the original network. To define the seeds of our super nodes, we apply the
'CoreHD' ranking from dismantling and decycling. We test our approach through
the analysis of two common methods for community detection: modularity
maximization with the Louvain algorithm and maximum likelihood optimization for
fitting a stochastic block model. Our results highlight that applying community
detection to the compressed network of super nodes is significantly faster
while successfully producing partitions that are more aligned with the local
network connectivity, more stable across multiple (stochastic) runs within and
between community detection algorithms, and overlap well with the results
obtained using the full network
A Kinetic Model for Grain Growth
We provide a well-posedness analysis of a kinetic model for grain growth
introduced by Fradkov which is based on the von Neumann-Mullins law. The model
consists of an infinite number of transport equations with a tri-diagonal
coupling modelling topological changes in the grain configuration.
Self-consistency of this kinetic model is achieved by introducing a coupling
weight which leads to a nonlinear and nonlocal system of equations.
We prove existence of solutions by approximation with finite dimensional
systems. Key ingredients in passing to the limit are suitable super-solutions,
a bound from below on the total mass, and a tightness estimate which ensures
that no mass is transported to infinity in finite time.Comment: 24 page
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