59,520 research outputs found
Lie systems: theory, generalisations, and applications
Lie systems form a class of systems of first-order ordinary differential
equations whose general solutions can be described in terms of certain finite
families of particular solutions and a set of constants, by means of a
particular type of mapping: the so-called superposition rule. Apart from this
fundamental property, Lie systems enjoy many other geometrical features and
they appear in multiple branches of Mathematics and Physics, which strongly
motivates their study. These facts, together with the authors' recent findings
in the theory of Lie systems, led to the redaction of this essay, which aims to
describe such new achievements within a self-contained guide to the whole
theory of Lie systems, their generalisations, and applications.Comment: 161 pages, 2 figure
A stochastic flow rule for granular materials
There have been many attempts to derive continuum models for dense granular
flow, but a general theory is still lacking. Here, we start with Mohr-Coulomb
plasticity for quasi-2D granular materials to calculate (average) stresses and
slip planes, but we propose a "stochastic flow rule" (SFR) to replace the
principle of coaxiality in classical plasticity. The SFR takes into account two
crucial features of granular materials - discreteness and randomness - via
diffusing "spots" of local fluidization, which act as carriers of plasticity.
We postulate that spots perform random walks biased along slip-lines with a
drift direction determined by the stress imbalance upon a local switch from
static to dynamic friction. In the continuum limit (based on a Fokker-Planck
equation for the spot concentration), this simple model is able to predict a
variety of granular flow profiles in flat-bottom silos, annular Couette cells,
flowing heaps, and plate-dragging experiments -- with essentially no fitting
parameters -- although it is only expected to function where material is at
incipient failure and slip-lines are inadmissible. For special cases of
admissible slip-lines, such as plate dragging under a heavy load or flow down
an inclined plane, we postulate a transition to rate-dependent Bagnold
rheology, where flow occurs by sliding shear planes. With different yield
criteria, the SFR provides a general framework for multiscale modeling of
plasticity in amorphous materials, cycling between continuum limit-state stress
calculations, meso-scale spot random walks, and microscopic particle
relaxation
Computations involving differential operators and their actions on functions
The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications
Scaling Algorithms for Unbalanced Transport Problems
This article introduces a new class of fast algorithms to approximate
variational problems involving unbalanced optimal transport. While classical
optimal transport considers only normalized probability distributions, it is
important for many applications to be able to compute some sort of relaxed
transportation between arbitrary positive measures. A generic class of such
"unbalanced" optimal transport problems has been recently proposed by several
authors. In this paper, we show how to extend the, now classical, entropic
regularization scheme to these unbalanced problems. This gives rise to fast,
highly parallelizable algorithms that operate by performing only diagonal
scaling (i.e. pointwise multiplications) of the transportation couplings. They
are generalizations of the celebrated Sinkhorn algorithm. We show how these
methods can be used to solve unbalanced transport, unbalanced gradient flows,
and to compute unbalanced barycenters. We showcase applications to 2-D shape
modification, color transfer, and growth models
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