2,001 research outputs found
The Filippov characteristic flow for the aggregation equation with mildly singular potentials
Existence and uniqueness of global in time measure solution for the
multidimensional aggregation equation is analyzed. Such a system can be written
as a continuity equation with a velocity field computed through a
self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)),
a well-posedness theory based on the geometric approach of gradient flows in
measure metric spaces has been developed for mildly singular potentials at the
origin under the basic assumption of being lambda-convex. We propose here an
alternative method using classical tools from PDEs. We show the existence of a
characteristic flow based on Filippov's theory of discontinuous dynamical
systems such that the weak measure solution is the pushforward measure with
this flow. Uniqueness is obtained thanks to a contraction argument in transport
distances using the lambda-convexity of the potential. Moreover, we show the
equivalence of this solution with the gradient flow solution. Finally, we show
the convergence of a numerical scheme for general measure solutions in this
framework allowing for the simulation of solutions for initial smooth densities
after their first blow-up time in Lp-norms.Comment: 33 page
An algebraic approach to general aggregation theory: Propositional-attitude aggregators as MV-homomorphisms
This paper continues Dietrich and List's [2010] work on propositional-attitude aggregation theory, which is a generalised unification of the judgment-aggregation and probabilistic opinion-pooling literatures. We first propose an algebraic framework for an analysis of (many-valued) propositional-attitude aggregation problems. Then we shall show that systematic propositional-attitude aggregators can be viewed as homomorphisms in the category of C.C. Chang's [1958] MV-algebras. Since the 2-element Boolean algebra as well as the real unit interval can be endowed with an MV-algebra structure, we obtain as natural corollaries two famous theorems: Arrow's theorem for judgment aggregation as well as McConway's [1981] characterisation of linear opinion pools.propositional attitude aggregation, judgment aggregation, linear opinion pooling, Arrow's impossibility theorem, many-valued logic, MV-algebra, homomorphism, Arrow's impossibility theorem, functional equation
Scaling Limits for Internal Aggregation Models with Multiple Sources
We study the scaling limits of three different aggregation models on Z^d:
internal DLA, in which particles perform random walks until reaching an
unoccupied site; the rotor-router model, in which particles perform
deterministic analogues of random walks; and the divisible sandpile, in which
each site distributes its excess mass equally among its neighbors. As the
lattice spacing tends to zero, all three models are found to have the same
scaling limit, which we describe as the solution to a certain PDE free boundary
problem in R^d. In particular, internal DLA has a deterministic scaling limit.
We find that the scaling limits are quadrature domains, which have arisen
independently in many fields such as potential theory and fluid dynamics. Our
results apply both to the case of multiple point sources and to the
Diaconis-Fulton smash sum of domains.Comment: 74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in
v2: added "least action principle" (Lemma 3.2); small corrections in section
4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version);
expanded section 6.
Existence of Ground States of Nonlocal-Interaction Energies
We investigate which nonlocal-interaction energies have a ground state
(global minimizer). We consider this question over the space of probability
measures and establish a sharp condition for the existence of ground states. We
show that this condition is closely related to the notion of stability (i.e.
-stability) of pairwise interaction potentials. Our approach uses the direct
method of the calculus of variations.Comment: This version is to appear in the J Stat Phy
Algorithms for Approximate Subtropical Matrix Factorization
Matrix factorization methods are important tools in data mining and analysis.
They can be used for many tasks, ranging from dimensionality reduction to
visualization. In this paper we concentrate on the use of matrix factorizations
for finding patterns from the data. Rather than using the standard algebra --
and the summation of the rank-1 components to build the approximation of the
original matrix -- we use the subtropical algebra, which is an algebra over the
nonnegative real values with the summation replaced by the maximum operator.
Subtropical matrix factorizations allow "winner-takes-it-all" interpretations
of the rank-1 components, revealing different structure than the normal
(nonnegative) factorizations. We study the complexity and sparsity of the
factorizations, and present a framework for finding low-rank subtropical
factorizations. We present two specific algorithms, called Capricorn and
Cancer, that are part of our framework. They can be used with data that has
been corrupted with different types of noise, and with different error metrics,
including the sum-of-absolute differences, Frobenius norm, and Jensen--Shannon
divergence. Our experiments show that the algorithms perform well on data that
has subtropical structure, and that they can find factorizations that are both
sparse and easy to interpret.Comment: 40 pages, 9 figures. For the associated source code, see
http://people.mpi-inf.mpg.de/~pmiettin/tropical
A stochastic min-driven coalescence process and its hydrodynamical limit
A stochastic system of particles is considered in which the sizes of the
particles increase by successive binary mergers with the constraint that each
coagulation event involves a particle with minimal size. Convergence of a
suitably renormalised version of this process to a deterministic hydrodynamical
limit is shown and the time evolution of the minimal size is studied for both
deterministic and stochastic models
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