3,872 research outputs found
Limit theorems for iterated random topical operators
Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively
homogeneous) operators. Let be defined by and
. This can modelize a wide range of systems including,
task graphs, train networks, Job-Shop, timed digital circuits or parallel
processing systems. When A(n) has the memory loss property, we use the spectral
gap method to prove limit theorems for . Roughly speaking, we show
that behaves like a sum of i.i.d. real variables. Precisely, we show
that with suitable additional conditions, it satisfies a central limit theorem
with rate, a local limit theorem, a renewal theorem and a large deviations
principle, and we give an algebraic condition to ensure the positivity of the
variance in the CLT. When A(n) are defined by matrices in the \mp semi-ring, we
give more effective statements and show that the additional conditions and the
positivity of the variance in the CLT are generic
Semigroup identities of tropical matrices through matrix ranks
We prove the conjecture that, for any , the monoid of all
tropical matrices satisfies nontrivial semigroup identities. To this end, we
prove that the factor rank of a large enough power of a tropical matrix does
not exceed the tropical rank of the original matrix.Comment: 13 page
Phase segregation for binary mixtures of Bose-Einstein Condensates
We study the strong segregation limit for mixtures of Bose-Einstein
condensates modelled by a Gross-Pitaievskii functional. Our first main result
is that in presence of a trapping potential, for different intracomponent
strengths, the Thomas-Fermi limit is sufficient to determine the shape of the
minimizers. Our second main result is that for asymptotically equal
intracomponent strengths, one needs to go to the next order. The relevant limit
is a weighted isoperimetric problem. We then study the minimizers of this limit
problem, proving radial symmetry or symmetry breaking for different values of
the parameters. We finally show that in the absence of a confining potential,
even for non-equal intracomponent strengths, one needs to study a related
isoperimetric problem to gain information about the shape of the minimizers
Optimized Schwarz waveform relaxation for Primitive Equations of the ocean
In this article we are interested in the derivation of efficient domain
decomposition methods for the viscous primitive equations of the ocean. We
consider the rotating 3d incompressible hydrostatic Navier-Stokes equations
with free surface. Performing an asymptotic analysis of the system with respect
to the Rossby number, we compute an approximated Dirichlet to Neumann operator
and build an optimized Schwarz waveform relaxation algorithm. We establish the
well-posedness of this algorithm and present some numerical results to
illustrate the method
On the Tightness of Bounds for Transients of Weak CSR Expansions and Periodicity Transients of Critical Rows and Columns of Tropical Matrix Powers
We study the transients of matrices in max-plus algebra. Our approach is
based on the weak CSR expansion. Using this expansion, the transient can be
expressed by , where is the weak CSR threshold and
is the time after which the purely pseudoperiodic CSR terms start to dominate
in the expansion. Various bounds have been derived for and ,
naturally leading to the question which matrices, if any, attain these bounds.
In the present paper we characterize the matrices attaining two particular
bounds on , which are generalizations of the bounds of Wielandt and
Dulmage-Mendelsohn on the indices of non-weighted digraphs. This also leads to
a characterization of tightness for the same bounds on the transients of
critical rows and columns. The characterizations themselves are generalizations
of those for the non-weighted case.Comment: 42 pages, 9 figure
Weak CSR expansions and transience bounds in max-plus algebra
This paper aims to unify and extend existing techniques for deriving upper
bounds on the transient of max-plus matrix powers. To this aim, we introduce
the concept of weak CSR expansions: A^t=CS^tR + B^t. We observe that most of
the known bounds (implicitly) take the maximum of (i) a bound for the weak CSR
expansion to hold, which does not depend on the values of the entries of the
matrix but only on its pattern, and (ii) a bound for the CS^tR term to
dominate. To improve and analyze (i), we consider various cycle replacement
techniques and show that some of the known bounds for indices and exponents of
digraphs apply here. We also show how to make use of various parameters of
digraphs. To improve and analyze (ii), we introduce three different kinds of
weak CSR expansions (named after Nachtigall, Hartman-Arguelles, and Cycle
Threshold). As a result, we obtain a collection of bounds, in general
incomparable to one another, but better than the bounds found in the
literature.Comment: 32 page
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