561 research outputs found
Resolvent Positive Linear Operators Exhibit the Reduction Phenomenon
The spectral bound, s(a A + b V), of a combination of a resolvent positive
linear operator A and an operator of multiplication V, was shown by Kato to be
convex in b \in R. This is shown here, through an elementary lemma, to imply
that s(a A + b V) is also convex in a > 0, and notably, \partial s(a A + b V) /
\partial a <= s(A) when it exists. Diffusions typically have s(A) <= 0, so that
for diffusions with spatially heterogeneous growth or decay rates, greater
mixing reduces growth. Models of the evolution of dispersal in particular have
found this result when A is a Laplacian or second-order elliptic operator, or a
nonlocal diffusion operator, implying selection for reduced dispersal. These
cases are shown here to be part of a single, broadly general, `reduction'
phenomenon.Comment: 7 pages, 53 citations. v.3: added citations, corrections in
introductory definitions. v.2: Revised abstract, more text, and details in
new proof of Lindqvist's inequalit
Symmetrizing quantum dynamics beyond gossip-type algorithms
Recently, consensus-type problems have been formulated in the quantum domain.
Obtaining average quantum consensus consists in the dynamical symmetrization of
a multipartite quantum system while preserving the expectation of a given
global observable. In this paper, two improved ways of obtaining consensus via
dissipative engineering are introduced, which employ on quasi local preparation
of mixtures of symmetric pure states, and show better performance in terms of
purity dynamics with respect to existing algorithms. In addition, the first
method can be used in combination with simple control resources in order to
engineer pure Dicke states, while the second method guarantees a stronger type
of consensus, namely single-measurement consensus. This implies that outcomes
of local measurements on different subsystems are perfectly correlated when
consensus is achieved. Both dynamics can be randomized and are suitable for
feedback implementation.Comment: 11 pages, 3 figure
Spectral analysis of semigroups and growth-fragmentation equations
The aim of this paper is twofold: (1) On the one hand, the paper revisits the
spectral analysis of semigroups in a general Banach space setting. It presents
some new and more general versions, and provides comprehensible proofs, of
classical results such as the spectral mapping theorem, some (quantified)
Weyl's Theorems and the Krein-Rutman Theorem. Motivated by evolution PDE
applications, the results apply to a wide and natural class of generators which
split as a dissipative part plus a more regular part, without assuming any
symmetric structure on the operators nor Hilbert structure on the space, and
give some growth estimates and spectral gap estimates for the associated
semigroup. The approach relies on some factorization and summation arguments
reminiscent of the Dyson-Phillips series in the spirit of those used in
[87,82,48,81]. (2) On the other hand, we present the semigroup spectral
analysis for three important classes of "growth-fragmentation" equations,
namely the cell division equation, the self-similar fragmentation equation and
the McKendrick-Von Foerster age structured population equation. By showing that
these models lie in the class of equations for which our general semigroup
analysis theory applies, we prove the exponential rate of convergence of the
solutions to the associated remarkable profile for a very large and natural
class of fragmentation rates. Our results generalize similar estimates obtained
in \cite{MR2114128,MR2536450} for the cell division model with (almost)
constant total fragmentation rate and in \cite{MR2832638,MR2821681} for the
self-similar fragmentation equation and the cell division equation restricted
to smooth and positive fragmentation rate and total fragmentation rate which
does not increase more rapidly than quadratically. It also improves the
convergence results without rate obtained in \cite{MR2162224,MR2114413} which
have been established under similar assumptions to those made in the present
work
On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems
Given a discrete-time linear switched system associated
with a finite set of matrices, we consider the measures of its
asymptotic behavior given by, on the one hand, its deterministic joint spectral
radius and, on the other hand, its probabilistic
joint spectral radii for Markov random
switching signals with transition matrix and a corresponding invariant
probability . Note that is larger than or
equal to for every pair . In
this paper, we investigate the cases of equality of with either a single or with the
supremum of over and we aim at
characterizing the sets for which such equalities may occur
Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts
We study the asymptotic behaviour of the following linear
growth-fragmentation equation and prove that under fairly general assumptions on the division
rate its solution converges towards an oscillatory function,explicitely
given by the projection of the initial state on the space generated by the
countable set of the dominant eigenvectors of the operator. Despite the lack of
hypo-coercivity of the operator, the proof relies on a general relative entropy
argument in a convenient weighted space, where well-posedness is obtained
via semigroup analysis. We also propose a non-dissipative numerical scheme,
able to capture the oscillations
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