A family of rotation numbers for discrete random dynamics on the circle

We revisit the problem of well-defining rotation numbers for discrete random dynamical systems on the circle. We show that, contrasting with deterministic systems, the topological (i.e. based on Poincar\'{e} lifts) approach does depend on the choice of lifts (e.g. continuously for nonatomic randomness). Furthermore, the winding orbit rotation number does not agree with the topological rotation number. Existence and conversion formulae between these distinct numbers are presented. Finally, we prove a sampling in time theorem which recover the rotation number of continuous Stratonovich stochastic dynamical systems on $S^1$ out of its time discretisation of the flow.Comment: 15 page

Escape from attracting sets in randomly perturbed systems

The dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science. Previous studies of escape in chaotic systems have rather focused on the case of unbounded noise, usually assumed to have Gaussian distribution. In this paper, we address the problem of escape induced by bounded noise. We show that the dynamics of escape from an attractor's basin is equivalent to that of a closed system with an appropriately chosen "hole". Using this equivalence, we show that there is a minimum noise amplitude above which escape takes place, and we derive analytical expressions for the scaling of the escape rate with noise amplitude near the escape transition. We verify our analytical predictions through numerical simulations of a two-dimensional map with noise.Comment: up to date with published versio

Rigidity of Curvature Bounds of Quotient Spaces Of Isometric Actions

Let $G\curvearrowright M$ be an isometric action of a Lie Group on a complete orientable Riemannian manifold. We disintegrate absolutely continuous measures with respect to the volume measure of $M$ along the principal orbits of $G\curvearrowright M$ and define a functional on the probability measures with support on the principal orbits of the action to further prove that the convexity properties of this functional guarantees necessary and sufficient conditions to the Ricci curvature of $M$ to be bound below by a given real number $K$.Comment: 22 page

Irreducible actions and compressible modules

Any finite set of linear operators on an algebra $A$ yields an operator algebra $B$ and a module structure on A, whose endomorphism ring is isomorphic to a subring $A^B$ of certain invariant elements of $A$. We show that if $A$ is a critically compressible left $B$-module, then the dimension of its self-injective hull $A$ over the ring of fractions of $A^B$ is bounded by the uniform dimension of $A$ and the number of linear operators generating $B$. This extends a known result on irreducible Hopf actions and applies in particular to weak Hopf action. Furthermore we prove necessary and sufficient conditions for an algebra A to be critically compressible in the case of group actions, group gradings and Lie actions

Random fluctuation leads to forbidden escape of particles

A great number of physical processes are described within the context of Hamiltonian scattering. Previous studies have rather been focused on trajectories starting outside invariant structures, since the ones starting inside are expected to stay trapped there forever. This is true though only for the deterministic case. We show however that, under finitely small random fluctuations of the field, trajectories starting inside Arnold-Kolmogorov-Moser (KAM) islands escape within finite time. The non-hyperbolic dynamics gains then hyperbolic characteristics due to the effect of the random perturbed field. As a consequence, trajectories which are started inside KAM curves escape with hyperbolic-like time decay distribution, and the fractal dimension of a set of particles that remain in the scattering region approaches that for hyperbolic systems. We show a universal quadratic power law relating the exponential decay to the amplitude of noise. We present a random walk model to relate this distribution to the amplitude of noise, and investigate this phenomena with a numerical study applying random maps.Comment: 6 pages, 6 figures - Up to date with corrections suggested by referee

Major accidents scenarios used for LUP and off-site emergency planning : importance of kinetic description

International audienceThe European States, France in particular, faced with several industrial accidents, like one of the most serious one, the AZF explosion in Toulouse (France) in 2001, which created the trauma of the stakeholders concerned by the chemical risk (industry, authorities and citizen). Learning from this experience, the French Ministry in charge of the Environment, with the help of INERIS, worked on upgrading its chemical risks knowledge and studied to improve new methods on risk assessment. In 2004, INERIS was in charge of developing a new approach to classify accidental scenarios in terms of probability, severity and response time. In this paper, INERIS proposes a prioritisation based on time of occurrence, of development of hazardous phenomena and of effects on targets. Then, INERIS shows a method enabling to integrate, as a second prioritisation criteria, the on-site and off-site response capabilities/abilities as well as the means for population protection in terms of time allowed

Emerging attractors and the transition from dissipative to conservative dynamics

The topological structure of basin boundaries plays a fundamental role in the sensitivity to the initial conditions in chaotic dynamical systems. Herewith we present a study on the dynamics of dissipative systems close to the Hamiltonian limit, emphasising the increasing number of periodic attractors and on the structural changes in their basin boundaries as the dissipation approaches zero. We show numerically that a power law with nontrivial exponent describes the growth of the total number of periodic attractors as the damping is decreased. We also establish that for small scales the dynamics is governed by \emph{effective} dynamical invariants, whose measure depends not only on the region of the phase space, but also on the scale under consideration. Therefore, our results show that the concept of effective invariants is also relevant for dissipative systems.Comment: 9 pages, 10 figures. Accepted and in press for PR

Representation of Markov chains by random maps: existence and regularity conditions

We systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory also tells us how convexity properties of the supports of the measures translate into regularity properties of the maps via Legendre transforms. Thus, from this scheme, we cannot only deduce the representation by measurable random maps, but we can also obtain conditions for the representation by continuous random maps. Finally, we present conditions for the representation of Markov chain by random diffeomorphisms.Comment: 22 pages, several changes from the previous version including extended discussion of many detail

High incidence of chromosomal numerical abnormalities by multicentromeric FISH in multiple myeloma patients

This study aimed to characterize genetic alterations by interphase multicentromeric FISH focusing on chromosomal numerical abnormalities and using some locus specific probes for the most frequent aberrations found in the disease, in a homogeneous cohort of 34 advanced stage, but recently diagnosed MM patients; 97% had numerical chromosomal abnormalities detected by FISH, being 75% hyperdiploid, 18% hypodiploid and 3% tri/tetraploid. Using locus specific probes, we found 13q deletion in 30% and IGH rearrangement in 25% of cases. Grouping hypodiploid patients together with del13q (unfavorable group) and comparing them to the remaining cases (non unfavorable group) we found a trend towards younger patients presenting more unfavorable abnormalities (p = 0.06) and significant lower hemoglobin level (Hb < 8.5 mg/dl, p = 0.03).Este estudo objetivou detectar as alterações genéticas em pacientes com mieloma múltiplo (MM), usando o método de hibridação in situ por fluorescência em interfases (FISH interfásico). Para detectar as alterações numéricas foram usadas sondas multicentroméricas e para os rearranjos mais freqüentemente observados na doença foram utilizadas as sondas lócus específicas para IGH, P53, ciclina D1 e RB1. Foram estudados 34 pacientes com MM em estágio avançado, ainda que recém-diagnosticados, 97% dos quais apresentaram anormalidades numéricas detectadas por FISH, sendo 75% hiperdiplóides, 18% hipodiplóides e 3% tri/tetraplóides. Em relação às demais anormalidades, a deleção 13q foi encontrada em 30% dos casos e o rearranjo IGH, em 25%. Agrupando os pacientes com hipodiploidia e com deleção 13q14 (grupo desfavorável) e comparando-os com os demais (grupo não-desfavorável), houve tendência a pacientes jovens no grupo desfavorável (p = 0,06) e níveis de hemoglobina (Hb) significativamente mais baixos (< 8,5 g/dl, p = 0,03).Universidade Federal de São Paulo (UNIFESP)UNIFESPSciEL