One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers

We study the one-dimensional Schr\"odinger equation with a disordered potential of the form $V (x) = \phi(x)^2+\phi'(x) + \kappa(x)$ where $\phi(x)$ is a Gaussian white noise with mean $\mu g$ and variance $g$, and $\kappa(x)$ is a random superposition of delta functions distributed uniformly on the real line with mean density $\rho$ and mean strength $v$. Our study is motivated by the close connection between this problem and classical diffusion in a random environment (the Sinai problem) in the presence of random absorbers~: $\phi(x)$ models the force field acting on the diffusing particle and $\kappa(x)$ models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponent $\Omega(E) = \gamma(E) - \mathrm{i} \pi N(E)$, where $N$ is the integrated density of states per unit length and $\gamma$ the reciprocal of the localisation length. By using the continuous version of the Dyson-Schmidt method, we find an exact formula, in terms of a Hankel function, in the particular case where the strength of the delta functions is exponentially-distributed with mean $v=2g$. Building on this result, we then solve the general case -- in the low-energy limit -- in terms of an infinite sum of Hankel functions. Our main result, valid without restrictions on the parameters of the model, is that the integrated density of states exhibits the power law behaviour N(E) \underset{E\to0+}{\sim} E^\nu \hspace{0.5cm} \mbox{where } \nu=\sqrt{\mu^2+2\rho/g}\:. This confirms and extends several results obtained previously by approximate methods.Comment: LaTeX, 44 pages, 17 pdf figure

Products of random matrices and generalised quantum point scatterers

To every product of $2\times2$ matrices, there corresponds a one-dimensional Schr\"{o}dinger equation whose potential consists of generalised point scatterers. Products of {\em random} matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in $\text{SL}(2, {\mathbb R})$. We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.Comment: 38 pages, 13 pdf figures. V2 : conclusion added ; Definition of function $\Omega$ change

Generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder:Some exact results

We consider the one-dimensional Schrödinger equation with a random potential and study the cumulant generating function of the logarithm of the wave function ⁡(), known in the literature as the “generalized Lyapunov exponent”; this is tantamount to studying the statistics of the so-called “finite-size Lyapunov exponent.” The problem reduces to that of finding the leading eigenvalue of a certain nonrandom non-self-adjoint linear operator defined on a somewhat unusual space of functions. We focus on the case of Cauchy disorder, for which we derive a secular equation for the generalized Lyapunov exponent. Analytical expressions for the first four cumulants of ln⁡|⁡()| for arbitrary energy and disorder are deduced. In the universal (weak-disorder and high-energy) regime, we obtain simple asymptotic expressions for the generalized Lyapunov exponent and for all the cumulants. The large deviation function controlling the distribution of ln⁡|⁡()| is also obtained in several limits. As an application, we show that, for a disordered region of size , the distribution of the conductance exhibits the power-law behavior ⁡()∼−1/2 as →0

Du transport intelligent à la route intelligente

- Une tendance mondiale, lourde et irréversible se dessine. Les conducteurs sont de plus en plus aidés (dans leurs efforts et dans leur tâche de conduite) et le droit change. L'information se diversifie, se structure et se personnalise. Cette tendante conduit à des évolutions. Aujourd'hui, le gestionnaire de la route la charge de donner aux conducteurs les informations (visuelles, kinesthésiques et parfois sonores) nécessaires à la conduite. Demain ces informations et les caractéristiques géométriques et physiques de la chaussée seront transmises par de nouveaux médias. Les conducteurs plus ou moins aidés en déduiront leur domaine de contrôlabilité. Les vitesses consigne, la répétition des feux et panneaux à l'intérieur des véhicules, les distances d'arrêt, les conditions de circulation des différents Itinéraires sont autant d'informations qu'attendent les conducteurs. Et ils ne comprendraient pas que l'infrastructure ne pas soit en mesure de leur donner. La notion de service devient prégnante et la justice, de plus en plus présente. Leur action conjuguée contribuera à inciter voire à obliger les gestionnaires à accompagner leur capacité à faire par une obligation de le faire et d'informer. L'article situe les enjeux et la nécessité d'une approche conceptuelle globale pour que l'infrastructure soit partie prenante des systèmes de transport intelligent et propose une ligne directrice pour y parvenir s'appuyant sur le concept de trajectoire. Il est une invitation à ouvrir le champ des applications du traitement du signal pour mieux cerner, définir et faire communiquer entre eux les sous systèmes du transport qui, à n'en pas douter deviendra de plus en plus « intelligent »

Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices

The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.Comment: LaTeX, 23 pages, 3 pdf figures ; review for a special issue on "Lyapunov analysis" ; v2 : typo corrected in eq.(3) & minor change

Autophosphorylation on S614 inhibits the activity and the transforming potential of BRAF

International audienceThe BRAF proto-oncogene serine/threonine-protein kinase, known as BRAF, belongs to the RAF kinase family. It regulates the MAPK/ERK signalling pathway affecting several cellular processes such as growth, survival, differentiation, and cellular transformation. BRAF is mutated in ~8% of all human cancers with the V600E mutation constituting ~90% of mutations. Here, we have used quantitative mass spectrometry to map and compare phosphorylation site patterns between BRAF and BRAF V600E. We identified sites that are shared as well as several quantitative differences in phosphorylation abundance. The highest difference is phosphorylation of S614 in the activation loop which is ~5fold enhanced in BRAF V600E. Mutation of S614 increases the kinase activity of both BRAF and BRAF V600E and the transforming ability of BRAF V600E. The phosphorylation of S614 is mitogen inducible and the result of autophosphorylation. These data suggest that phosphorylation at this site is inhibitory, and part of the physiological shut-down mechanism of BRAF signalling

A whole-genome sequence and transcriptome perspective on HER2-positive breast cancers.

HER2-positive breast cancer has long proven to be a clinically distinct class of breast cancers for which several targeted therapies are now available. However, resistance to the treatment associated with specific gene expressions or mutations has been observed, revealing the underlying diversity of these cancers. Therefore, understanding the full extent of the HER2-positive disease heterogeneity still remains challenging. Here we carry out an in-depth genomic characterization of 64 HER2-positive breast tumour genomes that exhibit four subgroups, based on the expression data, with distinctive genomic features in terms of somatic mutations, copy-number changes or structural variations. The results suggest that, despite being clinically defined by a specific gene amplification, HER2-positive tumours melt into the whole luminal-basal breast cancer spectrum rather than standing apart. The results also lead to a refined ERBB2 amplicon of 106 kb and show that several cases of amplifications are compatible with a breakage-fusion-bridge mechanism