How do random Fibonacci sequences grow?

We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0<p\le 1$). Our main result is that the exponential growth of $F_n$ for $0<p\le 1$ (linear case) or for $1/3\le p\le 1$ (non-linear case) is almost surely given by $$\int_0^\infty \log x d\nu_\alpha (x),$$ where $\alpha$ is an explicit function of $p$ depending on the case we consider, and $\nu_\alpha$ is an explicit probability distribution on \RR_+ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of $p$, since we prove that it is equal to zero for $0<p\le1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative

Graphene Nano-, Micro-and Macro-Photonics

ABSTRACT Graphene has already become an established medium for novel photonic devices and their applications. In some cases, e.g. the use of graphene as a non-linear medium with saturable absorption properties, it is experimentally convenient to use the readily available form that is known as graphene oxide. Moreover, technological and scientific developments that are advancing control of the properties of graphene for electronic applications are also likely to be applicable in photonic and optoelectronic devices. This presentation will review research in the field of graphene photonics across the world. It will address, in particular, its application as a saturable absorber, e.g. for pulsed operation of fibre lasers -as well as work on materials characterisation of deposited graphene films. Patterning of graphene films with precision at the microand nano-scales will be an important requirement -and will be considered in this presentation

Design of a high-performance optical tweezer for nanoparticle trapping

Integrated optical nanotweezers offer a novel paradigm for optical trapping, as their ability to confine light at the nanoscale leads to extremely high gradient forces. To date, nanotweezers have been realized either as photonic crystal or as plasmonic nanocavities. Here, we propose a nanotweezer device based on a hybrid photonic/plasmonic cavity with the goal of achieving a very high quality factor-to-mode volume (Q/V) ratio. The structure includes a 1D photonic crystal dielectric cavity vertically coupled to a bowtie nanoantenna. A very high Q/V ~ 107 (λ/n)−3 with a resonance transmission T = 29 % at λR = 1381.1 nm has been calculated by 3D finite element method, affording strong light–matter interaction and making the hybrid cavity suitable for optical trapping. A maximum optical force F = −4.4 pN, high values of stability S = 30 and optical stiffness k = 90 pN/nm W have been obtained with an input power Pin = 1 mW, for a polystyrene nanoparticle with a diameter of 40 nm. This performance confirms the high efficiency of the optical nanotweezer and its potential for trapping living matter at the nanoscale, such as viruses, proteins and small bacteria

Patient-specific cancer genes contribute to recurrently perturbed pathways and establish therapeutic vulnerabilities in esophageal adenocarcinoma

Abstract: The identification of cancer-promoting genetic alterations is challenging particularly in highly unstable and heterogeneous cancers, such as esophageal adenocarcinoma (EAC). Here we describe a machine learning algorithm to identify cancer genes in individual patients considering all types of damaging alterations simultaneously. Analysing 261 EACs from the OCCAMS Consortium, we discover helper genes that, alongside well-known drivers, promote cancer. We confirm the robustness of our approach in 107 additional EACs. Unlike recurrent alterations of known drivers, these cancer helper genes are rare or patient-specific. However, they converge towards perturbations of well-known cancer processes. Recurrence of the same process perturbations, rather than individual genes, divides EACs into six clusters differing in their molecular and clinical features. Experimentally mimicking the alterations of predicted helper genes in cancer and pre-cancer cells validates their contribution to disease progression, while reverting their alterations reveals EAC acquired dependencies that can be exploited in therapy