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

Demonstration of oxygen-enriched air staging at Owens-Brockway glass containers. Final technical report for the period April 1, 1995--February 28, 1997

The overall objective of this program was to demonstrate the use of a previously developed combustion modification technology to reduce NO{sub x} emissions from sideport regenerative container glass melters. Specific objectives were to: acquire baseline operating data on the host sideport furnace, evaluate secondary oxidant injection strategies based on earlier endport furnace results and through modeling of a single port pair, retrofit and test one port pair (the test furnace has six port pairs) with a flexible OEAS system, and select the optimal system configuration, use the results from tests with one port pair to design, retrofit, and test OEAS on the entire furnace (six port pairs), and analyze test results, prepare report, and finalize the business plan to commercialize OEAS for sideport furnaces

Demonstration of oxygen-enriched air staging at Owens-Brockway glass containers. Quarterly technical progress report for the period August 1, 1996--October 31, 1996

The objective of the program is to demonstrate the use of a previously developed combustion modification technology to reduce NO, emissions from sideport regenerative container glass melters. This technology, known as oxygen-enriched air staging (OEAS), has been demonstrated, and is now being commercialized for endport container glass furnaces. This report focuses on full furnace parametric and long-term testing

Mapping the sensitivity of split ring resonators using a localized analyte

Split ring resonator (SRR) based metamaterials have frequently been demonstrated for use as optical sensors of organic materials. This is made possible by matching the wavelength of the SRR plasmonic resonance with a molecular resonance of a specific analyte, which is usually placed on top of the metal structure. However, systematic studies of SRRs that identify the regions that exhibit a high electric field strength are commonly performed using simulations. In this paper we demonstrate that areas of high electric field strength, termed “hot-spots,” can be found by localizing a small quantity of organic analyte at various positions on or near the structure. Furthermore, the sensitivity of the SRR to the localized analyte can be quantified to determine, experimentally, suitable regions for optical sensing

A compact two-dimensional grating coupler used as a polarization splitter

We demonstrate a novel polarization splitter based on a two-dimensional grating etched in a silicon-on-insulator waveguide. The device couples orthogonal modes from a single-mode optical fiber into identical modes of two planar ridge waveguides. The extinction ratio is better than 18 dB in the wavelength range of 1530-1560 nm and the coupling efficiency is approximately 20%. The device is very compact and couples light only to transverse-electric modes of the planar waveguides. Therefore, it may be used in a polarization diversity configuration to implement a polarization insensitive photonic integrated circuit based on photonic crystal waveguides

Fermions and Loops on Graphs. I. Loop Calculus for Determinant

This paper is the first in the series devoted to evaluation of the partition function in statistical models on graphs with loops in terms of the Berezin/fermion integrals. The paper focuses on a representation of the determinant of a square matrix in terms of a finite series, where each term corresponds to a loop on the graph. The representation is based on a fermion version of the Loop Calculus, previously introduced by the authors for graphical models with finite alphabets. Our construction contains two levels. First, we represent the determinant in terms of an integral over anti-commuting Grassman variables, with some reparametrization/gauge freedom hidden in the formulation. Second, we show that a special choice of the gauge, called BP (Bethe-Peierls or Belief Propagation) gauge, yields the desired loop representation. The set of gauge-fixing BP conditions is equivalent to the Gaussian BP equations, discussed in the past as efficient (linear scaling) heuristics for estimating the covariance of a sparse positive matrix.Comment: 11 pages, 1 figure; misprints correcte

Flexible modelling of spatial variation in agricultural field trials with the R package INLA

The objective of this paper was to fit different established spatial models for analysing agricultural field trials using the open-source R package INLA. Spatial variation is common in field trials, and accounting for it increases the accuracy of estimated genetic effects. However, this is still hindered by the lack of available software implementations. We compare some established spatial models and show possibilities for flexible modelling with respect to field trial design and joint modelling over multiple years and locations. We use a Bayesian framework and for statistical inference the integrated nested Laplace approximations (INLA) implemented in the R package INLA. The spatial models we use are the well-known independent row and column effects, separable first-order autoregressive ( AR1⊗AR1 ) models and a Gaussian random field (Matérn) model that is approximated via the stochastic partial differential equation approach. The Matérn model can accommodate flexible field trial designs and yields interpretable parameters. We test the models in a simulation study imitating a wheat breeding programme with different levels of spatial variation, with and without genome-wide markers and with combining data over two locations, modelling spatial and genetic effects jointly. The results show comparable predictive performance for both the AR1⊗AR1 and the Matérn models. We also present an example of fitting the models to a real wheat breeding data and simulated tree breeding data with the Nelder wheel design to show the flexibility of the Matérn model and the R package INLA

Modelling of photonic wire Bragg Gratings

Some important properties of photonic wire Bragg grating structures have been investigate. The design, obtained as a generalisation of the full-width gap grating, has been modelled using 3D finite-difference time-domain simulations. Different types of stop-band have been observed. The impact of the grating geometry on the lowest order (longest wavelength) stop-band has been investigated - and has identified deeply indented configurations where reduction of the stop-bandwidth and of the reflectivity occurred. Our computational results have been substantially validated by an experimental demonstration of the fundamental stop-band of photonic wire Bragg gratings fabricated on silicon-on-insulator material. The accuracy of two distinct 2D computational models based on the effective index method has also been studied - because of their inherently much greater rapidity and consequent utility for approximate initial designs. A 2D plan-view model has been found to reproduce a large part of the essential features of the spectral response of full 3D models

A Bayesian General Linear Modeling Approach to Cortical Surface fMRI Data Analysis

Cortical surface functional magnetic resonance imaging (cs-fMRI) has recently grown in popularity versus traditional volumetric fMRI. In addition to offering better whole-brain visualization, dimension reduction, removal of extraneous tissue types, and improved alignment of cortical areas across subjects, it is also more compatible with common assumptions of Bayesian spatial models. However, as no spatial Bayesian model has been proposed for cs-fMRI data, most analyses continue to employ the classical general linear model (GLM), a “massive univariate” approach. Here, we propose a spatial Bayesian GLM for cs-fMRI, which employs a class of sophisticated spatial processes to model latent activation fields. We make several advances compared with existing spatial Bayesian models for volumetric fMRI. First, we use integrated nested Laplacian approximations, a highly accurate and efficient Bayesian computation technique, rather than variational Bayes. To identify regions of activation, we utilize an excursions set method based on the joint posterior distribution of the latent fields, rather than the marginal distribution at each location. Finally, we propose the first multi-subject spatial Bayesian modeling approach, which addresses a major gap in the existing literature. The methods are very computationally advantageous and are validated through simulation studies and two task fMRI studies from the Human Connectome Project. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement

Gene Regulatory Networks from Multifactorial Perturbations Using Graphical Lasso: Application to the DREAM4 Challenge

A major challenge in the field of systems biology consists of predicting gene regulatory networks based on different training data. Within the DREAM4 initiative, we took part in the multifactorial sub-challenge that aimed to predict gene regulatory networks of size 100 from training data consisting of steady-state levels obtained after applying multifactorial perturbations to the original in silico network