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

Unusual raptor nests around the world

From surveys in many countries, we report using unusual nesting materials (e.g., paper money, rags, metal, antlers, and large bones) and unusual nesting situations. For example, we documented nests of Steppe Eagles [Aquila nipalensis] and Upland Buzzards [Buteo hemilasius] on the ground beside well-traveled roads, Saker Falcon [Falco cherrug] eyries in attics and a cistern, and Osprey [Pandian haliaetus] nests on the masts of boats and on a suspended automobile. Other records include a Golden Eagle [A. chrysaelos] nest 7.0 m in height, believed to be the tallest nest ever described, and, for the same species, we report nesting in rudimentary, nests. Some nest sites are within a Few meters of known predators or competitors. These unusual observations may be important in revealing the plasticity of a species' behavioral repertoire

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

Metal–organic complexation in the marine environment

We discuss the voltammetric methods that are used to assess metal–organic complexation in seawater. These consist of titration methods using anodic stripping voltammetry (ASV) and cathodic stripping voltammetry competitive ligand experiments (CSV-CLE). These approaches and a kinetic approach using CSV-CLE give similar information on the amount of excess ligand to metal in a sample and the conditional metal ligand stability constant for the excess ligand bound to the metal. CSV-CLE data using different ligands to measure Fe(III) organic complexes are similar. All these methods give conditional stability constants for which the side reaction coefficient for the metal can be corrected but not that for the ligand. Another approach, pseudovoltammetry, provides information on the actual metal–ligand complex(es) in a sample by doing ASV experiments where the deposition potential is varied more negatively in order to destroy the metal–ligand complex. This latter approach gives concentration information on each actual ligand bound to the metal as well as the thermodynamic stability constant of each complex in solution when compared to known metal–ligand complexes. In this case the side reaction coefficients for the metal and ligand are corrected. Thus, this method may not give identical information to the titration methods because the excess ligand in the sample may not be identical to some of the actual ligands binding the metal in the sample

Vitamin status and cognitive function in a long-term care population

BACKGROUND: Ageing can be associated with poor dietary intake, reduced nutrient absorption, and less efficient utilization of nutrients. Loss of memory and related cognitive function are also common among older persons. This study aimed to measure the prevalence of inadequate vitamin status among long-term care patients and determine if an association exists between vitamin status and each of three variables; cognitive function, vitamin supplementation, and medications which alter gastric acid levels. METHODS: Seventy-five patients in a long-term care hospital in Guelph, Ontario were recruited to a cross-sectional study. 47 were female and the mean age was 80.7 (+/-11.5) years, ranging from 48 to 100 years. Blood was used to measure levels of vitamins B12 (cobalamin), B6 (pyridoxal-5'-phosphate/PLP), erythrocyte folate, vitamin B3 (niacin) and homocysteine (Hcy). The Standardized Mini-Mental State Examination (SMMSE) was administered to measure cognitive function. A list of medications and vitamin supplementation for each patient was provided by the pharmacy. RESULTS: The prevalence of low vitamin (B12, B6, erythrocyte folate, niacin) or high metabolite (homocysteine) levels among 75 patients were as follows: B12 <148 pmol/L in 5/75 (6.7%); B12 between 148 and 221 pmol/L in 26/75 (34.7%); B6 ≤30 nmol/L in 4/75 (5.3%); erythrocyte folate <370 nmol/L in 1/75 (1.3%); niacin ratio ≤1 in 20/75 (26.7%); homocysteine >13.3 μmol/L in 31/75 (41.3%). There was no significant difference among residents grouped into marked (n = 44), mild (n = 14), or normal (n = 9) cognitive function when evaluating the effect of vitamin status. There were no significant differences in mean B12 and homocysteine levels between users and non-users of drug therapy (Losec, Zantac, or Axid). Compared to vitamin supplement non-users, supplemented residents had significantly higher mean B12 (p < 0.0001) and erythrocyte folate (p < 0.05) concentrations and significantly lower mean homocysteine (p < 0.01) levels; 229.1 versus 423.6 pmol/L for B12, 882.9 versus 1043.6 nmol/L for erythrocyte folate and 14.4 versus 12.0 μmol/L for homocysteine. CONCLUSION: Given the prevalence data on vitamin status in this sample population, the possible benefits of vitamin supplementation should be considered in clinical intervention studies using these populations of elderly