Orange Peels and Fresnel Integrals

There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife

A uniformly accurate scheme for the numerical integration of penalized Langevin dynamics

In molecular dynamics, penalized overdamped Langevin dynamics are used to model the motion of a set of particles that follow constraints up to a parameter ε. The most used schemes for simulating these dynamics are the Euler integrator in Rd and the constrained Euler integrator. Both have weak order one of accuracy, but work properly only in specific regimes depending on the size of the parameter ε. We propose in this paper a new consistent method with an accuracy independent of ε for solving penalized dynamics on a manifold of any dimension. Moreover, this method converges to the constrained Euler scheme when ε goes to zero. The numerical experiments confirm the theoretical findings, in the context of weak convergence and for the invariant measure, on a torus and on the orthogonal group in high dimension and high codimension.publishedVersio

Solovay functions and K-triviality

As part of his groundbreaking work on algorithmic randomness, Solovay demonstrated in the 1970s the remarkable fact that there are computable upper bounds of prefix-free Kolmogorov complexity $K$ that are tight on infinitely many values (up to an additive constant). Such computable upper bounds are called Solovay functions. Recent work of Bienvenu and Downey~[STACS 2009, LIPIcs 3, pp 147-158] indicates that Solovay functions are deeply connected with central concepts of algorithmic randomness such as $Omega$ numbers, K-triviality, and Martin-Loef randomness. In what follows, among other results we answer two open problems posed by Bienvenu and Downey about the definition of $K$-triviality and about the Gacs-Miller-Yu characterization of Martin-Loef randomness. The former defines a sequence A to be K-trivial if K(A|n) =^+ n-K(n). So both involve the noncomputable function K. As our main results we show that in both cases K(n) can be equivalently replaced by any Solovay function, and, what is more, that among all computable functions such a replacement is possible exactly for the Solovay functions. Moreover, similar statements hold for the larger class of all right-c.e. in place of the computable functions. These full characterizations, besides having significant theoretical interest on their own, will be useful as tools when working with K-trivial and Martin-Loef random sequences

Optimal overlayer inspired by Photuris firefly improves light-extraction efficiency of existing light-emitting diodes

In this paper the design, fabrication and characterization of a bioinspired overlayer deposited on a GaN LED is described. The purpose of this overlayer is to improve light extraction into air from the diode's high refractive-index active material. The layer design is inspired by the microstructure found in the firefly Photuris sp. The actual dimensions and material composition have been optimized to take into account the high refractive index of the GaN diode stack. This two-dimensional pattern contrasts other designs by its unusual profile, its larger dimensions and the fact that it can be tailored to an existing diode design rather than requiring a complete redesign of the diode geometry. The gain of light extraction reaches values up to 55% with respect to the reference unprocessed LED.Comment: 9 pages, 9 Figures, published in Optics Expres

Exhaust emissions of regulated and unregulated pollutants of passenger cars

Exhaust emissions of VOC speciation, aldehydes and other carbonyl compounds, polyaromatics and regulated pollutants are measured using a vehicle bench on a sample of passenger cars. 30 diesel and gasoline cars are tested, complying with ECE 1504 to Euro 3 emission standards, according to 10 real-world driving cycles based on European driving behaviour, with some of them adapted to vehicle size. The emission results of this large-scale measurement campaign show the influence of vehicle technology and driving behaviour on the emission of 100 individual pollutants. In addition, the results are discussed per VOC group and compared with other studies. The influence of the successive emission standards on the emission factors is very positive in most of cases. However, whereas hot CO2 is almost stable, diesel hot NOx, diesel hot and cold VOC, and the 6 most carcinogenic gasoline PAH have increased with standards. Diesel vehicles are less pollutant for CO, HC, CO2, VOC, but more pollutant for NOx and PAH. The distribution of VOC species per molecular family highlights the fact that monoaromatics make up the biggest share (~88 and 62 % resp. for gasoline and diesel vehicles). The second family is the alkanes which contribute resp. 8 and 9% of the total mass of measured VOC. The majority of volatile PAH is observed in the gaseous phase, but the least volatile and the carcinogenic PAH are adsorbed more in particulate phase