Notes on the non-autonomous Lyness equation

We prove that the solutions of the non-autonomous Lyness equation x_n+1=(x_n+b_n)/x_n−1, b_n>0, x₀>0, x₁>0, are bounded away from zero and infinity if the sequence b_n is monotonic. We also give an example to show that the solutions need not be bounded even when the sequence b_n attains only two values.     

Morse–Smale Regression

This article introduces a novel partition-based regression approach that incorporates topological information. Partition-based regression typically introduces a quality-of-fit-driven decomposition of the domain. The emphasis in this work is on a topologically meaningful segmentation. Thus, the proposed regression approach is based on a segmentation induced by a discrete approximation of the Morse–Smale complex. This yields a segmentation with partitions corresponding to regions of the function with a single minimum and maximum that are often well approximated by a linear model. This approach yields regression models that are amenable to interpretation and have good predictive capacity. Typically, regression estimates are quantified by their geometrical accuracy. For the proposed regression, an important aspect is the quality of the segmentation itself. Thus, this article introduces a new criterion that measures the topological accuracy of the estimate. The topological accuracy provides a complementary measure to the classical geometrical error measures and is very sensitive to overfitting. The Morse–Smale regression is compared to state-of-the-art approaches in terms of geometry and topology and yields comparable or improved fits in many cases. Finally, a detailed study on climate-simulation data demonstrates the application of the Morse–Smale regression. Supplementary Materials are available online and contain an implementation of the proposed approach in the R package msr, an analysis and simulations on the stability of the Morse–Smale complex approximation, and additional tables for the climate-simulation study.     

Cyclic Hilbert Spaces and Connes’ Embedding Problem

Let M be a II ₁-factor with trace τ, the finite dimensional subspaces of L ²(M, τ) are not just common Hilbert spaces, but they have an additional structure. We introduce the notion of a cyclic linear space by taking these additional properties as axioms. In Sect. 3 we formulate the following problem: “does every cyclic Hilbert space embed into L ²(M, τ), for some M?”. An affirmative answer would imply the existence of an algorithm to check Connes’ embedding Conjecture. In Sect. 4 we make a first step towards the answer of the previous question.     

Two techniques for PIV-based aeroacoustic prediction and their application to a rod-airfoil experiment

In this paper, we discuss techniques by which the aeroacoustic properties of the turbulent flow in a rod-airfoil benchmark experiment can be inferred from time-resolved PIV measurement. While acoustic measurements can be made directly using microphones, the proposed techniques provide a means to directly link acoustic waves with specific flow events, which is invaluable in devising noise mitigation strategies. The approaches are possible thanks to recent improvements in digital and camera technology that can provide time-resolved measurements in air flows, necessary for the determination of unsteady flow quantities related to aeroacoustic production. Both techniques are based on Curle’s acoustic analogy, where one is based on Lagrangian determination of the required quantities, while the other requires all flow quantities to be converted into Fourier modes. Application of both techniques yields results that are in reasonable agreement with microphone noise measurements for the rod-airfoil experiment.     

On the angular shape of van der Waals dimers of small polar molecules

Ab initio calculations on twenty van der Waals dimers of small polar molecules at the experimentally observed intermolecular separation R, using Hartree-Fock molecular moments, show that the minima in the electrostatic interaction expanded up to R⁻⁶ converges to angular structures which are close to those observed experimentally for such complexes.     

Phase space geometry in scalar-tensor cosmology

We study the phase space of spatially homogeneous and isotropic cosmology in general scalar-tensor theories. A reduction to a two-dimensional phase space is performed when possible—in these situations the phase space is usually a two-dimensional curved surface embedded in a three-dimensional space and composed of two sheets attached to each other, possibly with complicated topology. The results obtained are independent of the choice of the coupling function of the theory and, in certain situations, also of the potential.     

Changes in the morphology of interstellar ice analogues after hydrogen atom exposure

The morphology of water ice in the interstellar medium is still an open question. Although accretion of gaseous water could not be the only possible origin of the observed icy mantles covering dust grains in cold molecular clouds, it is well known that water accreted from the gas phase on surfaces kept at 10 K forms ice films that exhibit a very high porosity. It is also known that in the dark clouds H(2) formation occurs on the icy surface of dust grains and that part of the energy (4.48 eV) released when adsorbed atoms react to form H(2) is deposited in the ice. The experimental study described in the present work focuses on how relevant changes of the ice morphology result from atomic hydrogen exposure and subsequent recombination. Using the temperature-programmed desorption (TPD) technique and a method of inversion analysis of TPD spectra, we show that there is an exponential decrease in the porosity of the amorphous water ice sample following D-atom irradiation. This decrease is inversely proportional to the thickness of the ice and has a value of ?(0) = 2 × 10(16) D-atoms cm(-2) per layer of H(2)O. We also use a model which confirms that the binding sites on the porous ice are destroyed regardless of their energy depth, and that the reduction of the porosity corresponds in fact to a reduction of the effective area. This reduction appears to be compatible with the fraction of D(2) formation energy transferred to the porous ice network. Under interstellar conditions, this effect is likely to be efficient and, together with other compaction processes, provides a good argument to believe that interstellar ice is amorphous and non-porous.