Magic of centroids

In this paper, we show some properties of centroids of geometric figures, such as triangles, quadrilaterals and tetrahedra. In particular, we will prove the properties by means of geometric transformations and by introducing extensions of triangles and quadrilaterals, i.e. by adding one, two or three new vertices to the figure. The study of these properties can be used, with profit, in a classroom activity supported by a dynamic geometry system.     

A Neyman–Pearson problem with ambiguity and nonlinear pricing

We consider a problem of the Neyman–Pearson type arising in the theory of portfolio choice in the presence of probability weighting, such as in markets with Choquet pricing (as in Araujo et al. in Econ Theory 49(1):1–35, 2011; Cerreia-Vioglio et al. in J Econ Theory 157(1):730–762, 2015; Chateauneuf and Cornet in Submodular financial markets with frictions. Working Paper, 2015; Chateauneuf et al. in Math Finance 6(3):323–330, 1996) and ambiguous beliefs about the payoffs of contingent claims (see Gilboa and Marinacci, in: Acemoglu, Arellano, Dekel (eds) Advances in economics and econometrics: theory and applications, tenth world congress of the econometric society, Cambridge University Press, Cambridge, 2013). Specifically, we consider a problem of optimal choice of a contingent claim so as to minimize a non-linear pricing functional (or a distortion risk measure), subject to a minimum expected performance measure (or a minimum expected return or utility), where expectations with respect to distorted probabilities are taken in the sense of Choquet. Such contingent claims are called cost-efficient. We give an analytical characterization of cost-efficient contingent claims under very mild assumptions on the probability weighting functions, thereby extending some of the results of Ghossoub (Math Financ Econ 10(1):87–111, 2016), and we provide examples of some special cases of interest. In particular, we show how a cost-efficient contingent claim exhibits a desirable monotonicity property: It is anti-comonotonic with the random mark-to-market value (or return, etc.) of the underlying financial position, and it is hence a hedge against such variability.     

High-efficiency liquid chromatography–mass spectrometry separations with 50 mm, 250 mm,and 1 m long polymer-based monolithic capillary columns for the characterization of complex proteolytic digests

In this study, high-efficiency LC–MS/MS separations of complex proteolytic digests are demonstrated using 50 mm, 250 mm, and 1 m long poly(styrene-co-divinylbenzene) monolithic capillary columns. The chromatographic performance of the 50 and 250 mm monoliths was compared at the same gradient steepness for gradient durations between 5 and 150 min. The maximum peak capacity of 400 obtained with a 50 mm column, increased to 485 when using the 250 mm long column and scaling the gradient duration according column length. With a 5-fold increase in column length only a 20% increase in peak capacity was observed, which could be explained by the larger macropore size of the 250 mm long monolith. When taking into account the total analysis time, including the dwell time, gradient time and column equilibration time, the 50 mm long monolith yielded better peptide separations than the 250 mm long monolithic column for gradient times below 80 min (n_c = 370). For more demanding separation the 250 mm long monolith provided the highest peak production rate and consequently higher sequence coverage. For the analysis of a proteolytic digest of Escherichia coli proteins a monolithic capillary column of 1 m in length was used, yielding a peak capacity of 1038 when applying a 600 min gradient.     

Nanostructured modified electrodes: role of ions and solvent flux in redox active polyelectrolyte multilayer films

Electrodes modified layer-by-layer by self-assembly of redox active polyelectrolytes comprised of osmium bipyridine-pyridine derivatized poly(allyl-amine) and poly(vinyl) sulfonate have been studied by EQCM, ellipsometry, cyclic voltammetry and electrochemical impedance spectroscopy in aqueous solutions of different anions and cations. Redox driven swelling by solvent exchange during oxidation, in excess to the hydration number, occurs by perturbation of the equilibrium between the osmotic and elastic forces as a result of the electrochemical injection of charge into the film. The exchanged mass and volume change during redox switching strongly depends on the nature of the anion under anion Donnan permselectivity conditions.     

Solvation structure and transport of acidic protons in ionic liquids: a first-principles simulation study

Ab initio simulations of a single molecule of HCl in liquid dimethyl imidazolium chloride [dmim][Cl] show that the acidic proton exists as a symmetric, linear ClHCl(-) species. Details of the solvation structure around this molecule are given. The proton-transfer process was investigated by applying a force along the antisymmetric stretch coordinate until the molecule broke. Changes in the free energy and local solvation structure during this process were investigated. In the reaction mechanism identified, a free chloride approaches the proton from the side. As the original ClHCl(-) distorts and the incoming chloride forms a new bond to the proton, one of the original chlorine atoms is expelled and a new linear molecule is formed.     

On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds

In this article we study the validity of the Whitney \(C^1\) extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the \(C^1\) extension property. We conclude by showing that the \(C^1\) extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.     

Optimally ending an epidemic

A stochastic and controlled version of the classic three-dimensional Kermack–McKendrick model for the spread of epidemics is considered. The aim is to end the epidemic as soon as possible, taking the quadratic control costs into account. An exact and explicit solution is found in a particular case by making use of the method of similarity solutions to solve the partial differential equation satisfied by the value function, subject to the appropriate conditions.     

The enhanced Sanov theorem and propagation of chaos

We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the ($k$-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean–Vlasov type limit, as shown in two corollaries.