Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

Possibilities of energy-resolved X-ray radiography for the investigation of paintings

X-ray radiographic images of paintings often show little or no contrast. In order to increase the contrast in radiographic images we measured the X-ray spectrum of a low power X-ray tube, after passing through the painting, with a high energy-resolution SDD detector. To obtain images, the detector is collimated with a 400 μm diameter pinhole and the painting was moved through the beam in the x and y-direction using a dwell time of a few seconds per pixel. The data obtained consists of a data cube of, typically, 200 × 200 pixels and a 512-channel X-ray spectrum for each pixel, spanning the energy range from 0 to 40 keV. Having the absorbance spectrum available for each pixel, we are able, a posteriori, to produce images by edge subtraction for any given element. In this way high contrast, element-specific, images can be obtained. Because of the high energy-resolution a much simpler edge subtraction algorithm can be applied. We also used principal-component imaging to obtain, in a more automated way, images with high contrast. Some of these images can easily be attributed to specific elements. It turns out that preprocessing of the spectral data is crucial for the success of the multivariate image processing.     

Hedging of Spatial Temperature Risk with Market-Traded Futures

Abstract

The main objective of this work is to construct optimal temperature futures from available market-traded contracts to hedge spatial risk. Temperature dynamics are modelled by a stochastic differential equation with spatial dependence. Optimal positions in market-traded futures minimizing the variance are calculated. Examples with numerical simulations based on a fast algorithm for the generation of random fields are presented.     

The Minimal Entropy Martingale Measure and Numerical Option Pricing for the Barndorff–Nielsen–Shephard Stochastic Volatility Model

Abstract

We develop and apply a numerical scheme for pricing options in the stochastic volatility model proposed by Barndorff–Nielsen and Shephard. This non-Gaussian Ornstein–Uhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy martingale measure. To numerically price options with respect to this risk neutral measure, one needs to consider a Black and Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black and Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.     

A Connection between Singular Stochastic Control and Optimal Stopping

We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.     

The challenge of being straight: explaining the linearity of a low-spin [FeNO]7 unit in a tropocoronand complex

We have carried out a density functional theory study of the S = 1/2 [FeNO]7 tropocoronand complex, Fe(5,5-TC)NO, as well as of some simplified models of this compound. The calculations accurately reproduce the experimentally observed trigonal-bipyramidal geometry of this complex, featuring a linear NO in an equatorial position and a very short Fe-N(NO) distance. Despite these unique structural features, the qualitative features of the bonding turn out to be rather similar for Fe(5,5-TC)NO and [FeNO]7 porphyrins. Thus, there is a close correspondence between the molecular orbitals (MOs) in the two cases. However, there is a critical, if somewhat subtle, difference in the nature of the singly occupied MOs (SOMOs) between the two. For square-pyramidal heme-NO complexes, the SOMO is primarily Fe d(z)2-based, which favors sigma-bonding interactions with an NO pi orbital, and hence a bent FeNO unit. However, for trigonal-bipyramidal Fe(5,5-TC)(NO), the SOMO is best described as primarily Fe d(x2-z2) in character, with the Fe-N(NO) vector being identified as the z direction. Apparently, such a d orbital is less adept at sigma bonding with NO and, as such, pi bonding dominates the Fe-NO interaction, leading to an essentially linear FeNO unit and a short Fe-N(NO) distance.     

Evaluation of energy-dispersive x-ray spectra with the aid of expert systems

Two recently developed Expert systems in the field of x-ray fluorescence are discussed. The first deals with the qualitative interpretation of x-ray spectra; it is characterized by a large knowledge base and a relatively small data base, both of which are of uniform structure. The second contains knowledge on quantitative spectrum evaluation and so has a much larger data base; its knowledge base is much smaller (at present), but contains several different types of rules.     

The morphological characterization of particles by automated scanning electron microscopy

Summary A method is described for the characterization and the classification according to shape of microscopic objects by automated SEM. The classification is performed by a hierarchical cluster analysis on a set of Fourier coefficients that are calculated from a set of radii, measured between a well defined centroid point and the contour lines of the object. This method is incorporated in existing commercial software for automated X-ray and size analysis of airborne particulate matter (PRC, Tracor Northern). Two examples demonstrate the possibilities and limitations of this method.

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Morphologische Charakterisierung mikroskopischer Objekte mit Hilfe der Raster-Elektronenmikroskopie

Zusammenfassung Eine Methode für die Charakterisierung und Klassifizierung mikroskopischer Objekte nach ihrer Form durch automatische Raster-Elektronenmikroskopie wurde beschrieben. Die Klassifizierung wird mittels einer hierarchischen Clusteranalyse unter Verwendung eines Satzes von Fourier-Koeffizienten durchgeführt, die aus einem Satz von Radien — gemessen zwischen einem exakt definierten Mittelpunkt und den Konturlinien der Objekte — berechnet werden. Diese Methode wird in eine kommerziell erhältliche Software für automatische Röntgen- und Größenverteilungs-analyse von luftgetragenen Staubteilchen eingebaut (PRC, Tracor Northern). An Hand von Beispielen werden die Möglichkeiten und Limitierungen dieser Methode dargestellt.