The Limiting Normal Cone to Pointwise Defined Sets in Lebesgue Spaces

We consider subsets of Lebesgue spaces which are defined by pointwise constraints. We provide formulas for corresponding variational objects (tangent and normal cones). Our main result shows that the limiting normal cone is always dense in the Clarke normal cone and contains the convex hull of the pointwise limiting normal cone. A crucial assumption for this result is that the underlying measure is non-atomic, and this is satisfied in many important applications (Lebesgue measure on subsets of \(\mathbb {R}^{d}\) or the surface measure on hypersurfaces in \(\mathbb {R}^{d}\)). Finally, we apply our findings to an optimization problem with complementarity constraints in Lebesgue spaces.     

Heisenberg’s Uncertainty Relation and Bell Inequalities in High Energy Physics

An effective formalism is developed to handle decaying two-state systems. Herewith, observables of such systems can be described by a single operator in the Heisenberg picture. This allows for using the usual framework in quantum information theory and, hence, to enlighten the quantum features of such systems compared to non-decaying systems. We apply it to systems in high energy physics, i.e. to oscillating meson–antimeson systems. In particular, we discuss the entropic Heisenberg uncertainty relation for observables measured at different times at accelerator facilities including the effect of CP\mathcal{CP} violation, i.e. the imbalance of matter and antimatter. An operator-form of Bell inequalities for systems in high energy physics is presented, i.e. a Bell-witness operator, which allows for simple analysis of unstable systems.     

Approximation of sparse controls in semilinear equations by piecewise linear functions

Semilinear elliptic optimal control problems involving the $L^1$ norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates.     

A high-throughput screening method for the exploration of optimal curing parameters and resistance to solvents of NANOMER®coating materials

A high-throughput screening method for the exploration of optimal curing parameters and resistance to solvents of NANOMER® coating materials based on the temperature-dependent swellability was developed. The screening method was first tested using a model sol made of pre-hydrolyzed (3-glycidoxypropyl)triethoxysilane (GPTES), tetraethoxysilane (TEOS), (heptadecafluoro-1,1,2,2-tetrahydrodecyl)triethoxysilane (FTS) and zirconium complex (prepared of zirconium-tert-butoxide complexed with acetylacetonate) charged with reactive diluent trimethylolpropan-triglycidether and defined amounts of fluorescein and cured at different temperatures. Afterwards, fluorescein was extracted with sodium hydroxide solution and the optical density of the supernatant of all samples was measured at 490 nm which is sensitive to the dye concentration. The optical density (OD) correlates with the degree of curing. According to this screening a temperature $CDATA \geq 140 ^\circA high-throughput screening method for the exploration of optimal curing parameters and resistance to solvents of NANOMER? coating materials based on the temperature-dependent swellability was developed. The screening method was first tested using a model sol made of pre-hydrolyzed (3-glycidoxypropyl)triethoxysilane (GPTES), tetraethoxysilane (TEOS), (heptadecafluoro-1,1,2,2-tetrahydrodecyl)triethoxysilane (FTS) and zirconium complex (prepared of zirconium-tert-butoxide complexed with acetylacetonate) charged with reactive diluent trimethylolpropan-triglycidether and defined amounts of fluorescein and cured at different temperatures. Afterwards, fluorescein was extracted with sodium hydroxide solution and the optical density of the supernatant of all samples was measured at 490 nm which is sensitive to the dye concentration. The optical density (OD) correlates with the degree of curing. According to this screening a temperature C is necessary for proper curing. The time dependence of extraction reveals information on resistance against sodium hydroxide solution, i. e. alkali resistance. The time dependent extraction of fluorescein at C of coatings cured at 100 and 140°C, respectively, shows a better resistance against 0.1 M sodium hydroxide solution for the one cured at 140°C, especially in the time range 10–60 min. The whole process — sol preparation, mixing of sols with dye, extraction, and optical spectroscopy—can be performed automatically. Further testes were made to proof the usability of this process. 96 hybrid materials were screened in regard to their alkali resistance and finally, a total number of 14 clear organic-inorganic hybrid coating systems with improved stability against sodium hydroxide solution were derived from this study.     

Elevated temperature and temperature programming in conventional liquid chromatography--fundamentals and applications

Temperature, as a powerful variable in conventional LC is discussed from a fundamental point of view and illustrated with applications from the author's laboratory. Emphasis is given to the influence of temperature on speed, selectivity, efficiency, detectability, and mobile phase composition (green chromatography). The problems accompanying the use of elevated temperature and temperature programming in LC are reviewed and solutions are described. The available stationary phases for high temperature operation are summarized and a brief overview of recent applications reported in the literature is given.     

Collective fluctuations in ordered fluids investigated by two-dimensional electron-electron double resonance spectroscopy

Two-dimensional electron-electron double resonance (2D-ELDOR) is a technique that is sensitive to the dynamical processes affecting spin labels in complex fluid environments. In ordered fluids, such as membrane vesicles, the 2D-ELDOR experiment is affected by the molecular tumbling in the locally ordered environment. This motion occurs on two different time scales, the faster molecular motion relative to the local director, and the slower collective fluctuations of the director field. In the experimental study of Patyal, Crepeau, and Freed (Biophys. J. 1997, 73, 2201), it was found that the widths of the autopeaks of the 2D-ELDOR spectrum increased as a function of the mixing time. In the present work, a theory is developed for the effects of director fluctuations on the autopeaks in the 2D-ELDOR experiment by employing an analytical solution of the stochastic Liouville equation for which the director field is treated as a multidimensional Gaussian process, as previously developed by Frezzato, Kothe, and Moro (J. Phys. Chem. B 2001, 105, 1281 and J. Phys. Chem. B 2004, 108, 9505). Good agreement is found between theory and experiment, notably the only adjustable parameter is k, the bending elastic modulus of the membrane. The values of k = 11 x 10(-20) J for 1,2-dipalmitoyl-sn-glycero-phosphatidylcholine (DPPC) vesicles and k = 15 x 10(-20) J for DPPC/gramicidin A (5:1) vesicles, both at 45 degrees C, were found from the analysis and agree well with previous related measurements by other physical techniques. This establishes 2D-ELDOR as a useful technique to study the elastic properties of biological membranes.     

Solubility of carbon dioxide in aqueous solutions of methanol. Predictions by molecular simulation and comparison with experimental data

The solubility of carbon dioxide in pure methanol, and in aqueous solutions of methanol, was computed using the Gibbs ensemble Monte Carlo (GEMC) technique for 313, 354, and 395 K at pressures up to 9 MPa. Three solvent mixtures (of methanol and water) with methanol mole fractions of 10, 50, and 75 mole percent (in the gas-free solvent mixture) were studied. The Monte Carlo simulations were conducted in an isothermal-isobaric ensemble applying effective pair potentials for the pure components from literature. Common mixing rules without any adjustable binary interaction parameters were used to describe the interactions between the mixture components. Overall, a good agreement between simulation results and recently published experimental data is achieved.     

Note on Vector Translation with Respect to Thompson’s Metric

It is known that vector translations are contractive with respect to Thompson’s part metric. Here, we give a simple proof, based on a representation of Thompson’s metric through positive functionals. Moreover, we use contractivity of translations to prove a fixed point result for mappings that are Lipschitz continuous with respect to Thompson’s metric with Lipschitz constant r>1. The case r = 1 for order preserving or order reversing mappings has been recently studied by Lawson and Lim. We apply our result to a nonlinear boundary value problem.