A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics

We consider the ordinary differential equation $$x^2 u''=axu'+bu-c \bigl(u'-1\bigr)^2, \quad x\in(0,x_0), $$ with $a\in\mathbb{R}, b\in\mathbb{R}$ , c>0 and the singular initial condition u(0)=0, which in financial economics describes optimal disposal of an asset in a market with liquidity effects. It is shown in the paper that if a+b<0 then no continuous solutions exist, whereas if a+b>0 then there are infinitely many continuous solutions with indistinguishable asymptotics near 0. Moreover, it is proved that in the latter case there is precisely one solution u corresponding to the choice x ₀=∞ which is such that 0≤u(x)≤x for all x>0, and that this solution is strictly increasing and concave.     

A difference between continuous and absolutely continuous norms in Banach function spaces

On examples we show a difference between a continuous and absolutely continuous norm in Banach function spaces.     

Energy dependence of {\bar{K}N} interaction in nuclear medium

When the $\bar{K}N$ system is submerged in nuclear medium the $\bar{K}N$ scattering amplitude and the final state branching ratios exhibit a strong energy dependence when going to energies below the $\bar{K}N$ threshold. A sharp increase of $\bar{K}N$ attraction below the $\bar{K}N$ threshold provides a link between shallow $\bar{K}$ -nuclear potentials based on the chiral $\bar{K}N$ amplitude evaluated at threshold and the deep phenomenological optical potentials obtained in fits to kaonic atoms data. We show the energy dependence of the in-medium K ^??? p amplitude and demonstrate the impact of energy dependent branching ratios on the Λ-hypernuclear production rates.     

Interaction between model poly(ethylene terephthalate) thin films and weakly ionised oxygen plasma

Model films of poly(ethylene terephthalate) were treated by oxygen plasma in order to quantify the etching rate and estimate the contribution of charged and neutral particles to the reaction probability. Model films with a thickness of 50 nm were deposited on a quartz crystal of a microbalance (QCM) by spin‐coating technique. The samples were exposed to oxygen plasma with the positive ion density of 4 × 10¹⁵ m^?3 and neutral oxygen atom density of 6 × 10²¹ m^?3. The etching rate was determined from the QCM signal and was 4.7 nm s^?1. The etching was found rather inhomogeneous as the atomic force microscopic images showed an increase of the surface roughness as a result of plasma treatment. The model films were completely removed from the surface of the quartz crystals in about 12 s. Knowing the etching rate and the flux of oxygen atoms to the surface allowed for calculation of the reaction probability which was found to be rather low at the value of 1.6 × 10^?4. Copyright © 2011 John Wiley & Sons, Ltd.     

Synthesis of Aromatic Compounds by Catalytic C?C Bond Activation of Biphenylene or Angular [3]Phenylene

Substituted phenanthrenes and picenes were easily prepared by reaction of biphenylene or angular [3]phenylene with various alkynes in the presence of a catalytic amount of [IrCl(cod)]₂/dppe (cod=1,5‐cyclooctadiene, dppe=1,2‐bis(diphenylphosphino)ethane). The reaction is based on C? C bond activation of the cyclobutane ring. The reaction tolerates the presence of bulky groups on the alkyne, such as the ferrocene moiety. In addition, a catalytic system based on [RhCl(cod)]₂/dppe enabled the, hitherto unreported, reaction of biphenylene with nitriles to provide phenanthridines.     

Measurement of bubble size distribution in protein foam fractionation column using capillary probe with photoelectric sensors

Bubble size is a key variable for predicting the ability to separate and concentrate proteins in a foam fraction ation process. It is used to characterize not only the bubble-specific interfacial a rea but also coalescence of bubbles in the foam phase. This article describes the development of a photoelectric method for measuring the bubble size distribution in both bubble and foam columns for concentrating proteins. The method uses a vacuum to withdraw a stream of gas-liquid dispersion from the bubble or foam column through a capillary tube with a funnel-shaped inlet. The resulting sample bubble cylinders are detected, and their lengths are calculated by using two pairs of infrared photoelectric sensors that are connected with a high-speed data acquisition system controlled by a microcomputer. The bubble size distributions in the bubble column 12 and 1 cm below the interface and in the foam phase 1 cm above the interface are obtained in a continuous foam fractionation process for concentrating ovalbumin. The effects of certain operating conditions such as the feed protein concentration, superficial gas velocity, liquid flow rate, and solution pH are investigated. The results may prove to be helpful in understanding the mechanisms controlling the foam fractionation of proteins.     

Adsorption of carboxymethyl cellulose on polymer surfaces: evidence of a specific interaction with cellulose

The adsorption of carboxymethyl cellulose (CMC), one of the most important cellulose derivatives, is crucial for many scientific investigations and industrial applications. Especially for surface modifications and functionalization of materials, the polymer is of interest. The adsorption properties of CMC are dependent not only on the solutions state, which can be influenced by the pH, temperature, and electrolyte concentration, but also on the chemical composition of the adsorbents. We therefore performed basic investigation studies on the interaction of CMC with a variety of polymer films. Thin films of cellulose, cellulose acetate, deacetylated cellulose acetate, polyethylene terephthalate, and cyclo olefin polymer were therefore prepared on sensors of a QCM-D (quartz crystal microbalance) and on silicon substrates. The films were characterized with respect to the thickness, wettability, and chemical composition. Subsequently, the interaction and deposition of CMC in a range of pH values without additional electrolyte were measured with the QCM-D method. A comparison of the QCM-D results showed that CMC is favorably deposited on pure cellulose films and deacetylated cellulose acetate at low pH values. Other hydrophilic surfaces such as silicon dioxide or polyvinyl alcohol coated surfaces did not adsorb CMC to a significant extent. Atomic force microcopy confirmed that the morphology of the adsorbed CMC layers differed depending on the substrate. On hydrophobic polymer films, CMC was deposited in the form of larger particles in lower amounts whereas hydrophilic cellulose substrates were to a high extent uniformly covered by adsorbed CMC. The chemical similarity of the CMC backbone seems to favor the irreversible adsorption of CMC when the molecule is almost uncharged at low pH values. A selectivity of the cellulose CMC interaction can therefore be assumed. All CMC treated polymer films exhibited an increased hydrophilicity, which confirmed their modification with the functional molecule.     

Existence of weak solutions to doubly degenerate diffusion equations

We prove existence of weak solutions to doubly degenerate diffusion equations