Linear differential equations with coefficients in weighted Bergman and Hardy spaces

Complex linear differential equations of the form

with coefficients in weighted Bergman or Hardy spaces are studied. It is shown, for example, that if the coefficient of belongs to the weighted Bergman space , where , for all , then all solutions are of order of growth at most , measured according to the Nevanlinna characteristic. In the case when all solutions are shown to be not only of order of growth zero, but of bounded characteristic. Conversely, if all solutions are of order of growth at most , then the coefficient is shown to belong to for all and .

Analogous results, when the coefficients belong to certain weighted Hardy spaces, are obtained. The non-homogeneous equation associated to is also briefly discussed.

     

Topochemical modification of cotton fibres with carboxymethyl cellulose

Adsorption of carboxymethyl cellulose (CMC) as a method to introduce charged (ionizable) groups onto cellulose cotton fibre surfaces was investigated. The method was based on application of a previously published method used for wood fibres. The amount of adsorbed ionizable groups was determined indirectly by analysis of CMC in solution by the phenol–sulphuric acid method and directly by conductometric titration of the fibres. Results from the two methods correlated well. The molecular weight and purity of the CMC had an influence on its adsorption onto cotton; high molecular weight CMC was preferentially adsorbed. The adsorbed charge correlated linearly with the amount of CMC adsorbed. The total charge of the cotton fibres could be increased by more than 50% by adsorption of CMC. It is expected that this modification procedure can be used in a wide spectrum of practical applications. Lidija Fras Zemljič and Karin Stana-Kleinschek are the members of the European Polysaccharide Network of Excellence (EPNOE).     

Photocycloaddition Reactions of 2‐(Alk‐3‐en‐1‐ynyl)cyclohex‐2‐enones

The newly synthesized 2‐(alk‐3‐en‐1‐ynyl)cyclohex‐2‐enones 4 undergo photodimerization (chemo‐ and regio‐)selectively at the exocyclic C?C bond to give diastereoisomeric mixtures of 1,2‐dialkynyl‐1,2‐dimethylcyclobutanes. On irradiation of 4 in the presence of 2‐chloroacrylonitrile, cyclobutane formation occurs again (chemo‐ and regio‐)selectively at the exocyclic C?C bond to afford diastereoisomeric mixtures of 2‐alkynyl‐1‐chloro‐2‐methylcyclobutanecarbonitriles. Similarly, compounds 4 undergo photoaddition to 2,3‐dimethylbuta‐1,3‐diene exclusively at the exocyclic C?C bond to afford mixtures of [2+2] and [4+2] cycloadducts.     

Kinetic energy operators in linearized internal coordinates

It is customary to describe molecular vibrations using as exact kinetic energy operators and as accurate potentials as possible. It has become a standard approach to express Hamiltonians in curvilinear internal displacement coordinates, because they offer a simple and physical picture of vibrational motions, including large amplitude changes in the shape. In the older normal mode model of molecular vibrations, the nuclei are thought to vibrate infinitesimally about the reference configuration, and the shape of the molecule is described using linearized approximations of the true geometrically defined internal displacement coordinates. It is natural to ask how the two approaches are related. In this work, I present a general yet practical way to obtain curvilinear displacement coordinates as closed function of their linearized counterparts, and vice versa. In contrast to the conventional power series approach, the body-frame dependency is explicitly taken into account, and the relations are valid for any value of the coordinates. The present approach also allows one to obtain easily exact kinetic energy operators in linearized shape coordinates.     

Self‐Assembly of the Cyclic Lipopeptide Daptomycin: Spherical Micelle Formation Does Not Depend on the Presence of Calcium Chloride

The cyclic lipopeptide Daptomycin, used as a treatment for infections where antimicrobial resistance is observed, is shown to self‐assemble into spherical micelles above a critical aggregation concentration. Micelles are observed either in the absence or presence of CaCl₂, in contrast to claims in the literature that CaCl₂ is required for micellization.     

Hierarchical Layer Engineering Using Supramolecular Double‐Comb Diblock Copolymers

The formation of unusual multilayered parallel lamellae‐in‐lamellae in symmetric supramolecular double‐comb diblock copolymers is presented. While keeping the concentration of surfactant fixed, the number of internal layers was found to increase with molecular weight M up to 34 for the largest block copolymer. The number of internal structures n was established to scale as M^0.67 and therefore enables easy design of such structures with great precision.     

Cellulose nanocrystal submonolayers by spin coating

Dilute concentrations of cellulose nanocrystal solutions were spin coated onto different substrates to investigate the effect of the substrate on the nanocrystal submonolayers. Three substrates were probed: silica, titania, and amorphous cellulose. According to atomic force microscopy (AFM) images, anionic cellulose nanocrystals formed small aggregates on the anionic silica substrate, whereas a uniform two-dimensional distribution of nanocrystals was achieved on the cationic titania substrate. The uniform distribution of cellulose nanocrystal submonolayers on titania is an important factor when dimensional analysis of the nanocrystals is desired. Furthermore, the amount of nanocrystals deposited on titania was multifold in comparison to the amounts on silica, as revealed by AFM image analysis and X-ray photoelectron spectroscopy. Amorphous cellulose, the third substrate, resulted in a somewhat homogeneous distribution of the nanocrystal submonolayers, but the amounts were as low as those on the silica substrate. These differences in the cellulose nanocrystal deposition were attributed to electrostatic effects: anionic cellulose nanocrystals are adsorbed on cationic titania in addition to the normal spin coating deposition. The anionic silica surface, on the other hand, causes aggregation of the weakly anionic cellulose nanocrystals which are forced on the repulsive substrate by spin coating. The electrostatically driven adsorption also influences the film thickness of continuous ultrathin films of cellulose nanocrystals. The thicker films of charged nanocrystals on a substrate of opposite charge means that the film thickness is not independent of the substrate when spin coating cellulose nanocrystals in the ultrathin regime (<100 nm).     

Surface modification and characterization of thermoplastic polyurethane

This work reports the modification of thermoplastic polyurethanes (TPUs) in order to enlarge their application range, for example, as biomaterials by increasing its hydrophilicity.A TPU was successfully modified by using three different strategies: ultra-violet irradiation (UV), gamma irradiation (GI) and interfacial modification (IM). The results suggested the possibility of modifying the polyurethane-based surface either with poly(ethylene glycol) (PEG) or hydroxylethyl methacrylate (HEMA) or hexamethylene diamine (HMD) or chitosan (CT) by using any of these methods. The properties of the grafted PU were evaluated by surface, structural and thermal analysis. The results suggest that, among the methods studied in this work, the modification by gamma irradiation (GI) seems to be the most promising, since this method gives high values of grafting yield and has the advantage of providing a clean modification, meaning that no initiator is needed.     

Spatial distribution and decrease of dye solar cell performance induced by electrolyte filling

The spatial performance variation of dye solar cell with standard liquid electrolyte was examined by dividing the cell into segments. Surprisingly large and permanent performance differences were found in different parts of the cell leading to significant losses in the overall cell efficiency. The decrease of open circuit voltage along the electrolyte filling direction suggests that 4-tert-butylpyridine is adsorbed non-uniformly as the electrolyte passes through the dyed TiO₂ layer during the filling process. The result indicates that non-uniform electrolyte adsorption may limit the up-scaling of dye solar cells, which calls for the examination of electrolyte filling techniques and electrolyte compositions less prone to this effect.     

Carboxymethyl cellulose on a fiber substrate: the interactions with cationic polyelectrolytes

High and low molecular weight (M_w) carboxymethyl celluloses (CMC) were adsorbed on a well-characterized fiber substrate (long fibers of a commercial bleached birch kraft pulp with the carboxylic acid groups in Na-form) to increase the charge of the fibers in a controlled fashion. The M_w played a role in the utilization of CMCs as a strength additive in paper sheets nearly doubling the tensile strength with the high M_w CMC. Swelling properties of the CMC treated fibers were measured with water retention value (WRV). The WRV increased more with the high M_w CMC. The swelling was further tuned by two highly cationic polyelectrolytes; high M_w poly(diallyldimethyl ammonium chloride) (PDADMAC) and low M_w polybrene (hexadimethrine bromide, [3,6]-ionene). They were chosen because of their known ability to neutralize the anionic charge either exclusively on the surface or in the whole fiber, respectively. Adsorption of PDADMAC could reduce WRV of the CMC pre-treated fibers to the level of the untreated reference, while polybrene adsorbed pulps with 3–10 times more cationic polyelectrolyte deswelled the fibers only slightly more than the surface neutralized fibers. These results indicated surface conformation differences with low and high M_w CMCs. While the conformation did play a role after physical alteration (drying and rewetting) of the fibers, the paper sheets produced from these fibers showed remarkable differences. In extreme cases, the strength of the paper could be retained after drying (low M_w CMC + PDADMAC) or paper, resistant to disintegration, could be achieved (CMC + polybrene).