Inequalities for the generalized trigonometric and hyperbolic functions

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p$p$-Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinovi?–Adamovi?’s inequality, Lazarevi?’s inequality, Huygens-type inequalities, Wilker-type inequalities, and Cusa–Huygens-type inequalities, are generalized to the case of generalized functions.     

Functional inequalities for the Teichmüller ring capacity

The geometric study of the action of quasiconformal mappings by means of conformal invariants often leads to inequalities for special functions. Examples of such special functions are the capacities of the Grötzsch and Teichmüller rings. Several new inequalities for these and related functions are given in the multidimensional case.     

Quantitative assessment of the enzymatic degradation of amorphous cellulose by using a quartz crystal microbalance with dissipation monitoring

The systematic evaluation of the degradation of an amorphous cellulose film by a monocomponent endoglucanase (EG I) by using a quartz crystal microbalance with dissipation monitoring (QCM-D) identified several important aspects relevant to the study the kinetics of cellulose degradation by enzymes. It was demonstrated that, to properly evaluate the mechanism of action, steady state conditions in the experimental set up need to be reached. Rinsing or diluting the enzyme, as well as concentration of the enzyme, can have a pronounced effect on the hydrolysis. Quantification of the actual hydrolysis was carried out by measuring the film thickness reduction by atomic force microscopy after the enzymatic treatment. The values correlated well with the frequency data obtained by QCM-D measurement for corresponding films. This demonstrated that the evaluation of hydrolysis by QCM-D can be done quantitatively. Tuning of the initial thickness of films enabled variation of the volume of substrate available for hydrolysis which was then utilized in establishing a correlation between substrate volume and hydrolytic activity of EG I as measured by QCM-D. It was shown that, although the amount of substrate affects the absolute rate of hydrolysis, the relative rate of hydrolysis does not depend on the initial amount of substrate in steady state system. With this experimental setup it was also possible to demonstrate the impact of concentration on crowding of enzyme and subsequent hydrolysis efficiency. This effort also shows the action of EG I on a fully amorphous substrate as observed by QCM-D. The enzyme was shown to work uniformly within the whole volume of swollen film, however being unable to fully degrade the amorphous film.     

Extremal problems and symmetrization for plane ring domains

We show that Teichmüller's classical lower bound for the capacity of a ring domain, obtained by circular symmetrization, can be replaced by an explicit one which is almost always better. The proof is based on a duplication formula for the solution of an associated extremal problem. Some inequalities are obtained for conformal invariants.

     

Some remarks on an existence problem for degenerate elliptic systems

The system which yields Beltrami's system if , is considered. Under a condition for the coefficients a non-existence theorem is proved.

     

Metrization of Weighted Graphs

We find a set of necessary and sufficient conditions under which the weight ${w: E \rightarrow \mathbb{R}^{+}}$ on the graph G = (V, E) can be extended to a pseudometric ${d : V \times V \rightarrow \mathbb{R}^{+}}$ . We describe the structure of graphs G for which the set ${\mathfrak{M}_{w}}$ of all such extensions contains a metric whenever w is strictly positive. Ordering ${\mathfrak{M}_{w}}$ by the pointwise order, we have found that the posets $({\mathfrak{M}_{w}, \leqslant)}$ contain the least elements ρ _0,w if and only if G is a complete k-partite graph with ${k \, \geqslant \, 2}$ . In this case the symmetric functions ${f : V \times V \rightarrow \mathbb{R}^{+}}$ , lying between ρ _0,w and the shortest-path pseudometric, belong to ${\mathfrak{M}_{w}}$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two.     

Comparative study of photodegradation of wood by a UV laser and a xenon light source

A detailed study of photodegradation of wood surfaces by xenon light source and a UV laser has been carried out. Silver birch, rubberwood, Scots pine and chir pine wood veneers were irradiated with a xenon light source or a 244 nm argon ion laser. The changes in chemical structure of wood surfaces were monitored by UV resonance Raman (UVRR), photoacoustic Fourier transform infrared (FTIR-PAS) and UV-vis reflectance spectroscopies. The depth profile of xenon lamp irradiated wood surfaces was carried out by measuring FTIR-PAS spectra at different moving mirror velocities. The UVRR and FTIR-PAS spectra of irradiated wood surfaces showed degradation of aromatic structure in lignin combined with strong formation of carbonyl structures. The FTIR-PAS spectra measured from xenon irradiated wood surfaces indicate that hardwood lignin degrades at a faster rate than softwood lignin. The UVRR spectra of xenon irradiated wood show a significant decrease in the intensities of aromatic structures at 1602 cm⁻¹. This is accompanied by a significant band broadening and notable shift towards longer wavenumbers, which has been attributed to the formation of o- and p-quinone structures as degradation products. The formation of quinone structures was also supported by the generation of a broad absorption band between 350 and 600 nm in UV-vis reflectance spectra of irradiated wood surfaces. There was a significant broadening in the region of 1500-1000 cm⁻¹ in UVRR spectra due to the formation of unsaturated structures as a result of lignin degradation. The UVRR spectra of laser irradiated wood showed similar behaviour i.e., overall broadening and a rapid reduction in the intensity of lignin aromatic structure. The rate of degradation by laser was very high. However, the extent of band broadening was higher in xenon irradiated wood indicating the generation of several different types of structures as compared to laser irradiation, which produces only particular type of structures. UVRR spectra of laser irradiated Whatman paper showed significant photodegradation of cellulose by UV laser. The UV degradation rate of lignin was much higher than cellulose.