Local Fixed Point Indices of Iterations of Planar Maps

Let \({f: U\rightarrow {\mathbb R}^2}\) be a continuous map, where U is an open subset of \({{\mathbb R}^2}\). We consider a fixed point p of f which is neither a sink nor a source and such that {p} is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations \({\{{\rm ind}(f^n,p)\}_{n=1}^\infty}\) is periodic, bounded from above by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem (Annals of Math., 146, 241–293 (1997)) onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional sphere.     

Analysis of dynamic pull-in voltage and response time for a micro-electro-mechanical oscillator made of power-law materials

Nonlinear Dynamics - Analysis of dynamic pull-in voltage for a micro-electro-mechanical oscillator of platform type is performed. The lumped mass model for the actuated micro-cantilever beam made...     

Re-examination of Nucleophilic Substitutionin Chlorokojic Acid

Summary.  Chlorokojic acid was reacted with S₂O₃ ²⁻, NO₃ ⁻, N₃ ⁻, I⁻, and SCN⁻. Only the three latter nucleophiles substituted the chlorine atom in the 2-CH₂Cl group of kojic acid. In none of the products nucleophilic substitution at position 6 of the 4-pyrone could be found. Regular substituion of chlorokojic acid with I⁻ (iodokojic acid), N₃ ⁻ (azidokojic acid), and SCN⁻ (thiocyanato and isothiocyanato kojic acids) was accompanied by formation of allomaltol. Reaction pathways for the formation of allomaltol and 6-substituted allomaltol derivatives are proposed. The latter has been formerly discovered in the reaction of chlorokojic acid with secondary amines. Received October 10, 1999. Accepted (revised) November 25, 1999     

Relative Entropy in Determining Degressive Proportional Allocations

The principle of degressively proportional apportionment of goods, being a compromise between equality and proportionality, facilitates the application of many different allocation rules. Agents with smaller entitlements are more interested in an allocation that is as close to equality as possible, while those with greater entitlements prefer an allocation as close to proportionality as possible. Using relative entropy to quantify the inequity of allocation, this paper indicates an allocation that neutralizes these two contradictory approaches by symmetrizing the inequities perceived by the smallest and largest agents participating in the apportionment. First, based on some selected properties, the set of potential allocation rules was reduced to those generated by power functions. Then, the existence of the power function whose exponent is determined so as to generate the allocation that symmetrizes the relative entropy with respect to equal and proportional allocations was shown. As a result, all agents of the apportionment are more inclined to accept the proposed allocation regardless of the size of their entitlements. The exponent found in this way shows the significant relationship between the problem under study and the well-known Theil indices of inequality. The problem may also be seen from this viewpoint.     

Formation and stability of poly-L-lysine/casein multilayers

The aim of our work was to investigate formation of multilayer films containing biocompatible polycation poly-L-lysine (PLL) and α- or β-casein. Since in the neutral pH casein is negatively charged, it has been used as a polyanionic layer for the film build-up. Casein containing films were formed at surface of Si/SiO₂ wafers and their thickness was measured by ellipsometry. The effect of ionic strength of PLL and casein solutions was investigated. After the multilayer films were formed, they were contacted with solutions having various pH and salts to determine film stability under these conditions. Additionally the response of the thickness of PLL/casein films to the temperature variation in the range of 5–45?°C was also analyzed. Formation and stability of casein containing films was also investigated on surfaces of titanium and stainless steel. We used fluorescently labeled protein to monitor the amount of casein in the film and its change after treatment with solutions containing calcium ions.     

An effective scaling frequency factor method for scaling of harmonic vibrational frequencies: Application to 1,2,4-triazole derivatives

Scaling of harmonic frequencies of a molecule is one of the methods of improving the agreement between the calculated from a quadratic force field and experimental vibrational spectrum. An application of the recently proposed effective scaling frequency factor (ESFF) method to the complicated 1,2,4-triazole derivatives is presented. The calculations are based on the DFT/B3LYP/6-311G** quadratic force fields. It is shown that the ESFF method is capable of providing the high-quality spectra with regard to the scaled frequencies, comparable to these obtained with the well-established scaled quantum mechanical (SQM) force field method. Using the recommended scaling factors for the 11-parameter calculations, the RMS value obtained for a set of 293 vibrational modes of four compounds is only 8.7 and 8.5 cm^?1, for SQM and ESFF, respectively, provided the hydrogen bonded CO bond was excluded from the general non-hydrogen XX stretch group, and the scaling factor attributed to this bond was optimized. The new, 9-parameter set of scaling factors provides SQM- and ESFF-scaled frequencies that are of comparable quality to those of the 11-parameter calculations. In addition, it provides (on average) more reliable band splittings in the middle region of the spectrum, and the order of the scaled frequencies corresponds to that of the experimental bands. The straightforward application of the ESFF method to estimate the value of the scaled frequency is also presented.     

Assessment of the content of phenolics and antioxidant action of inflorescences and leaves of selected species from the genus Sorbus sensu stricto

In order to find new sources of natural antioxidants, the antioxidant potential of 70% methanolic extracts from the inflorescences and leaves of 16 species from the genus Sorbus s.s. was evaluated using two complementary in vitro test systems: the DPPH (2,2-diphenyl-1-picrylhydrazyl) radical-scavenging assay and the AAPH [2,2￠-azobis-(2-amidinopropane)dihydrochloride]-induced linoleic acid (LA) peroxidation test. The radical-scavenging capacities of the extracts towards the DPPH radical were in the range of 0.25-0.86 millimolar Trolox? equivalents/g dry weight. They were significantly correlated (r=-0.8089, p<0.001) with the results of the LA-peroxidation test, indicating the Sorbus extracts to be universal antioxidants. Significant linear correlations were also found between the different antioxidant potentials and total phenolic contents as estimated by the Folin-Ciocalteu method and further verified by serial determinations of proanthocyanidins, chlorogenic acid isomers and flavonoids (?rêin the range of 0.71-0.95, p<0.001). Cluster analysis of the data matrix identified the ten samples (inflorescences of S. aucuparia, S. pohuashanensis, S. decora, S. koehneana, S. commixta, S. gracilis, and S. sitchensis, and the leaves of S. wilfordii, S. pogonopetala, and S. gracilis) exhibiting the highest antioxidant activity and total phenolic levels and therefore the greatest potential as effective sources for natural health products.