On the zero-stability of multistep methods on smooth nonuniform grids

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid \(\{t_n\}_{n=0}^N\) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., \(t_n = {\varPhi }(\tau _n)\), where \(\tau _n = n/N\) and the map \({\varPhi }\) is monotonically increasing with \({\varPhi }(0)=0\) and \({\varPhi }(1)=1\). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines \({\varPhi }\), and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an \(N^*\) such that the method is zero stable for \(N>N^*\), provided that \({\varPhi }\in C^2[0,1]\). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy \(h_n/h_{n-1} = 1 + {\mathrm {O}}(N^{-1})\) as \(N\rightarrow \infty \). The results are exemplified for BDF-type methods.     

A stability property of the densest circle packing

A new stability property of the densest circle packing in the plane is proved. This property is related to a conjecture ofL. Fejes Tóth.Research of this author was partially supported by Hungarian National Foundation for Scientific Research grant no. 1812.     

Generalized moments of additive functions

Elliott's generalization of the Turán-Kubilius inequality is further generalized by establishing an upper bound for the sum Σ_n≤xF(∣f(n) ? A∣), where f is a complex-valued additive arithmetical function, A an arbitrary number and F an arbitrary nonnegative-valued increasing function. A connected problem for group-valued functions is also considered.     

Improved quantitative thin-layer chromatographic method for the separation of cobalamin derivatives

Summary A new quantitative thin-layer chromatographic method was developed for the separation and determination of cobalamin derivatives. A mixture of acetone, acetonitrile, isopropanol, diethylamine and aqueous ammonium hydroxide as the eluent and precoated Kieselgel 60 chromatoplate as the TLC plate was used. The evaluation of the separated spots was carried out by densitometry. The method can be used for the determination of cyanocobalamin in fermentatic mixture without any cleaning or concentration procedure as well as for the purity test of cyanocobalamin and hydroxocobalamin.     

Positron annihilation in non-simple liquids. Normal hexane

Positron lifetime measurements were carried out in normal hexane vs. temperature in the range of 12.5–62.5°C. The evaluation of the lifetime spectra indicates the presence of four lifetime components, as published first by Jacobsen et al. An attempt is made to interprete the properties of the longest-living two components in terms of the microscopic structure of the liquid.     

Kegelschnitte auf der metrischen Ebene

Ohne ZusammenfassungHerrn Professor Ferenc Kárteszi zum 70-sten Geburtstag gewidmet     

Photocatalytic degradation of hydrocarbons by bentonite and TiO2 in aqueous suspensions containing surfactants

Organic substances (toluene, asphaltene dissolved in toluene) poorly soluble in water were solubilized by sodium dodecylsulfate (SDS) and the organic pollutants were mineralized together with the surfactants in diluted TiO₂ suspension by UV light. In concentrated aqueous suspension, crude oil was photodegraded on the surface of Na-bentonite and the photocatalyst TiO₂. Experiments have been carried out with Na-bentonite contaminated with crude oil mixed in a ratio of 1:1 with TiO₂ as well. The photooxidation processes were followed by total organic carbon content (TOC) and diffuse reflection infrared spectroscopy (DRIFT) measurements.     

Electronic properties and atomic structure of (Ti, Zr, Hf)---(Ni, Cu) metallic glasses

The available experimental and theoretical data on the electronic density of states at the Fermi level, n(E_F), of non-magnetic melt-quenched TE_100-xTL_x amorphous alloys (Te = Ti, Zr or Hf; TL = Ni or Cu) and for the corresponding crystalline counterparts is reviewed. First, the low-temperature specific heat and superconductive data are summarized, in order to derive experimental n(E_F) values. A comparison with n(E_F) data from theoretical band structure calculations shows good overall agreement, both qualitative and quantitative. By taking into account recent results on the different structural modifications of Ti, Zr and Hf metals, an extrapolation of the composition dependence of n(E_F) in TE-TL glasses to pure amorphous TE metals suggests that the local structure of these TE-TL amorphous alloys may be described by an fcc-like atomic arrangement. Some other reported results, which supply further evidence for the proposed fcc-like local structure in this type of metallic glass, are also discussed.     

Polaronic effects on interconfigurational fluctuation lifetimes in intermediate valence systems

It is shown that in systems exhibiting intermediate valence polaronic effects can lead to significant reductions in interconfiguration fluctuation frequency and to a shift in the electronic excitation spectrum. Numerical estimates for SmS give a shift in good accord with recent experiments and indicate a non-negligible lifetime enhancement factor.