There are infinitely many limit points of the fractional parts of powers

Suppose that α > 1 is an algebraic number and ξ > 0 is a real number. We prove that the sequence of fractional partsξα ⁿ , n = 1, 2, 3, …, has infinitely many limit points except when α is a PV-number and ξ ∈ ℚ(α). For ξ = 1 and α being a rational non-integer number, this result was proved by Vijayaraghavan.     

Convergence of multistep discretizations of DAEs

Standard ODE methods such as linear multistep methods encounter difficulties when applied to differential-algebraic equations (DAEs) of index greater than 1. In particular, previous results for index 2 DAEs have practically ruled out the use of all explicit methods and of implicit multistep methods other than backward difference formulas (BDFs) because of stability considerations. In this paper we embed known results for semi-explicit index 1 and 2 DAEs in a more comprehensive theory based on compound multistep and one-leg discretizations. This explains and characterizes the necessary requirements that a method must fulfill in order to be applicable to semi-explicit DAEs. Thus we conclude that the most useful discretizations are those that avoid discretization of the constraint. A freer use of e.g. explicit methods for the non-stiff differential part of the DAE is then possible.Dedicated to Germund Dahlquist on the occasion of his 70th birthdayThis author thanks the Centro de Estadística y Software Matemático de la Universidad Simón Bolivar (CESMa) for permitting her free use of its research facilities.Partial support by the Swedish Research Council for Engineering Sciences TFR under contract no. 222/91-405.     

Quantitative characterization of chaotic instabilities in semiconductors

A method for quantitative characterization of chaotic dynamical systems is discussed. An electronic instrument for determining the number of independent variablesk ^*, involved in the motion, is described. It allows one to obtain these in real time from a single observable. The suggested technique has been applied to quantification of strange attractors underlying chaotic instabilities in semi-insulating GaAsCr, and n-Ge, irradiated with high energy electrons. In n-Ge, for instance, the measured numbersk ^* range from 2 to 4 depending on control parameters. These measurements reveal the highly deterministic nature of the observed chaotic oscillations. The physical mechanisms responsible for the current instabilities and chaotic behaviour are discussed.     

Recyclization of 1,4-dihydropyridine derivatives in acidic medium

The recyclization of 1,4-dihydropyridines in aqueous-alcoholic hydrochloric acid medium proceeds with cleavage of a C-N bond and pyridine ring opening. Cyclohexenone derivatives are formed as a result of the subsequent intramolecular crotonic condensation of the acyclic intermediate. The leaving carbonyl substituents depart simultaneously with recyclization, depending on the acidity of the reaction medium. Dedicated to Prof. Dr. E. Lukevics on the occasion of his 70th birthday. __________ Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 1, pp. 49–58, January, 2007.     

A Study of Amorphous Precursors for PbZrO3-PbTiO3 Based Ceramic Materials

Amorphous precursors for PbZrO₃ and PbTiO₃ ceramics were prepared from lead acetate and the transition metaln-propoxide inn-propanol orn-butoxide inn-butanol and hydrolysed with an excess of water. According to GLC and TGA/EGA analyses, the type of alkoxide group influences distinctly the structure of heterometallic precursors, i.e., oxo or acetate bridging, and the amounts of hydroxyl and organic groups bound to the metal network. The local environments of metal atoms in the amorphous precursors were also studied by EXAFS. The analysis reveals that in Pb−Zr precursors alkoxide groups modify the coordination spheres of the zirconium atoms. Conversely, local environments of both lead and titanium atoms within the analysed range of 3.4 A depend weakly on the type of alkoxide used.     

Catalytic,asymmetric Mannich-type reactions of alpha-imino esters bearing readily removable substituents on nitrogen

[reaction: see text] Catalytic, enantioselective Mannich-type reactions of alpha-imino esters bearing readily removable substituents on nitrogen are described. Several N-carbamate-protected alpha-imino esters, which are readily prepared from 2-bromoglycine esters using a polymer-supported amine, reacted with silicon enolates to afford the desired adducts in high yields with high enantioselectivity using a copper(II)-diamine complex. Easy deprotection of the product amine and transformation to free alpha-amino acid derivatives have also been demonstrated.     

Transition metal salts-catalyzed aza-Michael reactions of enones with carbamates

Several transition metal salts were found to catalyze aza-Michael reactions of enones with carbamates efficiently. The catalytic activity was strongly dependent on the nature of the metal salts. While conventional Lewis acids such as BF(3).OEt(2), AlCl(3), or TiCl(4) showed lower activity, group 7-11 transition metal salts in higher oxidation states such as ReCl(5), Fe(ClO(4))(3).9H(2)O, RuCl(3).nH(2)O, OsCl(3).3H(2)O, RhCl(3).nH(2)O, PtCl(4).5H(2)O, or AuCl(3).H(2)O exhibited higher catalytic activity. [reaction: see text]     

Novel dye-attached macroporous films for cadmium, zinc and lead sorption: Alkali Blue 6B-attached macroporous poly(2-hydroxyethyl methacrylate)

Alkali Blue 6B-attached poly(2-hydroxyethyl methacrylate) (poly(HEMA)) microporous films were investigated as chelate forming sorbents for heavy metal removal. Poly(HEMA) microporous films were prepared by UV-initiated photo-polymerization of HEMA in the presence of an initiator (azobisisobutyronitrile (AIBN)). Alkali Blue 6B was attached covalently. These films with a swelling ratio of 58%, and carrying 14.8 mmol Alkali Blue 6B m(-2) which were then used in the removal of Cd(II), Zn(II) and Pb(II) from aqueous media. Adsorption rates were very high, equilibrium was achieved in about 30 min. The maximum adsorption of heavy metal ions onto the Alkali Blue 6B-attached films were 41.4 mmol m(-2) for Cd(II), 52.4 mmol m(-2) for Zn(II), and 64.5 mmol m(-2) for Pb(II). When the heavy metal ions competed during the adsorption from a mixture the adsorption values for Cd(II), Zn(II) and Pb(II) were quite close. Heavy metal ions were desorbed by using 0.1 M HNO(3). A significant amount of the adsorbed heavy metal ions (up to 95%) could be desorbed in 30 min. Repeated adsorption/desorption cycles showed the feasibility of these novel dye-attached microporous films for heavy metal removal.