On Transfer Operators for Continued Fractions with Restricted Digits

For any non-empty subset I of the natural numbers, let _I denotethose numbers in the unit interval whose continued fractiondigits all lie in I. Define the corresponding transfer operator for , where Re (rß) = _I is the abscissa of convergence of the series . When acting on a certain Hilbert space H_I, rß, weshow that the operator L_I, rß is conjugate to an integraloperator K_I, rß. If furthermore rß is real,then K_I, rß is selfadjoint, so that L_I, rß: H_I, rß H_I, rß has purely real spectrum.It is proved that L_I, rß also has purely real spectrumwhen acting on various Hilbert or Banach spaces of holomorphicfunctions, on the nuclear space C [0, 1], and on the Fréchetspace C [0, 1]. The analytic properties of the map rß L_I, rßare investigated. For certain alphabets I of an arithmetic nature(for example, I = primes, I = squares, I an arithmetic progression,I the set of sums of two squares it is shown that rß L_I, rß admits an analytic continuation beyond thehalf-plane Re rß > _I. 2000 Mathematics SubjectClassification 37D35, 37D20, 30B70.     

Molecular parameters describing intermolecular interactions in the electrooptical Kerr effect

Electrooptical Kerr effect has been studied in binary solutions of a dipolar liquid (-picoline, β-picoline) in a non-dipolar solvent (benzene, p-xylene) and 1,4-dioxane in the full range of concentrations of the dipolar component (0 ≤ f₂ ≤ 1). The experimental Kerr constant K_s of the solutions, refraction index n_s, density ρ_s and the dielectric constant ε_s have been measured and used for the calculation of the molar Kerr constants K_S^M within the Onsager local field model. Analysis of changes in the molar Kerr constants as a function of the solute concentration by fits of theoretical functions to the experimental ones has permitted a determination of the parameters characterizing intermolecular interactions in binary solutions.     

EPR and optical measurements of weakly doped LiNbO3: Er

Optical spectroscopy measurements of the congruent LiNbO₃ (LN) single crystals, weakly doped with Er (0.1–0.3 wt%) and Er (0.3 wt%) and Yb (0.5 wt%), have been carried out. The shape of the optical absorption and additional absorption bands registered after γ-irradiation suggests the presence of Er³⁺ ion pairs. EPR investigations were performed for LN single crystal doped with Er (0.1 wt%). Unusual behaviour of the temperature dependence of the intensity and linewidth of the main EPR line—corresponding to the fine transition of ^evenEr³⁺—ions, is reported. The main EPR line appears to be a superposition of several paramagnetic centres originating from isolated ^evenEr³⁺ ions and ^evenEr³⁺–^evenEr³⁺ pairs of ions. In low temperature region (below 20 K), the main EPR signal is dominated by signals arising from ^evenEr³⁺–^evenEr³⁺ pairs of ions. The inverse intensity of the EPR line at low temperature region fulfils the Curie–Weiss law and enabled to determine the Curie–Weiss constant Θ=1.5±0.5 K. The positive sign of Θ suggests that the ferromagnetic interactions arise in the system of ^evenEr³⁺–^evenEr³⁺ ion pairs in LN. Our results suggest that the distribution of Er ions in congruent LN is not homogeneous and Er impurity ions can form clusters in host lattice even in the case of weak erbium doping.     

Influence of rhodium additive on hydrogen electrosorption in palladium-rich Pd�CRh alloys

Hydrogen electrosorption into Pd-rich (>80?at.% Pd in the bulk) Pd?CRh alloys has been studied in acidic solutions (0.5?M H₂SO₄) using cyclic voltammetry and chronoamperometry. The influence of temperature (in the range between 283 and 328?K), electrode potential and alloy bulk composition on hydrogen electrosorption properties of Pd?CRh alloys is presented. It has been found that the additive of Rh to Pd?CRh alloys increases the maximum hydrogen solubility (for Rh bulk content below 10?at.%), decreases the potential of absorbed hydrogen oxidation peak and decreases the potential of the ???????? phase transition. Increasing temperature decreases the potential of absorbed hydrogen oxidation peak, the maximum hydrogen solubility, and the potential of the ???????? phase transition. The amounts of electrosorbed hydrogen for ??- and ??-phase boundaries, i.e., ??_max and ??_min, have been determined from the integration of the initial parts of current?Ctime responses in hydrogen absorption and desorption processes. The H/M ratio corresponding to ??_max increases with increasing Rh content, while for ??_min a maximum of H/M ratio is observed for the alloys containing ca. 95% Rh.     

Nuclear structure effects in the isotope shift with halo nuclei

Corrections to atomic energy levels due to nuclear structure effects are discussed. These are the finite nuclear size combined with relativistic and recoil corrections, and the nuclear polarizability. Good understanding of these effects is necessary for interpretation of high-precision measurements of the isotope shifts with neutron-rich nuclei ^6,8He, ¹¹Li and ¹¹Be. The summary of the results of the accurate atomic structure calculations is presented also.     

Relatively finite measure-preserving extensions and lifting multipliers by Rokhlin cocycles

We show that under some natural ergodicity assumptions, extensions given by Rokhlin cocycles lift the multiplier property if the associated locally compact group extension has only countably many L^∞-eigenvalues. We make use of some analogs of basic results from the theory of finite-rank modules associated to an extension of measure-preserving systems in the setting of a non-singular base.     

The equilibrium states for semigroups of rational maps

We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψ and a unique exp(P(ψ) − ψ)-conformal measure, where P(ψ) denotes the topological pressure of ψ. The research of M. Urbański was supported in part by the NSF Grant DMS 0400481. H. Sumi thanks University of North Texas for kind hospitality, during his stay there.     

Crystal structure of a novel synthetic inhibitor of HIV-1 protease

The crystal structure of a novel non-peptidic HIV-1 protease inhibitor derived by simple solid-state dimerization of 4-aryl-1,4-dihydropyridines, reveals a strained central cage and the conformation of its phenyl, benzyl, and hydroxymethylene substituents. The polycyclic cage includes two nearly flat cyclobutane rings and four fused piperidine rings in boat conformations. The cage geometry reveals two unexpected features, namely marked distortions of the valence angles in every second piperidine and a shortening of one of the cyclobutane bonds. The molecule displays exact centrosymmetry, but the central cage and the hydroxymethylene substituents also approximate the C₂-symmetry of the target enzyme. The two independent hydroxyl groups are involved in intermolecular hydrogen bonding, one as a donor, the other as an acceptor. The disposition of the hydroxyl groups in the molecular framework is compatible with the dual role of the inhibitor in the active-site cavity of HIV-1 protease, whereby one OH group is hydrogen-bonded to the catalytic aspartates, whereas another one provides an interface to the locked flaps of the enzyme.     

Coordination ability of insect kinin analogs

Spectroscopic data, including electronic absorption, CD and EPR results, as well as theoretical calculations have shown that the insertion of 4-aminopyroglutamate, a novel cis-peptide bond mimic, in the insect kinin peptide leads to an effective ligand towards Cu(II) ions at basic pH ranges. The 4-aminopyroglutamate motif induces a specific peptide conformation which favors the formation of one or two five-membered chelating rings stabilizing a bent structure, with the coordination of 3N-type or 4N-type in the metallopeptide molecule involving in metal bonding C-terminal of modified peptide. It is worth noting that the copper(II) bonding does not undergo hydrolysis even at a very high pH range.