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.     

Monotonicity and best approximation in Orlicz-Sobolev spaces with the Luxemburg norm

Criteria for strict monotonicity, upper (lower) locally uniform monotonicity and uniform monotonicity of Orlicz-Sobolev spaces with the Luxemburg norm are given. Some applications to best approximation are presented.     

Two 1-substituted 4-nitro­imidazoles

Crystalline 4-nitro-1-phenyl­imidazole, C₉H₇N₃O₂, (I), and 4′-­nitro-1-phenyl-4,1′-bii­imidazole, C₁₂H₉N₅O₂, (II), contain C—H⃛O and C—H⃛N hydrogen bonds, connecting the mol­ecules into infinite chains. The aromatic fragments in both compounds are nearly planar. The dihedral angles between the benzene and imidazole rings are 26.78 (5)° in (I) and 29.36 (8)° in (II).     

Caesium and hydrated lead chlorates studied by35Cl NQR spectroscopy

Molecular dynamics of caesium and lead chlorates (hydrated form) has been studied by³⁵Cl nuclear quadrupole resonance (NQR) spectroscopy. The occurrence of only one resonance line for the two compounds in the whole temperature range studied testifies to the crystallographically equivalent positions of their molecules in elementary cells. The relaxation processes in the two chlorates have been found to be analogous to those in the other chlorates not undergoing phase transitions. Analysis of the temperature dependence of the spin-lattice relaxation time has indicated that at low temperatures the dominant relaxation mechanism consists of torsion vibrations (ClO₃⁻ anion libration), whereas at 260 K the hindered rotation of the chlorate ion about the threefold axis sets on and its activation energy is 50.0 kJ/mol. The temperature dependence of the resonance frequency for the two chlorates studied is best described by the Brown model extended over the low-temperature range, which indicates large anharmonicity of the vibrations.     

Formally smooth bimodules

The notion of a formally smooth bimodule is introduced and its basic properties are analyzed. In particular it is proven that a B-A bimodule M which is a generator left B-module is formally smooth if and only if the M-Hochschild dimension of B is at most one. It is also shown that modules M which are generators in the category σ[M] of M-subgenerated modules provide natural examples of formally smooth bimodules.     

Equipment for digital recording of transient signals

The instrumentation for digital recording of transient signals was built and incorporated into a system for pulse radiolysis experiments. A single analog signal which is displayed on the screen of the oscilloscope is captured by a TV-camera in the measuring set up. The camera is connected to a Camac crate which is directly linked to a microcomputer. The analog signal can be digitized and memorized in 128×256 matrix points.     

Insight into the mechanism of three component condensation leading to aminomethylenebisphosphonates

Three-component reaction of a primary amine, diethyl phosphite and triethyl orthoformate followed by acid hydrolysis of the adduct provides N-substituted aminomethylenebisphosphonic acids in good yields. Being extremely versatile, it is commonly utilized for preparation of compounds possessing potential antiosteoporotic, antibacterial, anticancer, antiparasitic or herbicidal activity. However, the mechanism of the reaction remains unknown. p-Nitroaniline has been found an interesting tool to shed light on this matter. Its use allowed to separate and identify four intermediates, both non-phosphorus and phosphorus containing, and subsequently suggest the mechanism of the whole process. The acquired knowledge was helpful in explanation the route and the final product structure obtained for more complex reaction proceeding with the use of 4-aminopyridine. Additional alkylation of the pyridine nitrogen atom, leading to unexpected N-(1-alkylpyridinium-4-amino)methylenebisphosphonic acids was unambiguously proved.     

Modes in the Fabry-Perot resonator with the Kerr nonlinearity

Summary The symmetric modes in the cavity of Fabry-Perot type filled by the medium with the Kerr nonlinearity are constructed. The field appears to be double periodic along the longitudinal axis.

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Riassunto Si costruiscono i modi simmetrici nella cavità del tipo di Fabry-Perot riempita dal mezzo con la non linearità di Kerr. Il campo risulta doppiamene periodico lungo l’asse longitudinale.

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Резюме Конструируются симметричные моды в резонаторе Фабри-Перо, заполненном средой с нелинейностью Керра. Оказывается, поле имеет двойную периодичность вдоль продольной оси.

     

Syntheses and reactions of terpene β-hydroxyselenides and β-hydroxydiselenides

A convenient method for the synthesis of optically active trans-hydroxyselenides and trans-hydroxydiselenides from the bicyclic terpene group based on the reactions of sodium selenide or sodium diselenide with cis- and trans-(+)-3-carane, trans-(+)-2-carane and (?)-β-pinane epoxides is described. The corresponding cis-hydroxy and cis-methoxydiselenides were obtained in the reaction of sodium diselenide with β-hydroxy- and β-methoxytosylates. The influence of a hydroxy group at the β-position on the diastereomeric ratio of the products of the asymmetric methoxyselenenylation of styrene has been established by composition of the products, crystal structure analyses, and theoretical calculations using a DFT method on the B3LYP level (6-311G(d)).