Phase transitions for random walk asymptotics on free products of groups

Suppose we are given finitely generated groups Γ₁,…,Γ_m equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product Γ₁* … *Γ_m and give a complete classification of the possible asymptotic behaviour of the corresponding n‐step return probabilities. They either inherit a law of the form ?^nδn log n from one of the free factors Γ_i or obey a ?^nδn^?3/2‐law, where ? < 1 is the corresponding spectral radius and δ is the period of the random walk. In addition, we determine the full range of the asymptotic behaviour in the case of nearest neighbour random walks on free products of the form $\mathbb{Z}^{d_1}\ast \ldots \ast \mathbb{Z}^{d_m}Suppose we are given finitely generated groups Γ₁,…,Γ_m equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product Γ₁* … *Γ_m and give a complete classification of the possible asymptotic behaviour of the corresponding n‐step return probabilities. They either inherit a law of the form ?^nδn log n from one of the free factors Γ_i or obey a ?^nδn^?3/2‐law, where ? < 1 is the corresponding spectral radius and δ is the period of the random walk. In addition, we determine the full range of the asymptotic behaviour in the case of nearest neighbour random walks on free products of the form $\mathbb{Z}^{d_1}\ast \ldots \ast \mathbb{Z}^{d_m}$. Moreover, we characterize the possible phase transitions of the non‐exponential types n log n in the case Γ₁ * Γ₂. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Micro‐Raman and stratigraphic studies of the paintings on the ‘Cembalo’ model musical instrument (A.D. 1650) and laser‐induced degradation of the detected pigments

Micro‐fragments of the painted part of the ‘Cembalo’ model by Michele Todini (1625–1689) are investigated. The technique used for painting the terracotta base was studied via the stratigraphic analyses. No background layer of inorganic materials, e.g. gypsum, was found. To prevent absorption effects due to the terracotta porosity, a very thin layer of proteinaceous material was probably used. The micro‐Raman analyses have revealed the use of pigments currently used in the post‐Renaissance period (lead white, indigo, yellow of iron hydroxide, gypsum, hematite and carbon black) mixed with a pigment, the Prussian blue, discovered in A.D . 1704. This raises the authenticity problem of the work of art, a problem analysed and discussed in presenting the history of the work of art, and after the pigment study. The presence of degraded lead white is recognized via the laser‐induced degradation of the irradiated material. The possibility of a restoring action of the painted parts, as opposite to the non‐originality of the work, is considered and discussed. Since most part of the investigated pigments shows laser‐induced effects, a careful study of this phenomenon is performed by using the modern counterparts of the ancient pigments. For different laser powers, the temperatures of the investigated zones have been obtained via the detailed balance principle and connected to the laser‐induced degradation effects. Copyright © 2007 John Wiley & Sons, Ltd.     

Microwave-Assisted Synthesis: Can Transition Metal Complexes Take Advantage of This “Green” Method?

Microwave-assisted synthesis is considered environmental-friendly and, therefore, in agreement with the principles of green chemistry. This form of energy has been employed extensively and successfully in organic synthesis also in the case of metal-catalyzed synthetic procedures. However, it has been less widely exploited in the synthesis of metal complexes. As microwave irradiation has been proving its utility as both a time-saving procedure and an alternative way to carry on tricky transformations, its use can help inorganic chemists, too. This review focuses on the use of microwave irradiation in the preparation of transition metal complexes and organometallic compounds and also includes new, unpublished results. The syntheses of the compounds are described following the group of the periodic table to which the contained metal belongs. A general overview of the results from over 150 papers points out that microwaves can be a useful synthetic tool for inorganic chemists, reducing dramatically the reaction times with respect to traditional heating. This is often accompanied by a more limited risk of decomposition of reagents or products by an increase in yield, purity, and (sometimes) selectivity. In any case, thermal control is operative, whereas nonthermal or specific microwave effects seem to be absent.     

Photoinduced Energy- and Electron-Transfer Processes in a Side-to-Face RuII-Porphyrin/Perylene-bisimide Array

A side-to-face array DPy-gPBI[Ru(4-tBuTPP)(CO)]₂, based on a “green” perylene bisimide chromophore sandwiched between two Ru^II-porphyrins, has been prepared by self-assembly. Its photophysical properties have been characterized in detail by a combination of steady-state and time-resolved techniques upon selective excitation of the two different components. Different photoinduced processes are observed as a function of the excitation wavelength. Electron transfer quenching is attained upon “red light” excitation of the perylene unit, whilst an energy transfer pathway is followed upon “green light” excitation of the metallo-porphyrin moiety. Regardless of the excitation wavelength efficient population of the triplet excited state of the perylene chromophore is achieved. The photophysical results are discussed within the framework of classical electron transfer theory and compared with those of a previously reported system.     

Stochastic equation of population dynamics with diffusion on a domain

We consider Lotka-Volterra competition model with diffusion in a territorial domain with a stochastic perturbation which represents the random variations of environment conditions. We prove the existence, the uniqueness and the positivity of the solution. Moreover, the stochastic boundedness of the solution is analized.     

Semi-rigid analogues of the calcium antagonist verapamil: A molecular modelling study

Summary In this work the rigid-analogue approach has been used to obtain information on the active conformation(s) of the calcium antagonist verapamil. A series of semi-rigid analogues of verapamil were synthesized and their biological activities evaluated on guinea-pig heart and aorta. These molecules were analysed by means of molecular modelling techniques.On the basis of the pharmacological profile and conformational analysis of these compounds, two different models for negative inotropic and negative chronotropic activity are proposed. The two actions seem to be due to conformations of the molecules which differ in the orientation of their phenylethylamino groups.     

Tri-n-butyltin carboxylate derivatives of para-substituted phenyl-ethanoic acids: synthesis, characterization and X-ray structure determination

The tributyltin esters of 4-(ethyl)-phenyl-ethanoic acid and 4-(isopropyl)-phenyl-ethanoic acid have been prepared as model compounds of the repeating unit of the related stannylated polystyrenic derivatives. Both the products were fully characterized by proton and carbon NMR two-dimensional techniques. FT-IR spectra show in the solid state carboxylated moieties bridging R₃Sn groups with the metal atom expanding its coordination number, this structure being destroyed in solution. The supramolecular arrangement of the products in the solid state has been investigated by X-ray diffraction, which confirms the pentacoordination at tin in both the products, and indicates a different spatial arrangement of the alkylated aryl groups, as evidenced also by the slightly different thermal properties.