On an alternative cauchy equation in two unknown functions. Some classes of solutions

Summary In this paper we consider the alternative Cauchy functional equationg(xy) g(x)g(y) impliesf(xy) = f(x)f(y) wheref, g are functions from a topological group (X, ·) into a group (S,·). First we prove that, ifS is a Hausdorff topological group andX satisfies some weak additional hypotheses, then (f, g) is a continuous solution if and only if eitherf org is a homomorphism. Then we describe a more general class of solutions forX =R ⁿ .Partially supported by M.U.R.S.T. Research funds (40%)Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

An RQP algorithm using a differentiable exact penalty function for inequality constrained problems

In this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function. The algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an Armijo-type line search procedure that avoids the need to evaluate second order derivatives of the problem functions. We prove that the algorithm possesses global and superlinear convergence properties. Numerical results are reported.     

Towards a rigorous treatment of the Jeans-Landau-Teller method for the energy exchanges of harmonic oscillators

For classical Hamiltonian systems containing an harmonic oscillator of high frequency, one has the problem of controlling the energy exchange between the oscillator and the remaining slow degrees of freedom; under very general conditions, such an exchange turns out to be exponentially small with the frequency of the oscillator. In the Jeans-Landau-Teller method, one aims to prove the exponential dependence, and to estimate the coefficient of the exponential, by exploiting the analyticity properties of the solution of the differential equations describing the motion of the system. However, in practice, since the exact solution is not known, such properties are inferred from those of an approximate solution, with no control of the difference; this fact might a priori even invalidate the exponential dependence itself. In the present paper a rigorous treatment is given, for a particular model of interest in the domain of atomic collisions, by keeping control of the difference between the exact and the approximate solution.     

A stable binuclear complex containing PdHPd bonds

The preparations of the binuclear hydrido-bridged cations [(terdentate ligand)Pd(μ-H)Pd(terdentate ligand)]⁺ from [(terdentate ligand)Pd(acetone)]⁺ and NaO₂CH and [(terdentate ligand)Pd(μ-H)Pt(terdentate ligand)]⁺ from [(terdentate ligand)Pd(acetone)]⁺ and [(terdentate ligand)PtH] (terdentate ligand = 2,6-(Ph₂PCH₂)₂C₆H₃) are reported. The preparation of the cation [(terdentate ligand)Pt(μ-H)Pt(terdentate ligand)]⁺ is also reported.     

Endogenous growth of the population of reverse micelles

The coupling reaction between cetylbromide (CB) and trimethylamine (TMA) to yield the surfactant cetyltrimethylammonium bromide (CTAB) is studied in the system chloroform/isooctane (2/1,v/v)/water in which CTAB forms reverse micelles. This system affords an endogenous micelle population growth, i.e., an increase of the concentration of the micelles due to appearance of the surfactant in situ. The reaction is studied in the presence of preexisting CTAB reverse micelles. The rate of CTAB formation is measured by NMR spectroscopy, and the endogenous micelle population growth is directly monitored by time-resolved fluorescence quenching analysis. Under our experimental conditions, a 100% yield of the chemical reaction brings about a fourfold increase in the population of the reverse micelles. Since the water concentration is constant during chemical reaction, the newly formed water pools are formed at the expense of the initial ones, which brings about a decrease of the average water pool radius during micellar growth. The implication of the endogenous micelle population growth as a model for biological systems is briefly discussed.     

On an alternative functional equation related to the Cauchy equation

We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, bB,b0,M a positive integer; find all functionsf:G B such that for every (x, y) G ×G the Cauchy differencef(x+y)–f(x)–f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG.     

Crystal,molecular and electronic structure of 1,1-diaryl-2-halogenoethylenes

On the basis of semiempirical calculations, the conformation of the isolated molecule of 2-bromo-1,1-diphenylprop-1-ene corresponding to the minimum of total energy was determined. Assuming some information on the localization of the molecules in the crystal, also the conformation in a theoretical crystal was studied. The results have been compared with the crystal structure of 2-bromo-1,1-diphenylprop-1-ene, as obtained by X-ray diffraction. From the comparison a slight displacement of the C2 atom was suggested.

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Zusammenfassung Die dem Minimum der Gesamtenergie entsprechende Konformation des isolierten Moleküls von 2-Brom-1,1-diphenylprop-1-en wurde durch halb-empirische Methoden bestimmt. Gewisse Ergebnisse über die Anordnung der Moleküle im Kristall machten es möglich, die Konformation in einem theoretischen Kristall zu untersuchen. Nach Vergleich der Resultate mit der durch Röntgen-Analyse bestimmten Kristallstruktur des 2-Brom-1,1-diphenylprop-1-en wurde eine kleine Verschiebung des C2-Atoms vorgeschlagen.

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Résumé Par le moyen de calculs semiempiriques on a déterminé la conformation de la molécule isolée de 2-bromo-1, 1-diphénilprop-1-ène, correspondant au minimum d'énergie totale. On a aussi étudié la conformation dans le cristal théorique en se basant sur des informations concernant la localisation des molécules dans le cristal. On a comparé ces résultats avec ceux que nous avons obtenus par la détermination de la structure cristalline du 2-bromo-1, 1-diphénilprop-1-ène avec la diffraction des rayons X: cette comparaison nous a porté à déplacer un petit peu l'atome C 2.

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This work was supported by a grant from the Italian Consiglio Nazionale delle Ricerche.     

Synthesis of 4,5-dihydro-2,4-benzoxazepin-3( 1H)ones and 1,3,4,5-tetrahydro-2,4-benzoxazepines

o- Aminomethylbeiizyl alcohols (X) easily cyclize with phosgene in an aqueous alkaline medium to form 4,5-dihydro-2,4-benzoxazepin-3(1H)ones (IV) and with aldehydes in acidic conditions to yield 1,3,4,5-tetrahydro-2,4-benzoxazepines (V). The characteristics and chemical behaviour of these new heterocyclic ring systems are reported.     

The ring-opening reaction of chromenes: a photochemical mode-dependent transformation

The efficiency of the photochemical ring-opening of chromenes (or benzopyrans) depends on the vibronic transition selected by the chosen excitation wavelength. In the present work, ab initio CASPT2//CASSCF calculations are used to determine the excited-state ring-opening reaction coordinate for 2H-chromene (C) and 2,2-diethyl-2H-chromene (DEC) and provide an explanation for such an unusual mode-dependent behavior. It is shown that excited-state relaxation and decay occur via a multimodal and barrierless (or nearly barrierless) reaction coordinate. In particular, the relaxation out of the Franck-Condon involves a combination of in-plane skeletal stretching and out-of-plane modes, while the second part of the reaction coordinate is dominated exclusively by a different out-of-plane mode. Population of this last mode is shown to be preparatory with respect to both C-O bond breaking and decay via an S(1)/S(0) conical intersection. The observed mode-dependent ring-opening efficiency is explained by showing that the vibrational mode corresponding to the most efficient vibronic transition has the largest projection onto the out-of-plane mode of the reaction coordinate. To support the computationally derived mechanism, we provide experimental evidence that the photochemical ring-opening reaction of 2,2-dimethyl-7,8-benzo(2H)chromene, that similarly to DEC exhibits a mode-dependent photoreaction, has a low ( approximately 1 kcal mol(-1)) activation energy barrier.     

RuO4-promoted oxidative polycyclization of isoprenoid polyenes. A further insight into the stereochemistry of the process

Further studies on the RuO₄-catalyzed oxidative polycyclization of isoprenoid polyenes have been carried out. The configuration of the tris-THF product from the oxidation of geranylgeranyl acetate has been determined by a combination of spectral and chemical correlation methods. The oxidation of (E,Z)-farnesyl acetate, synthesized from nerol, has been carried out. This process stops at the first cyclization indicating that an E configuration of the Δ⁶ double bond is needed for the second cyclization to occur. The results are discussed in comparison with previous knowledge on the related Re(VII) polycyclization of polyenic bis-homoallylic alcohols.