On an identity of Ramanujan over finitely generated domains

Summary We show that a well-known identity of Ramanujan admits only a bounded number of solutions over general finitely generated domains. The bound is explicit and uniform in the sense that it depends only on the dimensions of the domains involved. Our method is constructive, and opens up a possibility to determine the solutions in concrete instances. In some special cases all solutions are determined. Our results can also be considered as a continuation of some theorems of Z. Daróczy and G. Hajdu, obtained over Z. We note that in case of Hosszú's equation, similar results were obtained by several authors.     

Looking for more or less hidden vector spaces: A strategy for investigating injective cogenerators

We investigate ad compare certain injective cogenerators of the category of left modules over any algebra or, more generally, over any ring. Si studiano e si confrontano alcuni cogeneratori iniettivi della categoria dei moduli sinitri su un'algebra o, più in generale, su di un anello.

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(Conferenza tenuta il 19 ottobre 19920     

Crystal structure of tris(2-pyridylthio-2-pyridinium)pentaisothiocyanato-dioxouranate(VI)

Summary The crystal structure of [DPSH] ₃ ⁺ [UO₂[NCS)₅]^3– (DPSH = 2 pyridylthio-2-pyridinium) has been determined by standard methods using diffractometer data. Crystals are monoclinic, space groupP2₁/a, witha=33.16 (3),b=16.68(2),c=7.85(1) Å, and=97.3(1); D_c=1.65 g · cm^–3 for Z=4. The structure has been refined to R=5.8% by least squares methods. Five thiocyanate groups are equatorially bonded to the linear uranyl group. The mean of the five U-N, N-C, and C-S bond distances are 2.45, 1.17, and 1.59 Å respectively. The protonated DPS molecules have the N,N-inside conformations.     

Glivenko-Cantelli type theorems: An application of the convergence theory of stochastic suprema

The uniform convergence of empirical processes on certain classes of sets follows from the convergence theory for random lower semicontinuous functions studies in the context of stochastic optimization. In the process, a richer class of sets for which one can prove this type of result is exhibited.Research supported in part by grants from Ministero Publica Istruzione and the National Science Foundation.     

Finitely additive random walks on infinitely generated free groups

We consider any purely finitely additive probability measure supported on the generators of an infinitely generated free group and the Markov strategy with stationary transition probability . As well as for the case of random walks (with countably additive transition probability) on finitely generated free groups, we prove that all bounded sets are transient. Finally, we consider any finitely additive measure (supported on the group generators) and we prove that the classification of the state space depends only on the continuous part of .     

Zur intramolekularen nucleophilen Addition von 2,3-Dihydrodipyrrin-1(10H)-onen

Model compound5 was prepared by acid catalyzedintramolecular nucleophilic addition of the alcohol (Z)-4. In comparison to theintermolecular addition productl-2, differences in structure and reactivity could be determined by X-ray crystal structure analysis and¹H-NMR-spectroscopic investigations. The interpretation of these differences are based on the assumption of stereoelectronic advantages in the intermolecular adductl-2. Its methoxy group was found to beperiplanar with respect to the -(NCO)-plane of the lactam unit enabeling efficient delocalization of the amide lone pair to the *-orbital of the C-OMe-bond. The orientation of the corresponding C-O-bond in5 is fixed by the rigid structure of the bicyclic ring system and can be classified assynclinal. Consequentlyl-2 eliminates methanol very easily under acidic or high temperature conditions, whereas5 is a stable compound.In the stereoelectronically favoured adductl-2 n *-delocalization increases elimination reactivity on the one hand and thermodynamic stability on the other hand. Therefore adducts of this kind may be good candidates for biological regulators—a matter of interest with regard to the protein-chromophore interaction in biliproteins.

     

Non-linear and non-constant variance calibration curves in analysis of volatile organic compounds for testing of water by the purge-and-trap method coupled with gas chromatography/mass spectrometry

Currently used operating conditions for analysing volatile organic compounds (VOCs) by purge-and-trap gas chromatography/mass spectrometry (GC/MS) produced non-linear calibration curves with non-uniform variance. Second-order polynomial models therefore had to be used in weighted regression analysis of measurements of replicates spiked at various concentrations. A transparent procedure based on a reported method of very low computational complexity allowed calculation of the parameters of second-order models, confidence bands of regression lines, prediction bands, and confidence intervals of discriminated analyte concentrations. Tolerance intervals were introduced for this last purpose. Critical, detection and quantification levels drawn from the calibration curves were compared with those calculated by the EPA method.     

Endocrine-disrupting pollutants properties affecting their bioactivity,remediation, and detection

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Calculation of the kinetic energy release of charge separation processes

Kinetic energy release (KER) was studied by experimental methods and semiempirical (MNDO and AM1) molecular orbital calculations in the case of various charge separation processes: loss of a methyl ion from [CH₃? C₄? CH₃]²⁺, [CH₃? C₃? CH₃]²⁺ and [N,N-dimethyl-p-phenylenediamine]²⁺. It was found that the KER corresponding to the width of a dish-topped peak at half-height is very close to the mean KER of the process. The calculated potential energy curves of these reactions show significant reverse critical energies, a large part of which was found to be due, in agreement with conventional assumptions, to electric repulsion between the two separating singly charged products. The bond order between the two separating ions is nearly zero in the transition state, so exchange of internal energy between them is unlikely. These explain the good agreement between the (calculated) reverse critical energy and the measured kinetic energy release.