The visibility function revisited

The visibility function of a compact setS E ^d assigns to eachx S the Lebesgue outer measure of its star inS. This function was introduced by G. Beer in 1972. In 1991, A. Forte Cunto characterized the points of discontinuity of the visibility function in the boundary of a planar Jordan domain. The basic intention of this paper is to extend this characterization to a compact subset ofE ^d . Under certain assumptions, it is proved here that the visibility function of such a set is continuous at a point if and only if the set of restricted visibility of this point has null Lebesgue outer measure.This paper was written as part of Argentine Research Project 01/TX38.     

A mixed-bridging ligand nonanuclear Ru(II) dendrimer containing a tris- chelating core. Synthesis and redox properties

The first ligand-cored dendrimer based on branching Ru(II) centers and containing mixed polypyridine bridging ligands has been prepared; redox experiments suggest that the redox-active core is not reduced at the expected potential, probably as a consequence of shielding induced by the rigid dendritic array.     

Ueber die Bereitung reinen Kalkes zum Gebrauche bei der Elementaranalyse organischer Substanzen

Ohne Zusammenfassung     

Formic and acetic acids in a nitrogen matrix: Enhanced stability of the higher-energy conformer

Formic acid (HCOOH, FA) and acetic acid (CH(3)COOH, AA) are studied in a nitrogen matrix. The infrared (IR) spectra of cis and trans conformers of these carboxylic acids (and also of the HCOOD isotopologue of FA) are reported and analyzed. The higher-energy cis conformer of these molecules is produced by narrowband near-IR excitation of the more stable trans conformer, and the cis-to-trans tunneling decay is evaluated spectroscopically. The tunneling process in both molecules is found to be substantially slower in a nitrogen matrix than in rare-gas matrices, the cis-form decay constants being approximately 55 and 600 times smaller in a nitrogen matrix than in an argon matrix, for FA and AA respectively. The stabilization of the higher-energy cis conformer is discussed in terms of specific interactions with nitrogen molecule binding with the OH group of the carboxylic acid. This model is in agreement with the observed differences in the IR spectra in nitrogen and argon matrices, in particular, the relative frequencies of the νOH and τCOH modes and the relative intensities of the νOH and νC=O bands.     

Expanded Pyridiniums: Bis‐cyclization of Branched Pyridiniums into Their Fused Polycyclic and Positively Charged Derivatives—Assessing the Impact of Pericondensation on Structural,Electrochemical, Electronic,and Photophysical Features

This study evaluates the impact of the extension of the π‐conjugated system of pyridiniums on their various properties. The molecular scaffold of aryl‐substituted expanded pyridiniums (referred to as branched species) can be photochemically bis‐cyclized into the corresponding fused polycyclic derivatives (referred to as pericondensed species). The representative 1,2,4,6‐tetraphenylpyridinium ( 1^H ) and 1,2,3,5,6‐pentaphenyl‐4‐(p‐tolyl)pyridinium ( 2^Me ) tetra‐ and hexa‐branched pyridiniums are herein compared with their corresponding pericondensed derivatives, the fully fused 9‐phenylbenzo[1,2]quinolizino[3,4,5,6‐def]phenanthridinium ( 1^H f ) and the hitherto unknown hemifused 9‐methyl‐1,2,3‐triphenylbenzo[h]phenanthro[9,10,1‐def]isoquinolinium ( 2^Me f ). Combined solid‐state X‐ray crystallography and solution NMR experiments showed that stacking interactions are barely efficient when the pericondensed pyridiniums are not appropriately substituted. The electrochemical study revealed that the first reduction process of all the expanded pyridiniums occurs at around ?1 V vs. SCE, which indicates that the lowest unoccupied molecular orbital (LUMO) remains essentially localized on the pyridinium core regardless of pericondensation. In contrast, the electronic and photophysical properties are significantly affected on going from branched to pericondensed pyridiniums. Typically, the number of absorption bands increases with extended activity towards the visible region (down to ca. 450 nm in MeCN), whereas emission quantum yields are increased by three orders of magnitude (at ca. 0.25 on average). A relationship is established between the observed differential impact of the pericondensation and the importance of the localized LUMO on the properties considered: predominant for the first reduction process compared with secondary for the optical and photophysical properties.     

Some a priori estimates for a class of operators in the Heisenberg group

In this note, we provide some a priori estimates for a class of operators that stem from the Laplacian in the Heisenberg group. We follow the idea contained in a proof given by Talenti, see (Ann Scuola Norm Sup Pisa Cl Sci (4) 3(4): 697–718, 1976), by adapting the classical notion of symmetrized rearrangement of a function to the framework of the Heisenberg group.     

Twist points of the von Koch snowflake

It is known that the set of twist points in the boundary of the von Koch snowflake domain has full harmonic measure. We provide a new, simple proof, based on the doubling property of the harmonic measure, and on the existence of an equivalent measure, invariant and ergodic with respect to the shift.