Using Clar sextets for two- and three-dimensional aromatic systems

After a brief history of the aromaticity concept, the use of Clar sextet circles is reviewed for explaining various aspects of planar aromatic systems (benzenoids, heterocycles) and of tridimensional carbon aggregates. When folding a graphene sheet for obtaining nanotubes, nanotori, or nanocones, the congruence of Clar sextet circles allows the classification of all such aggregates into two classes (congruent or incongruent) with marked differences in properties; this is in agreement with the well-known condition h - k ≡ 0 (mod 3), equivalent to congruence, in terms of the chiral vectors h and k for graphene sheets.     

A novel tungsten trioxide (WO3)/ITO porous nanocomposite for enhanced photo-catalytic water splitting

Hybrid nanocomposite films of ITO-coated, self-assembled porous nanostructures of tungsten trioxide (WO(3)) were fabricated using electrochemical anodization and sputtering. The morphology and chemical nature of the porous nanostructures were studied by Scanning Electron Microscopy (SEM) and X-ray Photoelectron Spectroscopy (XPS), respectively. The photoelectrochemical (PEC) properties of WO(3) porous nanostructures were studied in various alkaline electrolytes and compared with those of titania nanotubes. A new type of alkaline electrolyte containing a mixture of NaOH and KOH was proposed for the first time to the best of our knowledge and shown to improve the photocurrent response of the photoanodes. Here, we show that both the WO(3) nanostructures and titania nanotubes (used for comparison) exhibit superior photocurrent response in the mixture of NaOH and KOH than in other alkaline electrolytes. The WO(3) porous nanostructures suffered from surface corrosion resulting in a huge reduction in the photocurrent density as a function of time in the alkaline electrolytes. However, with a protective coating of ITO (100 nm), the surface corrosion of WO(3) porous nanostructures reduced drastically. A tremendous increase in the photocurrent density of as much as 340% was observed after the ITO was applied to the WO(3) porous nanostructures. The results suggest that the hybrid ITO/WO(3) nanocomposites could be potentially coupled with titania nanotubes in a multi-junction PEC cell to expand the light absorption capability in the solar spectrum for water splitting to generate hydrogen.     

Applying the conjugated circuits method to Clar structures of [n]phenylenes for determining resonance energies

We consider the aromaticity of biphenylene and structurally related linear or angular [n]phenylenes for which the direct application of the model of conjugated circuits does not offer valid expressions for resonance energy and aromaticity. We located the cause of this problem as being due to Kekulé valence structures in which neighboring benzenoid rings are connected by two CC double bonds. By restricting the selection of Kekulé valence structures to those that contribute to Clar structures of such systems, we were able to show that linear and angular [n]phenylenes have approximately similar resonance energies, with angular [n]phenylenes being slightly more stable due to second order contributions arising from disjoint conjugated circuits. Expressions for resonance energies of [n]phenylenes up to n = 8 are listed and recursion expressions for higher n values are outlined.     

On the Global Regularity for a Wave-Klein—Gordon Coupled System

In this paper we consider a coupled Wave-Klein—Gordon system in 3D, and prove global regularity and modified scattering for small and smooth initial data with suitable decay at infinity. This...     

Irreducibility Results for Compositions of Polynomials with Integer Coefficients

We obtain explicit upper bounds for the number of irreducible factors for a class of polynomials of the form f ○ g, where f,g are polynomials with integer coefficients, in terms of the prime factorization of the leading coefficients of f and g, the degrees of f and g, and the size of coefficients of f. In particular, some irreducibility results are given for this class of compositions of polynomials.     

Discrepancy of Fractions with Divisibility Constraints

We provide bounds for the absolute discrepancy of sequences of fractions with denominators streaming in a given arithmetic progression and satisfying divisibility constraints. Supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.     

A geometric approach to the classification of pencils of quadrics

Geometriae Dedicata -     

On a Problem of Brocard

It is proved that, if P is a polynomial with integer coefficients,having degree 2, and 1 > > 0, then n(n – 1) ...(n – k + 1) = P(m) has only finitely many natural solutions(m,n,k), n k > n, provided that the abc conjecture is assumedto hold under Szpiro's formulation. 2000 Mathematics SubjectClassification 11D75, 11J25, 11N13.