The Riemann Function for the Third-Order One-Dimensional Pseudoparabolic Equation

In this work, some classes of initial boundary-value problems are investigated in the Sobolev space for the third-order linear pseudoparabolic equation having, in general, nonsmooth coefficients. A new type of Riemann function concept is given for these problems, which is more natural than the classical Riemann-type function concept, and an integral form of the solutions of nonhomogeneous problems can be found more naturally using this concept.     

Analytic expression of the diffraction of a circular aperture

Recurring to the characteristic of Bessel function, we give the analytic expression of the Fresnel diffraction by a circular aperture, thus the diffractions on the propagation axis and along the boundary of the geometrical shadow are discussed conveniently. Since it is difficult to embody intuitively the physical meaning from this series expression of the Fresnel diffraction, after weighing the diffractions on the axis and along the boundary of the geometrical shadow, we propose a simple approximate expression of the circular diffraction, which is equivalent to the rigorous solution in the further propagation distance. It is important for the measurement of the parameter of the beam, such as the quantitative analysis of the relationship of the wave error and the divergence of the beam. In this paper, the relationship of the fluctuation of the transverse diffraction profile and the position of the axial point is discussed too.     

Copper(II) and cobalt(II) hydroxypyridinecarboxylates: Synthesis, crystal structures, spectral and magnetic properties

Synthesis and characterization of six copper(II) and cobalt(II) octahedral complexes [M(6-OHpic)₂(H₂O)₂] (6-OHpic is 6-hydroxypicolinato), [M(2-OHnic)₂(H₂O)₂] (2-OHnic is 2-hydroxynicotinato), [Cu(6-OHnic)₂(H₂O)₂] (6-OHnic is 6-hydroxynicotinato) as well as [Co(H₂O)₆](6-OHnic)₂ are reported. Their characterization was carried out using elemental analysis, infrared, and magnetic measurements. Based on IR spectra, N,O-coordination of 6-OHpic (via the oxygen atom of the carboxyl group and the nitrogen atom of the pyridine ring), O,O-asymmetrically chelating coordination of the carboxyl groups as well as ionic coordination of 6-OHnic and chelating O,O-coordination (through the oxygen atom of the carboxyl group and the oxygen atom of the amide group) of keto(amide) tautomer of 2-OHnic were supposed. Moreover, crystal structures of 2-OHnicH and the complex [Co(2-OHnic)₂(H₂O)₂]) were determined by X-ray single crystal structure analysis. The system of hydrogen bonds predominantly stabilizes the keto(amide) tautomer of both 2-hydroxynicotinic acid and 2-OHnic anion in the cobalt(II) complex. Intermolecular hydrogen bonds (between the oxygen atom of the amide group and the hydrogen atom of the NH group) interconnect two neighbouring molecules of 2-OHnicH forming dimers. Cobalt(II) in complex [Co(2-OHnic)₂(H₂O)₂] has nearly a regular compressed tetragonal bipyramidal arrangement. Presented at the 1st International Conference “Applied Natural Sciences” on the occasion of the 10th anniversary of the University of Ss. Cyril and Methodius, Trnava, 7–9 November 2007.     

Ethyl 2‐methoxy‐6‐[(triphenylphosphoranylidene)amino]nicotinate and ethyl 2‐methylsulfanyl‐6‐[(triphenylphosphoranylidene)amino]nicotinate

The molecules of ethyl 2‐methoxy‐6‐[(triphenylphosphoranylidene)amino]nicotinate, C₂₇H₂₅N₂O₃P, (I), and ethyl 2‐methylsulfanyl‐6‐[(triphenylphosphoranylidene)amino]nicotinate, C₂₇H₂₅N₂O₂PS, (II), have almost identical bond lengths and molecular conformations, and both show evidence for polarized electronic structures. However, the crystal structures, as illustrated by the weak hydrogen bonds linking the molecules, are significantly different. The significance of this study lies in the observation that two compounds which are almost identical in constitution, configuration and conformation nonetheless adopt different crystal structures.     

Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces

For discrete Hecke pairs (G,H), we introduce a notion of covariant representation which reduces in the case where H is normal to the usual definition of covariance for the action of G/H on c₀(G/H) by right translation; in many cases where G is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations of c₀(G/H) which are multiples of the multiplication representation on ?²(G/H), and more generally, we prove an imprimitivity theorem for regular representations of certain crossed products by coactions of homogeneous spaces. We thus obtain new criteria for extending unitary representations from H to G.     

Linear and non-linear optical properties of some donor–acceptor oxadiazoles by ab initio Hartree-Fock calculations

The molecular hyperpolarizability of some donor–acceptor oxadiazoles was investigated using ab initio methods. Ab initio optimizations were performed at the Hartree-Fock level using different basis sets, starting with the minimal basis set, and then split valence sets. The first hyperpolarizabilities were calculated at the Hartree-Fock level employing the corresponding basis sets using Gaussian 98W. In general, the first hyperpolarizability is dependent on the choice of method and basis set. In order to understand this phenomenon in the context of molecular orbital picture, we examined the molecular HOMOs and molecular LUMOs generated via HF/6-31G level. It has also been calculated the polarizability, anisotropy of polarizability and ground state dipole moment of all the molecules. Several of these oxadiazoles display significant second-order molecular nonlinearity, β(8.57–195.05 × 10⁻³⁰ esu) and provide the basis for future design of efficient nonlinear optical materials having the oxadiazole core.