For a number q bigger than 1, we consider a q-difference version of a second-order singular differential operator which depends on a real parameter. We give three exact parameter intervals in which the operator is semibounded from above, not semibounded, and semibounded from below, respectively. We also provide two exact parameter sets in which the operator is symmetric and self-adjoint, respectively. Our model exhibits a more complex behavior than in the classical continuous case but reduces to it when q approaches 1.

To study molecular features connected with chirality, a procedure for the quantitative estimation of the chirality level of compounds of different classes is needed. A procedure for estimating the molecular asymmetry level relative to mirror‐reflection axes of symmetry, S1, S2, S4 and S6, has been developed. The geometrical mean of these parameters is the disymmetry function (DF). To calculate the DF, the molecule must be fixed in the coordinate system, transferred to the main axes of inertia.

We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl-Titchmarsh matrix-valued functions as well as a new version of the functional model for such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions, we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its selfadjoint extension which generate a periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach, we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.  2004 Elsevier Inc. All rights reserved.

We consider a two-dimensional autonomous system of rational difference equations with three positive parameters. It was conjectured by Ladas that every positive solution of this system converges to a finite limit. Here we confirm this conjecture.

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We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl -Titchmarsh matrixvalued functions as well as a new version of the functional model in such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extension generating periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.

The efficiency of the developed method of dissymmetry functions was studied using various model systems. The alteration of the dissymmetry function was analysed with systematic variations of the bond lengths, valence angles and masses of atoms in model tetrahedra. The behaviour of the dissymmetry function was studied for conformationally labile systems and chiral polyhedra. In general, it was found that in all cases the alteration of the dissymmetry function is in agreement with the speculative representations of the changes in degree of chirality.