Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized coherent states for oscillators associated with the Charlier q-polynomials

We continue studying the generalized coherent states for oscillator-like systems associated with a given family of orthogonal polynomials. We consider the case of generalized oscillators generated by the Charlier q-polynomials. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 39–46, April, 2008.     

Symmetric bent-shaped liquid crystal dimers showing transitions between optically uniaxial and biaxial smectic phases

We report the synthesis of a new homologous series of symmetric bent-shaped liquid crystal dimers bearing oxybiphenylcarboxylate mesogens linked by a pentamethylene spacer. The dimers were characterised by optical, thermal, X-ray diffraction and computational techniques. Shorter dimers showed an I-N-SmA-SmC phase sequence with strong first-order phase transitions. For the longer dimers, the unconventional uniaxial SmA phase present in shorter members collapsed. A 9:1 binary mixture of a bent dimer with a non-mesogenic dimer exhibited upon the SmA-SmC phase transition a maximum layer contraction of 1%.     

Method of orthogonal simplexes and its applications to convex programming

Numerical methods for solving a convex programming problem are considered whose guaranteed convergence rate depends only on the space dimension. On average, the ratio of the corresponding geometric progression is better than that in the basis model of ellipsoids or simplexes. Results of numerical experiments are presented.     

Projection onto polyhedra in outer representation

The projection of the origin onto an n-dimensional polyhedron defined by a system of m inequalities is reduced to a sequence of projection problems onto a one-parameter family of shifts of a polyhedron with at most m + 1 vertices in n + 1 dimensions. The problem under study is transformed into the projection onto a convex polyhedral cone with m extreme rays, which considerably simplifies the solution to an equivalent problem and reduces it to a single projection operation. Numerical results obtained for random polyhedra of high dimensions are presented.     

On groups with the almost orthogonality condition

The concept of the orthogonality is important in the study of lattice-ordered groups. The purpose of this paper is to investigate a generalization of this concept to directed groups. The idea of an AO-group is introduced here. Some results are obtained concerning convex directed subgroups and Archimedean extensions of AO-groups. Interpolation AO-groups are also described.     

Asymptotic Approximations between the Hahn-Type Polynomials and Hermite, Laguerre and Charlier Polynomials

It has been shown in Ferreira et al. (Adv. Appl. Math 31:61–85, [2003]), López and Temme (Methods Appl. Anal. 6:131–196, [1999]; J. Cpmput. Appl. Math. 133:623–633, [2001]) that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic expansions. In this paper we continue with that investigation and establish asymptotic connections between the fourth level and the two lower levels: we derive twelve asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Hermite, Charlier and Laguerre polynomials. From these expansions, several limits between polynomials are derived. Some numerical experiments give an idea about the accuracy of the approximations and, in particular, about the accuracy in the approximation of the zeros of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of the zeros of the Hermite, Charlier and Laguerre polynomials.      

Hydrogen Radical Additions to Unsaturated Hydrocarbons and the Reverse β‐Scission Reactions: Modeling of Activation Energies and Pre‐Exponential Factors

The group additivity method for Arrhenius parameters is applied to hydrogen addition to alkenes and alkynes and the reverse β‐scission reactions, an important family of reactions in thermal processes based on radical chemistry. A consistent set of group additive values for 33 groups is derived to calculate the activation energy and pre‐exponential factor for a broad range of hydrogen addition reactions. The group additive values are determined from CBS‐QB3 ab‐initio‐calculated rate coefficients. A mean factor of deviation of only two between CBS‐QB3 and experimental rate coefficients for seven reactions in the range 300–1000 K is found. Tunneling coefficients for these reactions were found to be significant below 400 K and a correlation accounting for tunneling is presented. Application of the obtained group additive values to predict the kinetics for a set of 11 additions and β‐scissions yields rate coefficients within a factor of 3.5 of the CBS‐QB3 results except for two β‐scissions with severe steric effects. The mean factor of deviation with respect to experimental rate coefficients of 2.0 shows that the group additive method with tunneling corrections can accurately predict the kinetics and is at least as accurate as the most commonly used density functional methods. The constructed group additive model can hence be applied to predict the kinetics of hydrogen radical additions for a broad range of unsaturated compounds.