Some generalizations of the classical moment problem

The article is devoted to two generalizations of the classical power moment problem, namely: 1) instead of representing the moment sequence by ⁿ , a representation by polynomialsP _n (), ¹, connected with a Jacobi matrix, appears; 2) in the representation, instead of ⁿ , the expression ⁿ figures, where is a real generalized function (i.e., we investigate some infinite-dimensional moment problem).The work is partially supported by the DFG, Project 436 UKR 113/39/0 and by the CRDF, Project UM1-2090.     

Hyperbolicity of linear partial differential equations with delay

Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed.The author thanks W. Desch (Graz), I. Gyri (Veszprém) and R. Schnaubelt (Halle) for helpful discussions.     

Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights

We present two approaches to the spectral studies for infinite Jacobi matrices with monotonic or near-to-monotonic weights. The first one is based on the subordination theory due to Khan and Pearson [17] combined with the detailed analysis of the transfer matrices for the solutions of the formal eigenequation. The second one uses an extension of the commutator approach developed by Putnam in [19]. Applying these methods we prove the absolute continuity for several classes of weights and diagonals. For some other cases we prove the emptiness of the point spectrum. The results are illustrated with examples and compared with the results of Dombrowski [7]-[13], Clark [2] and of Máté and Nevai [18]. We show that some of our results are stronger.The research of the first author has been supported by the KBN grant PB 2 P03A 002 13.     

Sub-optimal Hankel norm approximation for the analytic class of infinite-dimensional systems

The sub-optimal Hankel norm approximation problem is solved under the assumptions that the system is given in terms of a triple of operators (–A, B, C), where–A is the infinitesimal generator of an exponentially stable, analytic semigroup on the Hilbert spaceZ,B L ( ^m ,Z where –1<0,C L is obtained in terms of the system parameters–A, B, C. (Z, ^p ), and the system is approximately controllable. An explicit parameterization of all solutions     

On Fuzzy Probability Theory

Our main aim from this work is to see which theorems in classical probability theory are still valid in fuzzy probability theory. Following Gudder's approach [Demonestratio Mathematica 31(3), 1998, 235–254; Foundations of Physics, 30, 1663–1678] to fuzzy probability theory, the basic concepts of the theory, that is of fuzzy probability measures and fuzzy random variables (observables), are presented. We show that fuzzy random variables extend the usual ones. Moreover, we prove that for any separable metrizable space, the crisp observables coincide with random variables. Then we prove the existence of a joint observable for any collection of observables, and we prove the weak law of large numbers and the central limit theorem in the fuzzy context. We construct a new definition of almost everywhere convergence. After proving that Gudder's definition implies ours and presenting an example that indicates that the converse is not true, we prove the strong law of large numbers according to this definition.     

Gas chromatographic and IR spectroscopic characteristics of tri- and tetramethylcyclohexenyl butenyl ketones

Infrared spectroscopic characteristics and gas chromatographic retention indices of tri- and tetramethylcyclohexenyl butenyl ketones were determined at different temperatures of analysis. Thermodynamic characteristics of sorption on the apolar stationary phase were calculated. The sorption characteristics of tri- and tetramethylcyclohexenyl butenyl ketones show that under conditions of capillary gas chromatography these compounds are retained due to the dispersive energy. It was shown by FTIR spectroscopy that the formation of conjugation systems of double bonds is responsible for the higher retention of the -isomers compared with that of the -isomers. The shortening of the distance between the carbonyl and trimethylcyclohexenyl groups was found to be accompanied by a decrease in the retention of the isomeric compounds.     

Injective hulls are not natural

In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are injective. In particular, assigning to a field its algebraic closure, to a poset or Boolean algebra its Mac-Neille completion, and to an R-module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation. Received January 21, 2000; accepted in final form August 10, 2001. RID="h1" RID="h2" RID="h3" ID="h1"The hospitality of York University is gratefully acknowledged by the first author. ID="h2"Third author partially supported by the Grant Agency of the Czech Republic under Grant no. 201/99/0310, and the hospitality of York University is also acknowledged. ID="h3"Partial financial assistance by the Natural Sciences and Engineering Councel of Canada is acknowledged by the fourth author.     

Photoacoustic gas sensing employing fundamental and frequency-doubled radiation of a continuously tunable high-pressure CO2 laser

We present the first photoacoustic spectrometer for gas sensing employing both the fundamental and the frequency-doubled radiation of a continuously tunable high-pressure CO₂ laser with room temperature operation. A quasi-phase-matched diffusion-bonded GaAs crystal is used in the system for second-harmonic generation. A pulsed photoacoustic detection scheme with a non-resonant cell, equipped with an 80-microphone array, is employed. The wide continuous tuning range in the fundamental (9.2–10.7 μm) and the frequency-doubled (4.6–5.35 μm) regimes, together with the narrow linewidth of 540 MHz (0.018 cm^-1) for the 10-μm region and of 1050 MHz (0.0315 cm^-1) for the 5-μm region, allow the measurement of gas mixtures, individual species and isotope discrimination. This is illustrated with measurements on NO and CO₂. The measured isotope ratio ¹⁵ NO/¹⁴ NO=(3.58±0.55)×10^-3 agrees well with the literature (3.700×10^-3) and demonstrates the good selectivity of the system. Received: 30 April 2002 / Revised version: 10 June 2002 / Published online: 2 September 2002 RID="*" ID="*"Corresponding author. Fax: +41-1/633-1077, E-mail: sigrist@iqe.phys.ethz.ch     

Laser-based photoacoustic ammonia sensors for industrial applications

An industrial trace-ammonia sensor based on photoacoustic spectroscopy and CO₂ lasers has been developed for measuring ammonia with a 1σ detection limit of 220 parts-per-trillion (ppt) in an integration time of 30 s. The instrument response time for measuring ammonia was 200 s, limited by adsorption effects due to the polar nature of ammonia. The minimum detectable fractional absorbance was 2.0×10^-7, and the minimum normalized detectable absorption coefficient for this system was 2.4×10^-7 W cm^-1/z. The 9R(30) transition of the CO₂ laser at 9.22 μm with 2 W of output power was used to probe the strong sR(5,K) multiplet of ammonia at the same wavelength. This sensor was demonstrated with an optically multiplexed configuration for simultaneous measurement in four cells. Received: 3 April 2002 / Revised version: 31 May 2002 / Published online: 21 August 2002 RID="*" ID="*"Corresponding author. Fax: +1-310/458-0171, E-mail: webber@pranalytica.com