The stability of the exponential equation

We generalize the well-known Baker's superstability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary commutative complex semisimple Banach algebra. It was shown by Ger that the superstability phenomenon disappears if we formulate the stability question for exponential complex-valued functions in a more natural way. We improve his result by showing that the maximal possible distance of an -approximately exponential function to the set of all exponential functions tends to zero as tends to zero. In order to get this result we have to prove a stability theorem for real-valued functions additive modulo the set of all integers .

     

Mosco approximation of integrands and integral functionals

It is shown that, given a lower semicontinuous convex integrandf satisfying a suitable integrability condition, there exists a sequence of Lipschitz simple integrands which Mosco converges tof and such that the sequence of conjugate integrands Mosco converges tof ^*. Moreover, this sequence can be chosen so that the sequence of associated integral functionals, respectively defined onL ¹(X) andL (X ^*), Mosco converges as well.We wish to thank Professor Erik J. Balder for interesting remarks on the first version of this work.     

On hydrogen-bonded complexes: the case of (HF)2

The hydrogen fluoride dimer (HF)₂ is the most completely characterized hydrogen bonded species incorporating a donor molecule and an acceptor molecule. We provide a summary of experimental and theoretical information pertinent to the fundamental and harmonic vibrational frequencies, equilibrium geometry and dissociation energiesD _e andD ₀ as well as a brief critical discussion including some new results on the potential function and current best estimates of experimental quantities.     

Strukturuntersuchungen an naßersponnenen Polyacrylnitrilfasern nach der Gefriertrocknung

Zusammenfassung An unterschiedlich verstreckten und hydrothermisch nachbehandelten, danach gefriergetrockneten Polyacrylnitril-Gelfasern (PAN) wurden Messungen der Röntgenkleinwinkelstreuung (RKWS), Röntgenweitwinkelstreuung (RWWS) und Tieftemperatur-Gassorption (TTGA) durchgeführt. Die RKWS-Messungen zeigen, daß sowohl in den isotropen als auch in den orientierten Proben eine exponentialstatistische Verteilung der Form und Größe der Inhomogenitäten (Hohlräume, Cluster) vorliegt, welche durch die Kohärenzlängel _c bestimmt wird. Eine erhöhte Röntgenabsorption wurde bei den verstreckten Proben festgestellt. Die Analyse der aus der Kohärenzlänge und den Volumsanteilen der Phasen berechneten charakteristischen RK WS-Parameter (z.B. innere Oberfläche, Durchschußlänge) ergibt, daß jede Behandlung der Gelfasern eine Zu- oder Abnahme der etwa gleich großen Hohlräume verursacht (nucleotic change). Das relative Ausmaß dieser Veränderungen wird besprochen.Der Vergleich mit den Ergebnissen der TTGA an den gleichen Fasern zeigt, daß die Tendenzen bei der spezifischen Oberfläche und dem Porenvolumen (Zu- bzw. Abnahme) bei beiden Methoden annähernd gleich sind. Die Absolutwerte des Hohlraumvolumens unterscheiden sich aber z.T. um den Faktor 40. Dies führt zur Schlußfolgerung, daß mit der RK WS Faserbereiche als Hohlräume erfaßt werden, die bei der TTGA nicht angezeigt werden, weil sie für das Meßgas unzugänglich sind.

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Structural studies on wet-spun freeze-dried polyacrylonitrile fibers

Summary Polyacrylonitrile fibers (PAN) were drawn to different extents, hydrothermally aftertreated, and at last freeze-dried. The products were studied by small angle X-ray scattering (SAXS), wide angle X-ray scattering (WAXS), and low temperature gassorption (LTGS). The SAXS data show that the shape and the size of the inhomogeneities (voids, clusters) have an exponential statistical distribution determined by the correlation lengthl _c, both in the isotropic fibers and in the fiber-oriented ones. It has also been observed that stretching always causes a higher X-ray absorption. An analysis of the characteristic SAXS parameters (e.g., internal surface, mean chord length), which were calculated from the correlation length and the volume fractions of the phases, evidences that in all samples an augmentation or diminution of the number of the voids with nearly the same shape and size takes place (nucleotic change). Its relative number has been pointed out.Comparison with the results of the LTGS investigations at the same fibers shows that the tendencies of the specific surface and porosity (increase, decrease) are approximately the same with both methods. However, the absolute values of void volume differ in some cases by the factor 40. This leads to the conclusion that the SAXS method records domains of these fibers as voids which are not registered by LTGS (voids not accessible for the gas).

Herrn em.o.Univ. Prof. Dr.Josef Schurz zum 70. Geburtstag gewidmet     

Computational complexity of LCPs associated with positive definite symmetric matrices

Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of ordern forn 2—he has shown that to solve the problem of ordern, either of the two methods goes through 2 ⁿ pivot steps before termination.However that paper leaves it as an open question to show whether or not the same property holds if the matrix,M, in the LCP is positive definite and symmetric. The class of LCPs in whichM is positive definite and symmetric is of particular interest because of the special structure of the problems, and also because they appear in many practical applications.In this paper, we study the computational growth of each of the two methods to solve the LCP, (q, M), whenM is positive definite and symmetric and obtain similar results.This research is partially supported by Air Force Office of Scientific Research, Air Force Number AFOSR-78-3646. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon.     

Stochastic integration,trace and the skeleton of wiener functionals

It is shown that the notion of trace induced by a given complete orthonormal system relates the Skorohod integral with a corresponding Ogawa‐type integral evaluated with respect to the same orthonormal systems. Similarly the multiple Wiener‐Ito integral is shown to be related to a multiple Ogawa‐type integral induced by a complete orthonormal system via the Hu‐Meyer formula with suitably defined multiple traces. The notion of skeleton of a Wiener functional relative to a given orthonormal system is defined and yields what seems to be a “natural” extension of Wiener functionals to the Cameron Martin space and the Wiener processes with a different scale.     

Generalizations of a Theorem of Datko and Pazy

This note gives necessary and sufficient conditions for exponential stability of semigroups of linear operators in Banach spaces. Generalizations of a well-known result due to Datko, Pazy and Neerven are obtained for the case of semigroups of operators that are not strongly continuous.     

Long-time simulation of a highly oscillatory Vlasov equation with an exponential integrator

We change a previous time-stepping algorithm for solving a multi-scale Vlasov–Poisson system within a Particle-In-Cell method, in order to perform accurate long-time simulations. As an exponential integrator, the new scheme allows us to use large time steps compared to the size of the oscillations in the solution.     

Local Sensitivity Analysis in Estimation Problems

This article deals with the problem of local sensitivity analysis, that is, how sensitive are the results of a statistical analysis to changes in the data? A general methodology of sensitivity analysis is applied to some statistical problems. The proposed methods are applicable to any statistical problem that can be expressed as an optimization problem or that involves solving a system of equations. As some particular examples, the methodology is applied to the maximum likelihood method, the standard and constrained methods of moments and the standard and constrained probability weighted moments methods. Unlike the standard method of moments, the constrained method of moments ensures that the obtained estimates are always consistent with the data. Closed analytical formulas for the calculation of these local sensitivities are derived. The obtained sensitivities include: (a) the objective function sensitivities to data points and (b) the sensitivities of the estimated parameters to data points. The derived formulas for the sensitivities are based on recent results in the area of mathematical programming. Several examples of parameter estimation problems are used to illustrate the methodology.     

Measure and integration: The basic extension and representation theorems in terms of new inner and outer envelopes

The work of the author in measure and integration is based on new inner and outer envelope formations, which replace the traditional Carathéodory outer measure and certain simple suprema and infima. The new formations lead to essential improvements in both the extent and the adequacy of the basic results. However, they did not find an entrance into the recent textbook literature. The present paper seeks to demonstrate their power with the examples of the basic inner and outer extension and representation theorems for set functions and functionals.