Resonance type dielectric behavior in solids

Experimental evidence now indicates that the microwave dielectric properties of solids are strongly influenced by the infrared active phonon modes. The “real dielectric constant”, ε′, of polar materials, i. e. those in which ?′ is noticeably greater than the square of the optical index of refraction, exhibits a temperature dependence dominated by the change of integrated infrared absorption. The microwave dielectric loss, ?″, is influenced by the low frequency “tail” of the infrared resonance type absorption, as may be inferred from the monotonic increase with temperature and proportionality to frequency. It is suggested that the one dimensional coupling of the electric field to the acoustic modes of a dielectric solid would lead to additional low frequency “background” dielectric losses which are essentially frequency independent, in agreement with experimental observations.     

Stereochemistry of cyclobutanone resulted from cycloadditions of t-butylcyanoketene to bicyclo[2.2.1]heptene derivatives, as evidence for a π2s+π2a reaction mode

The stereochemistry of the cyclobutanones 1-7, resulted from the reaction of t-butylcyanoketene with bicyclo[2.2.1]heptene, bicyclo[2.2.1]heptadiene, 1,4 - dihydro - 1,4 - methanonaphthalene, 1,4 - dihydro - 9 - (1 - methylethylidene) - 1,4 - methanonaphthalene, 1,4 - dihydro - 1,4 - epoxynaphthalene, l,4,4a,8b - tetrahydro - 1,4 - methanobiphenylene (l,4,4a,8b) and 1,4,4a,8b - tetrahydro - 1,4 - methanobiphenylene(1,4,4aβ,8bβ) was established as having the cyclobutanone ring exo and the t-Bu group in the configuration. These findings represent a stereochemical argument in favour of a _π2_s + _π2_a reaction mode of t-butylcyanoketene to the above mentioned bicyclo[2.2.1]heptene derivatives. Observations regarding preservation of the original configurations of alkenes as well as the geometrical distorsion of the cyclobutanones are shortly discussed.     

121Sb-Mössbauer-Spektren von Antimon(V)-Verbindungen. II

¹²¹Sb Mössbauer Spectra of Antimony (V) Compounds. II The Mössbauer resonance effect of ¹²¹Sb has been studied at 4.2 K in a series of 11 antimony(V) compounds which are in most cases bridged. Isomer shift and quadrupole splitting variations are considered in the light of results of X-ray and vibrational spectroscopy. Preparation and the vibrational spectra of [NMe₄][SbCl₄(n₃)₂] are described. Within the limitations of available structural data related tin compounds are compared with those of corresponding ¹¹⁹Sn Mössbauer spectra.     

Approximations of inventory models

Summary A general procedure is presented for constructing approximations of discrete review single product dynamic inventory models. Bounds are derived for the approximations and compared with the ones ofHinderer [1978],Whitt [1978] for the approximation of a general dynamic program. Good order-policies are constructed. Finally, conditions are given under which the sequence of bounds associated with a sequence of approximating models converges to zero.

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Zusammenfassung Es wird ein allgemeines Verfahren zur Approximation periodisch inspizierter dynamischer Ein-Produkt-Lagerhaltungsmodelle betrachtet. Dazu werden Fehlerschranken gegeben. Diese werden verglichen mit denen vonHinderer [1978],Whitt [1978] für ein allgemeines dynamisches Entscheidungsmodell. Außerdem werden gute Bestellpolitiken konstruiert und hinreichende Bedingungen gegeben, unter denen die Fehlerschranken einer Folge von approximierenden Modellen gegen Null konvergiert.

     

Periodic,multiplicative number-theoretical functions

D. Leitmann andD. Wolke [3] proved that multiplicative functions, which are restrictions of continuous, periodic functionsF: with irrational period, are constant equal to 0 or 1. In this note a simpler proof for this result is given.     

Banach convolution algebras of functions II

LetG be a noncompact, locally compact group. By means of generalized dyadic decompositions ofG, translation invariant Banach spacesF(B, B, X) of (classes of) measurable functions onG are constructed, e. g. certain weighted amalgams ofL ^p -spaces. Basic properties of these spaces are derived and connections with spaces considered in the literature are indicated. As a main result, sufficient conditions are given which imply that a space of this type is a Banach algebra with respect to convolution.With 1 Figure     

A method of bisection for discounted Markov decision problems

We describe in this paper how to introduce the concept of bisection into Dynamic Programming of stationary Markovian decision processes with infinite horizon. The combination of these two methods delivers a new class of algorithms. A basic bisection algorithm is given. Its rate of convergence is rather independent of the input data, especially the discount factor or contracting factor. If in any method of successive approximations we have > 0.5, then the bisection method has a faster convergence than the successive approximations. Bisection is especially advantageous if is close to unity. The computation of two inventory control problems with the basic algorithm of bisection is presented. The second example has a discount factor=0.9994. Using Howard's automobile replacement problem the new method is compared with value iteration, policy iteration andPorteus' method of extrapolations. Among them bisection turned out to be the most powerful procedure. The combination withPorteus' method of extrapolation is not as efficient as the bisection method alone. We also investigate how to combine the bisection method with monotone extrapolation when the transition matrices do not have equal row sums.

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Zusammenfassung In dieser Arbeit wird die Idee der Bisektion in die Dynamische Optimierung stationärer Markovscher Entscheidungsprozesse mit unendlichem Planungshorizont eingeführt. Die Verbindung dieser beiden Methoden schafft eine neue Klasse von Algorithmen. Ein Basisalgorithmus, der selbst Ausgangspunkt zahlreicher Verfeinerungen und Spezialisierungen bildet, wird vorgestellt. Seine Konvergenzgeschwindigkeit ist nahezu unabhängig von Input-Daten, speziell dem Diskontfaktor bzw. der Lipschitz-Konstanten. Die Bisektion ist immer dann schneller als die Methode der sukzessiven Approximation, falls>0,5 ist. Deshalb ist das neue Verfahren besonders bei Problemen mit schwacher Kontraktionseigenschaft, d.h. nahe bei eins, geeignet. Der Rechenverlauf des Basisalgorithmus wird anhand zweier Lagerhaltungsmodelle diskutiert von denen das zweite eine Lipschitz-Konstante=0.9994 besitzt. Am Beispiel des Howardschen Autoersatzproblems zeigt sich, daß das Bisektionsverfahren im Vergleich mit der Wertiteration, Politikiteration undPorteus' Extrapolationsverfahren das schnellste Verfahren ist. Die Kombination der Bisektion mit der Extrapolationsmethode ist ungünstiger als die Bisektion allein. Es wird noch untersucht, wie die Bisektion mit der monotonen Extrapolation verbunden werden kann, wenn die Zustands-übergangsmatrizen keine konstanten Zeilensummen besitzen.

     

A comparison of two approaches to the optimal positioning of a new product in an attribute space

This paper is concerned with the optimal positioning of a new product in an attribute space. A deterministic optimization model that relies on the axiom of choice is compared with the model byShocker/Srinivasan [1974] that incorporates a probabilistic measure of choice. Investigating the assumptions and the construction of both models, it is shown that the probability function byShocker/Srinivasan is based on a controversial assumption and incorporates a parameter that has not been given an economic explanation. Furthermore, both models are applied to complex configurations. It is demonstrated that the optimal positions for the new product resulting from the model byShocker/Srinivasan depend heavily on the parameter just mentioned and may run contrary to plausible consequences of the axiom of choice.

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Zusammenfassung Für die optimale Gestaltung neuer Produkte, d.h. ihrer Positionierung in einem Raum von Produkteigenschaften, wird ein Modell angeboten, das auf einer deterministischen Kaufentscheidung beruht, ein anderes, das eine stochastische Kaufentscheidung unterstellt. Es wird gezeigt, daß der zuletzt genannte Modelltyp stark von der Spezifikation eines ökonomisch bisher nicht erklärten Parameters abhängt und z.T. Konsequenzen aufweist, die dem Wahlaxiom zuwider laufen, auf das das erstgenannte Modell aufbaut. Durch einen Vergleich auch komplexer Konfigurationen von Käuferbeurteilungen neuer Produkte wird gezeigt, welche Gestaltungsentscheidungen auf der Grundlage der beiden Modelle zur Optimierung einer Zielfunktion zu treffen sind.

     

Über die optimale Beleuchtung einer geraden Straße

The following problem is due toL. Fejes Tóth: For a given sort of lamps letf(x) be the intensity of brightness in each point having distancex from the foot of the lamp. How must infinitely many lamps of this kind be arranged on a both-side infinite rectilinear road, such that the infimum of the intensities of brightness, extended over the whole street, is maximal, subject to the condition that the average number of lamps per kilometer is bounded by a fixed number (“Best distributions”)? The main result of this paper is: Iff is strictly convex, the equidistant distribution is best.     

On the Fourier transformation on spaces of (p,q)-multipliers

LetG denote a locally compact abelian topological group. The aim of the present paper is to prove an “intermediate” result between two well-known results ofL. Hörmander andG. I. Gaudry concerning the structure of the spaces ?_G?μ?t ^p,q (G).