Zum Cauchy-Problem bei der verallgemeinerten Wärmeleitungsgleichung

Cauchy's problem for the equationu _xx +x ^–1 u _x =u _t ( real) was discussed byD. Colton if –1,–2,–3, ... Now existence and uniqueness theorems and representations of the solutions are given for the cases =–1,–2, –3,... The methods ofD. Colton and of this paper are different but the results are similar.     

Constrainedn-person games

The usual properties of a characteristic function game were derived byvon Neumann andMorgenstern from the properties of a game in normal form. In this paper we give a linear programming principle for the calculation of the characteristic function. The principle is a direct application ofCharnes' linear programming method for the calculation of the optimal strategies and the value of a two-person zero-sum game. The linear programming principle gives another method for proving the standard properties of a characteristic function when it is derived from a game in normal form. Using an idea originated byCharnes for two person games, we develop the concept of a constrainedn-person game as a simple, practical extension of ann-person game. However the characteristic function for a constrainedn-person game may not satisfy properties, such as superadditivity, usually associated with a characteristic function.     

Zur Konstruktion eindimensionaler homogener Markoffscher Prozesse

Summary The aim of this paper is the construction of a class of semi-groups of homogeneous onedimensional Markov processes with respect to a given infinitesimal operator. This is done namely by the method of stochastic integration represented in the book of Skorokhod. The Lipschitzian conditions — needed there — are weakened to uniform continuity by a method of approximation of semi-groups. Also to these semi-groups we can construct Markov processes as solutions of stochastic integral equations. The representation is connected with the result of Meyer, Watanabe, Motoo.     

A generalisation of systematic relaxation methods for consistently ordered matrices

Summary A systematic relaxation method is analysed for consistently ordered matrices as defined by Broyden (1964). The method is a generalisation of successive over-relaxation (S.O.R.). A relation is derived between the eigenvalues of the iteration matrix of the method and the eigenvalues of the Jacobi iteration matrix. Forp-cyclic matrices, the method corresponds to using a special type of diagonal matrix instead of a single relaxation factor. For certain choices of this diagonal matrix, the method has a better asymptotic rate of convergence than S.O.R. and requires less calculations and computer store.     

On the eigenproblem for displacement integral equations

Summary This paper is concerned with the eigenproblem for displacement integral equations which have kernels that are finite cosine transforms. Previous work is extended to a broader class of kernels and the eigenfunctions are treated for the first time.This work was supported by the United States Atomic Energy Commission.     

Über ein Problem der Fließbandfertigung bei stochastischer Nachfrage

Zusammenfassung Bei der Fließbandfertigung mit stochastischer Nachfrage interessiert man sich für die Anpassung der Produktion an die Marktgegebenheiten. Es wird die Produktion eines Gutes betrachtet. Der Unternehmung stehen mehrere gleichwertige Fertigungsstraßen zur Verfügung. Es wird angenommen, daß alle in Betrieb befindlichen Fertigungsstraßen den gleichen und unveränderlichen Output pro Zeiteinheit haben. Gesucht ist ein optimales Verhalten, das die diskontierte Summe der zu erwartenden Kosten minimiert. Es ist naheliegend, Verfahren der dynamischen Programmierung zu verwenden. Für das Modell wird der Fall zyklischer Kontrolle betrachtet. Die Verteilung der Nachfrage während der Kontrollperioden sei bekannt, konstant, und es gelte stochastische Unabhängigkeit für verschiedene Perioden. Überschreitet die Nachfrage den Lagerbestand, dann erfolgen für den fehlenden Output Vormerkungen. Die Belieferung findet später statt.

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Summary In assembly line production with stochastic demand one is interested in the adaption of the output to the market situation. The production of one product is considered. The company has several assembly lines of the same type. It is assumed that the assembly lines in action have the same, and a constant output rate. An optimal policy, minimizing the expected value of the total discounted costs, has to be found. It is concievable that the method of dynamic programming should be used. In the model the case of cyclic control is considered. The distribution of demand during each control period is given, constant, and there is stochastic independence for different periods. If the demand exceeds the stock the deficient output is noted and delivery takes place later.

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Vorgel. v.:W. Wittmann, Heidelberg.     

A duality theorem for continuous-time linear programming problems

Summary In dealing with dynamic economic policy models one encounters optimization problems whose objective function is an integral of a linear function of a finite number of continuous variables and whose constraints are linear integral inequalities. A set of intertemporal efficiency conditions (equilibrium conditions) yielding the optimal policy are given. By approximating the continuous problem by a set of discrete problems and appealing to a well known convergence theorem in functional analysis a continuous analog of the duality theorem is proved.

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Zusammenfassung Bei der Beschäftigung mit dynamischen Modellen der ökonomischen Politik stößt man auf Optimierungsprobleme, deren Zielfunktion ein Integral einer linearen Funktion von einer endlichen Anzahl stetiger Variablen ist und deren Beschränkungen lineare Integral-Ungleichungen sind. Eine Menge intertemporaler Effizienz-Bedingungen (Gleichgewichtsbedingungen), die zur optimalen Politik führen, sind gegeben. Durch Approximation des kontinuierlichen Problems mittels einer Menge von diskreten Problemen und Berufung auf einen wohlbekannten Konvergenzsatz aus der Funktionalanalysis wird ein stetiges Analogon des Dualitätstheorems bewiesen.

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The author is indebted to Mr.Arnold Faden for helpful suggestions and to ProfessorKarl A. Fox andGerhard Tintner for encouragement during the preparation of the paper. This research has been partially supported by a grant from the Ford Foundation to the School of Business Administration administered by the Center for Research in Management Science, University of California, Berkeley.

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Vorgel. v.:G. Tintner.     

Ein verallgemeinertes deterministisches Netzplanmodell

Zusammenfassung Bei der Netzplantechnik und ihren Anwendungen werden unterschiedliche Modelle und Verfahren behandelt. Unter Verwendung der Graphentheorie wird ein verallgemeinertes Netzplanmodell beschrieben, das die bekannten Modelle (Vorgangsknotennetzplan, Vorgangspfeilnetzplan, Ereignisknotennetzplan) als Sonderfälle enthält.

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Summary In network-analysis and its applications one usually distinguishes between activity-on-arrow, activity-on-node and event-on-node networks. Using the theory of graphs, a generalized deterministic network model is described, containing the well known techniques as special cases.