Stabilitätsaussagen Über Klassen Von Matrizen Mit Verschwindenden Zeilensummen
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Den Anstoss zu dieser Arbeit gab das von mir entwickelte "Dynamic Social Choice"-Modell, welches das Entscheidungs- verhalten in einer Gruppe zum Gegenstand hat.Um das Verstandnis fur die vorliegende Arbeit zu erhohen, mochte ich deshalb kurz auf die Grundzuge dieses Modells, welches in [6] ausfuhrlich beschrieben ist, eingehen. Das "Dynamic Social Choice"-Modell setzt eine Gruppe von n (n = beliebige, naturliche Zahl) Individuen mit individuellen, reellen Nutzenfunktionen x.(t), i=l, ., n, voraus.Die Aenderungen der individuellen Nutzen werden durch eine Interaktionsmatrix A mit reellen Matrixelementen (fur i j ist a .. gleich dem Grad des J Einflusses des j-ten Individuums auf den Nutzen des i-ten - dividuums) gemass dem Differentialgleichungssystem x.(t) = La .. ( x, (t) - x, (t) ), i=l, ., n, bestimmt. Hi J J In diesem Zusammenhang interessiert dann die frage, fur welche Interaktionsmatrizen - bei beliebigen aber festen Anfangsnutzen x.(O), i=l, ., n - alle individuellen Nutzenfunktionen x.(t), i=l, ., n, fur t.OO gegen denselben endlichen Wert konvergieren. Dies bedeutet, dass ein Konsens uber alle individuellen Nutzen erfolgt.Interaktionsmatrizen, welche diese Eigenschaft besitzen, werden stabil genannt. Die vorliegende Arbeit befasst sich daher mit Stabilitatsproblemen von Differentialgleichungssystemen.Es werden lineare, homogene Differentialgleichungssysteme mit konstanten Koeffizienten, fur welche die Zeilensummen der Koeffizientenmatrix verschwinden, zu- grunde gelegt.Wie bekanntlich aus der Theorie der Differential- gleichungen folgt, ist die Losung eines solchen Differential- gleichungssystems mit der Losung eines entsprechenden Matrix- eigenwertproblems gleichbedeutend.
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