February 26, 2017

Tietze - Einfuehrung in die angewandte Wirtschaftsmathematik by Jürgen Tietze

By Jürgen Tietze

Mathematik in den Wirtschaftswissenschaften bedeutet einerseits das challenge, mathematische Ideen zu verstehen, um die dazugehörigen Techniken zu beherrschen und andererseits, diese zunächst abstrakten Techniken zielgerichtet und sinnvoll für ökonomische Anwendungen nutzbar zu machen. Das nun in der 17. Auflage vorliegende Buch ist als Lehr-, Arbeits- und Übungsbuch vorrangig zum Selbststudium konzipiert. Es berücksichtigt beide Aspekte durch ausführliche Darstellung, Begründung und Einübung mathematischer Grundelemente und ökonomisch relevanter mathematischer Techniken.

Die aktuelle Auflage enthält erstmals einen Intensiv-Brückenkurs zur elementaren Algebra mit mehr als 500 Übungselementen (in Übungsaufgaben, Selbstkontroll-Tests, Eingangstest, Schlusstest).

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Fig. 6). 6 We now can give a visual explanation of the condition that the integral f is a Bott function on the isoenergy surface Q = fH = constg . Take the line H = const in the two-dimensional plane R2 (H f ). Suppose that this line does not pass through singular points of and intersects the smooth pieces of transversally. Then f will be a Bott function on Q (notice that Q is the preimage of this line in M 4 ). , among them there will be neither tori nor Klein bottles. , x is a xed point of the Poisson action of R2 .

If H (x) = h , then we denote the corresponding isoenergy surface by Qh . As we already know, it is always invariant with respect to the Hamiltonian eld v . Consider an isoenergy three-dimensional surface Q = Q3 . In what follows, we shall assume it to be a smooth compact submanifold in M 4 . In particular, we shall consider only those 3-surfaces on which dH 6= 0. In the case of two degrees of freedom, for integrability of the system v it is su cient to have just one additional integral f functionally independent of the energy integral H .

17. , the set = F (K ) Rn , is called the bifurcation diagram. Thus, the bifurcation diagram is the set of all critical values of F . According to the Sard theorem, the set has zero measure in Rn . In most examples of integrable systems appearing in physics and mechanics, the bifurcation diagram is a manifold with singularities. In other words, it consists of several strata (pieces) i each of which is a smooth i -dimensional surface in Rn . 4 as = 0 + 1 + : : : + n;1 , where di erent strata do not intersect with each other and the union of them gives the whole of .

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