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(IR+)n and log : (18+)n -+ RI be defined componenentwise: exp(x)i = exp(xi) and log(x)i = log(xi). These are mutually inverse bijections between Rn and (R+)n. Let A : (R+)n - (R+)n be any function on the positive cone and let £(A) : Rn -> Rn denote the function log(A(exp)).

The * operation cannot now be defined on all of R((A)) but does exist on those series whose constant term is 0. 2]. It is rather peculiar that there are two generalisations of Kleene's original result, in one of which idempotency plays a crucial role while in the other it is the free monoid structure of A*. It would be very interesting to have a single formulation which includes both contexts and yet retains the clarity of Kleene's original result. 2 The tropical dioid The star operation is not a simple one because of its infinitary nature.

They must usually be closed, or autonomous, in that they require only an initial condition to generate a sequence of occurrence times. Open systems, in contrast, require input to be regularly provided. To model this, it is more convenient to work, not with vectors in Rn, but with appropriate functions R -> Rn, which represent input, or output, histories. This requires an extension of the theory of topical functions to infinite dimensional spaces, a problem studied in [Kola]. 2. 2, Gaubert has made initial investigations in this direction, [Gau].

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Applied Functional Analysis: Numerical Methods, Wavelet Methods, and Image Processing by Abul Hasan Siddiqi

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