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In order to provide the platform functions in a correct and consistent way, the platform servers need to have certain items of knowledge about the infrastructure: For searching mobile clients as well as for the handover mechanism, the platform server needs to know all its neighbors, that is, leaf servers with geographical regions that are adjacent to its own region For the search mechanism, it needs to know the platform server in the next hierarchy level, which is responsible for the domain to which the platform server itself belongs In order to create a new object, it needs to know at least one object register To store and retrieve object pro les, it needs to know at least one pro le server.

From a numerical point of view, the general question of ef cient procedures to compute approximately the moments of the number of crossings remains wide open We will come back to this subject in s 4 and 5, adding some more or less recent results This will be done in the context of relating crossings to the distribution of the maximum of a stochastic process, even though it has an independent interest Finiteness of moments of crossings of Gaussian processes has been considered by Belayev (1966), Miroshin (1977), and Cuzick (1975) For stationary Gaussian processes, the suf cient condition for niteness of the variance of N0 (X, I ) in Exercise 36 is in Cram r and Leadbetter s book (1967), where e an explicit formula for the variance is also given Geman (1972) proved that this condition is also necessary for niteness at the level u = 0.

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Since the PLASMA infrastructure can be deployed in many different con gurations, the above-mentioned information has to be provided dynamically, that is, the platform servers need a way to retrieve all necessary information independent of the actual setting Therefore, we decided to use Jini lookup services [21] to provide and maintain this information For each domain, one logical lookup service exists Each lookup service contains the following information: All PLASMA servers in the domain of the Jini lookup service, including the geographical region belonging to each server The addresses of object registers available in this domain The addresses of data bases available in this domain Whether the domain is a search domain or not How to contact a lookup service of the next hierarchy level upwards How to contact a lookup service of the next hierarchy level downwards, which is responsible for a particular domain.

In a recent paper, Kratz and Le n (2006) proved that the same condition is necessary and suf cient o for any level u and also for the number of crossings with some differentiable curves For non-Gaussian processes, suf cient conditions have been given by Besson and Wschebor (1983) The next theorem gives suf cient conditions, which are reasonably easy to check in speci c cases, to be able to assure the niteness of k E Nu It is taken from Nualart and Wschebor (1991) Theorem 36 Let m be a positive real number Consider a real-valued stochastic process X = {X(t) : t I } de ned on a compact interval I of the real line, with paths of class C p+1 , p > 2m.

So, every platform server can access the information necessary to contact other PLASMA servers, such as pro le servers from the lookup services The lookup service in the top hierarchy knows all root servers with the corresponding lookup services in the domains directly below the root servers An automatic recon guration of PLASMA is supported by the fact that all platform servers register appropriate lookup services, requesting to be noti ed if changes in the platform structure occur 1332 Platform Components In order to provide useful location-aware applications, it is necessary that the current position of a user can be queried by the application This means a suitable positioning system has to be part of the infrastructure and must be able to submit its information to PLASMA This information has to be processed in a suitable manner.

We assume also that for each t I the random variable X(t) has a density and that for some > 0, C = sup{pX(t) (x) : t I, x [u , u + ]} < Then.

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