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We analyze the spectral properties of the Perron-Frobenius operator, U, associated with some simple highly chaotic maps. We obtain a spectral decomposition of U in terms of generalized eigenfunctions of U and its adjoint. The corresponding eigenvalues are related to the decay rates of correlation functions and have magnitude less than one, so that physically measurable quantities manifestly approach equilibrium. To obtain decaying eigenstates of unitary and isometric operators it is necessary to extend the Hilbert space formulation of dynamical systems. We describe and illustrate a method to obtain the decomposition explicitly.

This paper describes current activities of the MPI-2 Forum. The MPI-2 Forum is a group of parallel computer vendors, library writers, and application specidits working together to define a set of extensions to MPI (Message Passing Interface). MPI was defined by the same process and now has many implementations, both vendor-proprietary and publicly available, for a wide variety of parallel computing environments. In this paper we present the salient aspects of the evolving MPI-2 document as it now stands. We discuss proposed extensions and enhancements to MPI in the a r e s of dynamic process management, one-sided operations, collective operations, new language binding, real-time computing, external interfaces, and miscellaneous topics. 1 Introduction During 1993 and 1994, a group of parallel computer vendors, library writers, and application scientists met regularly to define a standard interface for message-passing libraries. The result of this effort was MPI (Message-Passing Interface) [8]. Implementations of MPI are now widely available, including portable and freely available implementations [2, 4, 91 and specialized versions from vendors. General information on MPI is available at [I]. For the purposes of this paper, it will be useful to refer to the result of the initial MPI standardization effort as "MPI-1." MPI-I defined an interface for a specific message-passing model of parallel computation, in which a fixed number of processes with disjoint a Jdi=ss spaces communicate throzgh a cooperative mechanism (when two processes commurikate, one sends and the other receives). MPI provides many types of point-to-point communication, to incorporate requirements for robustness, expressivity, and performance. Messages are strictly typed and scoped, allowing for communication in a heterogeneous environment. MPI also contains an extensive set of collective operations, process topology functions, and a profiling interface. The most distinctive feature of the current MPI-2 proposals described in this paper is that they go beyond the strict message-passing model defined above. In MPI-2, processes may create other processes, so that the number of processes in an MPI computation can change dynamically (Section 2). Processes can interact directly with the memory of other by a contractor of the U. S. Government under contracf No. W-31-104ENG-38. Accordingly, tfm U. S. Government r a i n s a nonexclusive, royalty-free license to prbllrh or raproduce the published form of this contribution, w allow others m do 10. for 1 A Ew fs ~~~~D DISCLAIMER Portions of this document m a y be illegible in electronic image products. Images are produced from the best available original dOCUment processes (Section 3). Extensions, semantic modifications, and subset definitions in support of real-time and embedded systems (Section 4) also represent changes to the computational model. Other topics being discussed in MPI-2 include extending MPI-1's collective operations to intercommunicators and nonblocking operations (Section 5) , bindings for C...

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