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By Barnsley M.F., Demko S.

Iterated functionality platforms (i.f.ss) are brought as a unified approach of producing a vast type of fractals. those fractals are frequently attractors for i.f.ss and take place because the helps of chance measures linked to useful equations. The life of convinced 'p-balanced' measures for i.f.ss is confirmed, and those measures are uniquely characterised for hyperbolic i.f.ss. The Hausdorff-Besicovitch size for a few attractors of hyperbolic i.f.ss is expected as a result of p-balanced measures. What seems to be the broadest framework for the precisely computable second idea of p-balanced measures - that of linear i.f.ss and of probabilistic combos of iterated Riemann surfaces - is gifted. This generally generalizes past paintings on orthogonal polynomials on Julia units. An instance is given of fractal reconstruction with using linear i.f.ss and second idea.

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Henzinger, B. Horowitz, R. Majumdar, and H. Wong-Toi, Beyond HyTech: Hybrid System Analysis Using Interval Numerical Methods, AAAI Spring Symposium on Hybrid Systems, Stanford University, 1999. A. -H. Ho, and H. Wong-Toi, HyTech: A Model Checker for Hybrid Systems, Software Tools for Technology Transfer 1, 110-122, 1997. Approximate Reachability Analysis of Piecewise-Linear Dynamical Systems KV97. MV99. PLY99. PSK99. VS99. V98. 31 A. Kurzhanski ans I. Valyi, Ellipsoidal Calculus for Estimation and Control, Birkhauser, 1997.

So far we were able to calculate rather easily the reachable states of non-trivial systems with up to 6 dimensions (in fact, the measure of complexity for such problems depends on the dimensionality, the coupling of the variables and the granularity of the discretization). 5 Fig. 3. Calculating reachable states for a 3-dimensional system. 1 Extensions and Applications Piecewise-Linear Systems For purely continuous linear systems there are classical methods, more efficient than ours, for solving certain problems such as stability or controller synthesis.

GM99. HHMW99. HHW97. R. Alur, C. Courcoubetis, N. A. -H. Ho, X. Nicollin, A. Olivero, J. Sifakis and S. Yovine, The Algorithmic Analysis of Hybrid Systems, Theoretical Computer Science 138, 3–34, 1995. R. L. Dill, A Theory of Timed Automata, Theoretical Computer Science 126, 183–235, 1994. E. Asarin, O. Bournez, T. Dang, A. Pnueli and O. Maler, Effective Synthesis of Switching Controllers for Linear Systems, submitted for publication, 2000. O. Bournez, O. Maler and A. Pnueli, Orthogonal Polyhedra: Representation and Computation, in [VS99], 46-60.

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