# Bernhard Riemann's Gesammelte mathematische Werke und by Weber H., Dedekind R. (eds.)

By Weber H., Dedekind R. (eds.)

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Topics in Hyperplane Arrangements, Polytopes and Box-Splines (Universitext)

A number of mathematical components which were constructed independently during the last 30 years are introduced jointly revolving round the computation of the variety of crucial issues in appropriate households of polytopes. the matter is formulated right here by way of partition services and multivariate splines. In its easiest shape, the matter is to compute the variety of methods a given nonnegative integer may be expressed because the sum of h mounted optimistic integers.

Mathematical logic and applications. Proc.meeting, Kyoto, 1987

Those court cases contain the papers offered on the good judgment assembly held on the learn Institute for Mathematical Sciences, Kyoto college, in the summertime of 1987. The assembly customarily lined the present examine in a number of components of mathematical good judgment and its functions in Japan. a number of lectures have been additionally awarded by means of logicians from different nations, who visited Japan in the summertime of 1987.

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36 L. Hong et al. A collision with a chaotic saddle in a fractal boundary is the typical mechanism by which hyerchaotic attractors can be suddenly destroyed. In the hyperchaotic crises, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary.

The boundary is shown in (b) of figures. The color coding and symbol hold throughout the chapter. As a increases, the chaotic saddle in the basin interior becomes bigger and closer to the boundary. 0:1648; 0:1649/. In the case, the chaotic saddle is touching the saddle in the boundary when a D 0:1648, creating a chaotic saddle in a fractal boundary when a D 0:1649. The chaotic saddle in the fractal boundary has a complicated structure and plays an extremely important role in an upcoming hyperchaotic crisis.

The following statements hold. N / ˇ ˇ ˇ 1. xkCN /) are stable. N / ˇ ˇ ˇ 2. xkCN /) are unstable. ˇ ˇ ˇ ˇ ˇ < 1 (or 1 and ˇ C 1 and ˇ 2 ˇ < 1), 3. xkCN /) occurs. ˇ ˇ ˇ ˇ C ˇ ˇ < 1 and ˇ 1), then 4. xkCN /) occurs. ˇ ˇ ˇ ˇ ˇ ˇ ˇ 5. xkCN /) occurs. 3 Illustrations A numerical prediction of the periodic solutions of the Henon map is presented with varying parameter b for a D 0:85, as shown in Fig. 1. The dashed vertical lines give the bifurcation points. 5 −2 −1 0 1 2 Parameter b Fig. J. Luo and Y.