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In this paper, a translation of the visual description technique HyCharts to Hybrid Data-Flow Graphs (HDFG) is given. While HyCharts combine a data-flow and a control-flow oriented formalism for the specification of the architecture and the behavior of hybrid systems, HDFG allow the efficient and homogeneous internal representation of hybrid systems in computers and their automatic manipulation. HDFG represent a system as a data-flow network built from a set of fundamental functions.
The translation permits to combine the advantages of the different description techniques: The use of HyCharts for specification supports the abstract and formal interactive specification of hybrid systems, while HDFG permit the tool based optimization of hybrid systems and the synthesis of mixed-signal prototypes.
New conditions of solvability based on a general theorem on the calculation of the index at infinity for vector fields that have degenerate principal linear part as well as degenerate ... next order ... terms are obtained for the 2 Pi-periodic problem for the scalar equation x'' +n2x=g(|x|)+f(t,x)+b(t) with bounded g(u) and f(t,x) -> 0 as |x| -> 0. The result is also applied to the solvability of a two-point boundary value problem and to resonant problems for equations arising in control theory.
AMS subject classifications: 47Hll, 47H30.
Let G be a group of prime order q with generator g. We study hardcore subsets H is include in G of the discrete logarithm (DL) log g in the model of generic algorithms. In this model we count group operations such as multiplication, division while computations with non-group data are for free. It is known from Nechaev (1994) and Shoup (1997) that generic DL-algorithms for the entire group G must perform p2q generic steps. We show that DL-algorithms for small subsets H is include in G require m/ 2 + o(m) generic steps for almost all H of size #H = m with m <= sqrt(q). Conversely, m/2 + 1 generic steps are su±cient for all H is include in G of even size m. Our main result justifies to generate secret DL-keys from seeds that are only 1/2 * log2 q bits long.