By Viktor A. Sadovnichiy, Mikhail Z. Zgurovsky

As within the past quantity at the subject, the authors shut the distance among summary mathematical ways, akin to utilized tools of contemporary algebra and research, primary and computational mechanics, nonautonomous and stochastic dynamical platforms, at the one hand and useful purposes in nonlinear mechanics, optimization, determination making conception and keep an eye on concept at the other.

Readers also will enjoy the presentation of contemporary mathematical modeling equipment for the numerical answer of complex engineering difficulties in biochemistry, geophysics, biology and climatology. This compilation might be of curiosity to mathematicians and engineers operating on the interface of those fields. It provides chosen works of the joint seminar sequence of Lomonosov Moscow nation collage and the Institute for utilized process research at nationwide Technical collage of Ukraine “Kyiv Polytechnic Institute”. The authors come from Brazil, Germany, France, Mexico, Spain, Poland, Russia, Ukraine and america.

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**Extra info for Continuous and Distributed Systems II: Theory and Applications (Studies in Systems, Decision and Control)**

**Example text**

2 cx exp − c 2 Moreover, for any ν we have Sν ∼ Hence, pk (n) ∼1+ p(n) ∞ ν=1 ν . 2 cx (−1)ν Sν ∼ exp − e− 2 . 2. C. Auluck et al. [3] formulated the following hypotheses. , for every k we have pk (n) ≤ pk0 (n). Then, for n → ∞ asymptotic relation k0 ∼ c−1 n 1/2 log n holds. e. for every k we obtain pk (n) ≤ pk0 (n). Then for k1 < k2 ≤ k0 we have pk1 (n) ≤ pk2 (n) and for k0 < k1 < k2 we have pk2 (n) < pk1 (n). They proved that for any δ > 0 and for sufficiently large n, we have n 1/2 < k0 ≤ (1 + δ)c−1 n 1/2 log n.

9) The mark n has a more complicated definition (see [4–6, 14]). 5 If all the conditions of the Theorem 4 are satisfied, then the marked molecules of the systems with gravitational potential consist of the ribs of the following types: (a) the rib A−A with the mark r = 0, if Q3 = {H = h < 1}; and with the mark r = 1/2, if Q3 = {H = h > 1}. The mark ε = +1 in both cases; (b) the rib A−B with the mark r = 0 and the mark ε = +1; (c) the rib B−B with the mark r = ∞ and the mark ε = −1, if the rib is symmetric relative to the axis OH; and the mark ε = +1 in other case; (d) if the system admits the atoms of type B, then there exist marks of type n.

O. Kantonistova 2 Topological Classification of Geodesic Flows … 15 Fig. 2 Complicated Atoms and Molecules Recall that an atom is complicated if critical connected level surface of function f contains several critical points. Such objects naturally arise in many problems in geometry and physics (see Fig. 6). We now give a simple example. Suppose that a finite group G acts smoothly on a surface X 2 , and let f be a G-invariant Morse function; then, as a rule, such function will be complicated. Indeed, if, for instance, the orbit of a critical point x entirely belongs to a connected component of the level line {f (x) = const}, then this level contains several critical points.