[en] Using the (generalized) Darboux transformation in the case of the Clifford torus, we construct for all Pythagorean triples (p,q,n) ELEMENT OF z³ a CP³-family of Willmore tori in S⁴ with Willmore energy 2(nπ)².

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[en] This letter restudies the Nosé-Hoover oscillator. Some new averagely conservative regions are found, each of which is filled with different sequences of nested tori with various knot types. Especially, the dynamical behaviors near the border of “chaotic region” and conservative regions are studied showing that there exist more complicated and thinner invariant tori around the boundaries of conservative regions bounded by tori. Our results suggest an infinite number of island chains in a “chaotic sea” for the Nosé-Hoover oscillator

[en] This paper is devoted to the classical Knotting Problem: for a given manifold N and number m describe the set of isotopy classes of embeddings N→S^m. We study the specific case of knotted tori, that is, the embeddings S^p×S^q→S^m. The classification of knotted tori up to isotopy in the metastable dimension range m ≥ p + 3/2q + 2, p≤q, was given by Haefliger, Zeeman and A. Skopenkov. We consider the dimensions below the metastable range and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension. Bibliography: 35 titles.

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[en] We study the variation of the Mumford quotient by the action of a maximal torus T on a flag variety G/B as we change the projective embedding G/B→P(V(χ)), where the T-linearization is induced by the standard G-linearization. To do this, we describe the linear spans of the supports of the semistable orbits. This enables us to calculate the rank of the Picard group of the quotient (G/B)^ss//T in the case when G contains no simple components of type A_n

[en] We discuss several open problems in the theory of Hamiltonian systems. They are all related to the Hamiltonian systems with divided phase space, where Kolmogorov–Arnold–Moser tori coexist with ergodic components of positive measure. (open problem)

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