Lemma 5. If the class 7) is determinate, then each space X E 7 ) \ "P2n is 7)2n_l(O)-preuniversal.Proof. We suppose that the class :P is determinate and fix spaces X E 7~\:P2n and A E 7~2,~-1(0). Let us embed the space X into the Hilbert cube Q and the space A into the Cantor cube 2 ~. Let a: 2 ~ --+ Q be an arbitrary surjective mapping. By [6, 37.1], we have a-l(X) E P. Moreover, a-l(X) r P=n (the assumption that a-l(X) E :P2n implies 2 `0 \ a-l(X) E P2n-1, and therefore, Q \ X = a(2" \ a-l(X)) E P2n-1, whence X E 7~2n, a contradiction). Since 7)2n_1 C P2n+2 and the class 7 ) is determinate,

No abstract

No abstract

No abstract

Extended states for polyharmonic operators with quasi-periodic potentials in dimension two J. Math. Phys. 53, 103512 (2012) Mean first-passage time for random walks in general graphs with a deep trap J. Chem. Phys. 137, 124104 (2012) Smallest eigenvalue distributions for two classes of β-Jacobi ensembles J. Math. Phys. 53, 103301 (2012) An analytic criterion for generalized synchronization in unidirectionally coupled systems based on the auxiliary system approach Chaos 22, 033146 (2012) Third order integrability conditions for homogeneous potentials of degree −1We give a definition and discuss main properties of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems. The definition relies on a technical construction which is a definition of a relative Hopf invariant for divergence-free vector fields in multiconnected domains of the following type: ͑flat domain in R 2 ͒ ϫ ͑circle͒. We prove correctness of the definitions, and discuss ergodic interpretation of the Hamiltonian Hopf invariant, together with its relation with the Calabi invariant.