In the paper we describe obstructions for the existence of symplectic and Hamiltonian symplectic circle actions on closed compact manifolds in terms of Hirzebruch genera and relations between differential and homotopic invariants of such manifolds. All manifolds considered are assumed to be closed and compact.Theorem 1. The Todd genus of a manifold equipped with a symplectic circle action with isolated fixed points is either equal to zero and then the action is non-Hamiltonian, or equal to one and then the action is Hamiltonian. Any symplectic circle action on a manifold with the positive Todd genus is Hamiltonian.Proof. A symplectic circle action is Hamiltonian if and only if there is such a connected component of the fixed point set that its weights of the S 1representation in the normal bundle are positive. This connected component is unique and corresponds to the global minimum of the moment map (see details in [1]). Let us apply the fixed point formula for Hirzebruch genus χ y corresponding to the power series x(1+ye −x(1+y) ) 1−e −x(1+y) . Let M s be connected components of the fixed point set, d s be the number of negative weights of the S 1 -representation in the normal bundle of M s , then(see [2]) χ y (M) = s (−y) ds χ y (M s ),here the sum is taken over all connected components of the fixed point set.Since χ y (pt) = 1, the constant term of χ y of a manifold equipped with a Hamiltonian circle action with isolated fixed points is equal to one. Vice versa, if the constant term of χ y (M) is not equal to zero, then any symplectic circle action on M is Hamiltonian. The statement of the theorem follows from T d(M) = χ y (M)| y=0 .The result of Theorem 1 is a particular case of the restrictions imposed on manifolds with a circle action by the Conner-Floyd equations. Let us fix * The author was supported by Seggie Brown fellowship.

A new method of diagonalisation of the XY-Hamiltonian of inhomogeneous open linear chains with periodic (in space) changing Larmor frequencies and coupling constants is developed. As an example of application, multiple quantum dynamics of an inhomogeneous chain consisting of 1001 spins is investigated. Intensities of multiple quantum coherences are calculated for arbitrary inhomogeneous chains in the approximation of the next nearest interactions.

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