doi: 10.1007/s00222-010-0240-4pmid: N/A
We show that the Rost nilpotence theorem is true for geometrically rational surfaces over fields of characteristic zero.
doi: 10.1007/s00222-010-0241-3pmid: N/A
We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T.
doi: 10.1007/s00222-010-0242-2pmid: N/A
We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ≪1,K≫1,s>1, we construct smooth initial data u 0 with $\|u_{0}\|_{{H}^{s}}<\delta$ , so that the corresponding time evolution u satisfies $\|u(T)\|_{{H}^{s}}>K$ at some time T. This growth occurs despite the Hamiltonian’s bound on $\|u(t)\|_{\dot{H}^{1}}$ and despite the conservation of the quantity $\|u(t)\|_{L^{2}}$ .
doi: 10.1007/s00222-010-0243-1pmid: N/A
We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic (“center”) behavior, e.g. partially hyperbolic diffeomorphisms. A number of applications illustrate its power.
doi: 10.1007/s00222-010-0245-zpmid: N/A
The Frobenius number F(a) of an integer vector a with positive coprime coefficients is defined as the largest number that does not have a representation as a positive integer linear combination of the coefficients of a. We show that if a is taken to be random in an expanding d-dimensional domain, then F(a) has a limit distribution, which is given by the probability distribution for the covering radius of a certain simplex with respect to a (d−1)-dimensional random lattice. This result extends recent studies for d=3 by Arnold, Bourgain-Sinai and Shur-Sinai-Ustinov. The key features of our approach are (a) a novel interpretation of the Frobenius number in terms of the dynamics of a certain group action on the space of d-dimensional lattices, and (b) an equidistribution theorem for a multidimensional Farey sequence on closed horospheres.
doi: 10.1007/s00222-010-0246-ypmid: N/A
In this article we study the Lyapunov exponent for random matrix products of positive matrices and express them in terms of associated complex functions. This leads to new explicit formulae for the Lyapunov exponents and to an efficient method for their computation.
Showing 1 to 6 of 6 Articles