9,512 research outputs found
On the Littlewood conjecture in fields of power series
Let \k be an arbitrary field. For any fixed badly approximable power series
in \k((X^{-1})), we give an explicit construction of continuum many
badly approximable power series for which the pair
satisfies the Littlewood conjecture. We further discuss the Littlewood
conjecture for pairs of algebraic power series
New proofs of determinant evaluations related to plane partitions
We give a new proof of a determinant evaluation due to Andrews, which has
been used to enumerate cyclically symmetric and descending plane partitions. We
also prove some related results, including a q-analogue of Andrews's
determinant.Comment: 25 page
Differential Geometry of Hydrodynamic Vlasov Equations
We consider hydrodynamic chains in dimensions which are Hamiltonian
with respect to the Kupershmidt-Manin Poisson bracket. These systems can be
derived from single equations, here called hydrodynamic Vlasov
equations, under the map For these
equations an analogue of the Dubrovin-Novikov Hamiltonian structure is
constructed. The Vlasov formalism allows us to describe objects like the
Haantjes tensor for such a chain in a much more compact and computable way. We
prove that the necessary conditions found by Ferapontov and Marshall in
(arXiv:nlin.SI/0505013) for the integrability of these hydrodynamic chains are
also sufficient.Comment: 24 page
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