2,270 research outputs found
Euler characteristics in relative K-groups
Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tate's papers [17, 18], we have had a simple and explicit formula for the EulerāPoincarĆ© characteristic of the cohomology of M. In this note we are interested in a refinement of this formula when M also carries an action of some algebra [script A], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative K-group attached to [script A] (see Section 2). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes (see Section 3). This representability may be of independent interest in other contexts.
Our formula for the equivariant Euler characteristic over [script A] implies the āisogeny invarianceā of the equivariant conjectures on special values of the L-function put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tate's original paper [18]. I am very grateful to J-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper
Tamagawa Numbers for Motives with (Non-Commutative) Coefficients
Let be a motive which is defined over a number field and admits an action of a finite dimensional semisimple \bq-algebra . We formulate and study a conjecture for the leading coefficient of the Taylor expansion at of the -equivariant -function of . This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou et al. and also the central conjectures of classical Galois module theory as developed by Frƶhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order \A in for which there exists a `projective \A-structure' on . The existence of such a structure is guaranteed if \A is a maximal order, and also occurs in many natural examples where \A is non-maximal. In each such case the conjecture with respect to a non-maximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in by making use of the category of virtual objects introduced by Deligne
Resonant ratcheting of a Bose-Einstein condensate
We study the rectification process of interacting quantum particles in a
periodic potential exposed to the action of an external ac driving. The
breaking of spatio-temporal symmetries leads to directed motion already in the
absence of interactions. A hallmark of quantum ratcheting is the appearance of
resonant enhancement of the current (Europhys. Lett. 79 (2007) 10007 and Phys.
Rev. A 75 (2007) 063424). Here we study the fate of these resonances within a
Gross-Pitaevskii equation which describes a mean field interaction between many
particles. We find, that the resonance is i) not destroyed by interactions, ii)
shifting its location with increasing interaction strength. We trace the
Floquet states of the linear equations into the nonlinear domain, and show that
the resonance gives rise to an instability and thus to the appearance of new
nonlinear Floquet states, whose transport properties differ strongly as
compared to the case of noninteracting particles
Slow Relaxation and Phase Space Properties of a Conservative System with Many Degrees of Freedom
We study the one-dimensional discrete model. We compare two
equilibrium properties by use of molecular dynamics simulations: the Lyapunov
spectrum and the time dependence of local correlation functions. Both
properties imply the existence of a dynamical crossover of the system at the
same temperature. This correlation holds for two rather different regimes of
the system - the displacive and intermediate coupling regimes. Our results
imply a deep connection between slowing down of relaxations and phase space
properties of complex systems.Comment: 14 pages, LaTeX, 10 Figures available upon request (SF), Phys. Rev.
E, accepted for publicatio
On the Existence of Localized Excitations in Nonlinear Hamiltonian Lattices
We consider time-periodic nonlinear localized excitations (NLEs) on
one-dimensional translationally invariant Hamiltonian lattices with arbitrary
finite interaction range and arbitrary finite number of degrees of freedom per
unit cell. We analyse a mapping of the Fourier coefficients of the NLE
solution. NLEs correspond to homoclinic points in the phase space of this map.
Using dimensionality properties of separatrix manifolds of the mapping we show
the persistence of NLE solutions under perturbations of the system, provided
NLEs exist for the given system. For a class of nonintegrable Fermi-Pasta-Ulam
chains we rigorously prove the existence of NLE solutions.Comment: 13 pages, LaTeX, 2 figures will be mailed upon request (Phys. Rev. E,
in press
Breather initial profiles in chains of weakly coupled anharmonic oscillators
A systematic correlation between the initial profile of discrete breathers
and their frequency is described. The context is that of a very weakly
harmonically coupled chain of softly anharmonic oscillators. The results are
structurally stable, that is, robust under changes of the on-site potential and
are illustrated numerically for several standard choices. A precise genericity
theorem for the results is proved.Comment: 12 pages, 4 figure
AC field induced quantum rectification effect in tunnel junctions
We study the appearance of directed current in tunnel junctions, quantum
ratchet effect, in the presence of an external ac field f(t). The current is
established in a one-dimensional discrete inhomogeneous "tight-binding model".
By making use of a symmetry analysis we predict the right choice of f(t) and
obtain the directed current as a difference between electron transmission
coefficients in opposite directions, . Numerical
simulations confirm the predictions of the symmetry analysis and moreover, show
that the directed current can be drastically increased by a proper choice of
frequency and amplitudes of the ac field f(t).Comment: 4 pages, 3 figures, to be published in Physical Review
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