1,336 research outputs found
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
Obtaining Breathers in Nonlinear Hamiltonian Lattices
We present a numerical method for obtaining high-accuracy numerical solutions
of spatially localized time-periodic excitations on a nonlinear Hamiltonian
lattice. We compare these results with analytical considerations of the spatial
decay. We show that nonlinear contributions have to be considered, and obtain
very good agreement between the latter and the numerical results. We discuss
further applications of the method and results.Comment: 21 pages (LaTeX), 8 figures in ps-files, tar-compressed uuencoded
file, Physical Review E, in pres
Statistics of wave interactions in nonlinear disordered systems
We study the properties of mode-mode interactions for waves propagating in
nonlinear disordered one-dimensional systems. We focus on i) the localization
volume of a mode which defines the number of interacting partner modes, ii) the
overlap integrals which determine the interaction strength, iii) the average
spacing between eigenvalues of interacting modes, which sets a scale for the
nonlinearity strength, and iv) resonance probabilities of interacting modes.
Our results are discussed in the light of recent studies on spreading of wave
packets in disordered nonlinear systems, and are related to the quantum many
body problem in a random chain.Comment: 7 pages, 7 figure
Interaction-induced connectivity of disordered two-particle states
We study the interaction-induced connectivity in the Fock space of two
particles in a disordered one-dimensional potential. Recent computational
studies showed that the largest localization length of two interacting
particles in a weakly random tight binding chain is increasing unexpectedly
slow relative to the single particle localization length , questioning
previous scaling estimates. We show this to be a consequence of the approximate
restoring of momentum conservation of weakly localized single particle
eigenstates, and disorder-induced phase shifts for partially overlapping
states. The leading resonant links appear among states which share the same
energy and momentum. We substantiate our analytical approach by computational
studies for up to . A potential nontrivial scaling regime sets in
for , way beyond all previous numerical attacks.Comment: 5 pages, 4 figure
Breathers on lattices with long range interaction
We analyze the properties of breathers (time periodic spatially localized
solutions) on chains in the presence of algebraically decaying interactions
. We find that the spatial decay of a breather shows a crossover from
exponential (short distances) to algebraic (large distances) decay. We
calculate the crossover distance as a function of and the energy of the
breather. Next we show that the results on energy thresholds obtained for short
range interactions remain valid for and that for (anomalous
dispersion at the band edge) nonzero thresholds occur for cases where the short
range interaction system would yield zero threshold values.Comment: 4 pages, 2 figures, PRB Rapid Comm. October 199
Energy thresholds for discrete breathers in one-, two- and three-dimensional lattices
Discrete breathers are time-periodic, spatially localized solutions of
equations of motion for classical degrees of freedom interacting on a lattice.
They come in one-parameter families. We report on studies of energy properties
of breather families in one-, two- and three-dimensional lattices. We show that
breather energies have a positive lower bound if the lattice dimension of a
given nonlinear lattice is greater than or equal to a certain critical value.
These findings could be important for the experimental detection of discrete
breathers.Comment: 10 pages, LaTeX, 4 figures (ps), Physical Review Letters, in prin
Intermittent many-body dynamics at equilibrium
The equilibrium value of an observable defines a manifold in the phase space of an ergodic and equipartitioned many-body system. A typical trajectory pierces that manifold infinitely often as time goes to infinity. We use these piercings to measure both the relaxation time of the lowest frequency eigenmode of the Fermi-Pasta-Ulam chain, as well as the fluctuations of the subsequent dynamics in equilibrium. The dynamics in equilibrium is characterized by a power-law distribution of excursion times far off equilibrium, with diverging variance. Long excursions arise from sticky dynamics close to q-breathers localized in normal mode space. Measuring the exponent allows one to predict the transition into nonergodic dynamics. We generalize our method to Klein-Gordon lattices where the sticky dynamics is due to discrete breathers localized in real space.We thank P. Jeszinszki and I. Vakulchyk for helpful discussions on computational aspects. The authors acknowledge financial support from IBS (Project Code No. IBS-R024-D1). (IBS-R024-D1 - IBS)Published versio
Acoustic breathers in two-dimensional lattices
The existence of breathers (time-periodic and spatially localized lattice
vibrations) is well established for i) systems without acoustic phonon branches
and ii) systems with acoustic phonons, but also with additional symmetries
preventing the occurence of strains (dc terms) in the breather solution. The
case of coexistence of strains and acoustic phonon branches is solved (for
simple models) only for one-dimensional lattices.
We calculate breather solutions for a two-dimensional lattice with one
acoustic phonon branch. We start from the easy-to-handle case of a system with
homogeneous (anharmonic) interaction potentials. We then easily continue the
zero-strain breather solution into the model sector with additional quadratic
and cubic potential terms with the help of a generalized Newton method. The
lattice size is . The breather continues to exist, but is dressed
with a strain field. In contrast to the ac breather components, which decay
exponentially in space, the strain field (which has dipole symmetry) should
decay like . On our rather small lattice we find an exponent
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