69 research outputs found
Estimates in Beurling--Helson type theorems. Multidimensional case
We consider the spaces of functions on the
-dimensional torus such that the sequence of the Fourier
coefficients belongs to
. The norm on is defined by
. We study the rate of
growth of the norms as
for -smooth real
functions on (the one-dimensional case was investigated
by the author earlier). The lower estimates that we obtain have direct
analogues for the spaces
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
Concentration inequalities for random fields via coupling
We present a new and simple approach to concentration inequalities for
functions around their expectation with respect to non-product measures, i.e.,
for dependent random variables. Our method is based on coupling ideas and does
not use information inequalities. When one has a uniform control on the
coupling, this leads to exponential concentration inequalities. When such a
uniform control is no more possible, this leads to polynomial or
stretched-exponential concentration inequalities. Our abstract results apply to
Gibbs random fields, in particular to the low-temperature Ising model which is
a concrete example of non-uniformity of the coupling.Comment: New corrected version; 22 pages; 1 figure; New result added:
stretched-exponential inequalit
Decoherence produces coherent states: an explicit proof for harmonic chains
We study the behavior of infinite systems of coupled harmonic oscillators as
t->infinity, and generalize the Central Limit Theorem (CLT) to show that their
reduced Wigner distributions become Gaussian under quite general conditions.
This shows that generalized coherent states tend to be produced naturally. A
sufficient condition for this to happen is shown to be that the spectral
function is analytic and nonlinear. For a rectangular lattice of coupled
oscillators, the nonlinearity requirement means that waves must be dispersive,
so that localized wave-packets become suppressed. Virtually all harmonic
heat-bath models in the literature satisfy this constraint, and we have good
reason to believe that coherent states and their generalizations are not merely
a useful analytical tool, but that nature is indeed full of them. Standard
proofs of the CLT rely heavily on the fact that probability densities are
non-negative. Although the CLT generally fails if the probability densities are
allowed to take negative values, we show that a CLT does indeed hold for a
special class of such functions. We find that, intriguingly, nature has
arranged things so that all Wigner functions belong to this class.Comment: Final published version. 17 pages, Plain TeX, no figures. Online at
http://astro.berkeley.edu/~max/gaussians.html (faster from the US), from
http://www.mpa-garching.mpg.de/~max/gaussians.html (faster from Europe) or
from [email protected]
Quasisymmetric graphs and Zygmund functions
A quasisymmetric graph is a curve whose projection onto a line is a
quasisymmetric map. We show that this class of curves is related to solutions
of the reduced Beltrami equation and to a generalization of the Zygmund class
. This relation makes it possible to use the tools of harmonic
analysis to construct nontrivial examples of quasisymmetric graphs and of
quasiconformal maps.Comment: 21 pages, no figure
On absolute continuity of the spectrum of a periodic magnetic Schr\"odinger operator
We consider the Schr\"odinger operator in , , with
the electric potential and the magnetic potential being periodic
functions (with a common period lattice) and prove absolute continuity of the
spectrum of the operator in question under some conditions which, in
particular, are satisfied if
and , .Comment: 25 page
PAC learning using Nadaraya-Watson estimator based on orthonormal systems
Regression or function classes of Euclidean type with compact support and certain smoothness properties are shown to be PAC learnable by the Nadaraya-Watson estimator based on complete orthonormal systems. While requiring more smoothness properties than typical PAC formulations, this estimator is computationally efficient, easy to implement, and known to perform well in a number of practical applications. The sample sizes necessary for PAC learning of regressions or functions under sup norm cost are derived for a general orthonormal system. The result covers the widely used estimators based on Haar wavelets, trignometric functions, and Daubechies wavelets
Bounded and compact multipliers between Bergman and Hardy spaces
This paper studies the boundedness and compactness of the coefficient multiplier operators between various Bergman spaces A p and Hardy spaces H q . Some new characterizations of the multipliers between the spaces with exponents 1 or 2 are derived which, in particular, imply a Bergman space analogue of the Paley-Rudin Theorem on sparse sequences. Hardy and Bergman spaces are shown to be linked using mixed-norm spaces, and this linkage is used to improve a known result on ( A p , A 2 ), 1< p <2.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42949/1/20_2005_Article_BF01225524.pd
GENERATION OF SHOCK WAVES IN DEUTERIUM IN THE EXPLOSIVE VOITENKO GENERATOR
Dans cette conférence on présente les résultats des études sur la création de fortes ondes de choc dans le deutérium, par un générateur Voitenko. Pour des vitesses maximales diférentes / de 4,7 à 6,5 mm/”s/ des plaques métalliques / en duraluminium/ entrainées par explosion, on a obtenu un caractére variable de l'allure de l'onde de choc en fonction de la distance. On a obtenu des valeurs maximales de cette vitesse dans les limites entre 58 et 63 mm/”s dans des distances différentes de la coupe d'embouchure de la chambre de compression sphérique.Results of investigations on receiving of strong shock waves in deuterium, generated in the so called Voitenko generator have been presented in this paper. A variable character of shock wave velocity profile versus path has been obtained fordifferent maximum velocities /4,7 + 6,5 mm/”s/ of explosively driven liners /duralumin plates/. Maximum obtained values of this velocity were within 58 + 63 mm/”s at different distances from the output cross-section of the spherical compression chamber
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