2,625 research outputs found
Arithmetic harmonic analysis for smooth quartic Weyl sums: three additive equations
We establish the non-singular Hasse principle for systems of three diagonal
quartic equations in 32 or more variables, subject to a certain rank condition.
Our methods employ the arithmetic harmonic analysis of smooth quartic Weyl sums
and also a new estimate for their tenth moment.Comment: 22 page
The Biharmonic mean
We briefly describe some well-known means and their properties, focusing on
the relationship with integer sequences. In particular, the harmonic numbers,
deriving from the harmonic mean, motivate the definition of a new kind of mean
that we call the biharmonic mean. The biharmonic mean allows to introduce the
biharmonic numbers, providing a new characterization for primes. Moreover, we
highlight some interesting divisibility properties and we characterize the
semi--prime biharmonic numbers showing their relationship with linear recurrent
sequences that solve certain Diophantine equations
Diophantine conditions and real or complex Brjuno functions
The continued fraction expansion of the real number x=a_0+x_0, a_0\in
{\ZZ}, is given by 0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in
{\NN}, for The Brjuno function is then
and the number
satisfies the Brjuno diophantine condition whenever is bounded.
Invariant circles under a complex rotation persist when the map is analytically
perturbed, if and only if the rotation number satisfies the Brjuno condition,
and the same holds for invariant circles in the semi-standard and standard maps
cases. In this lecture, we will review some properties of the Brjuno function,
and give some generalisations related to familiar diophantine conditions. The
Brjuno function is highly singular and takes value on a dense set
including rationals. We present a regularisation leading to a complex function
holomorphic in the upper half plane. Its imaginary part tends to the Brjuno
function on the real axis, the real part remaining bounded, and we also
indicate its transformation under the modular group.Comment: latex jura.tex, 6 files, 19 pages Proceedings on `Noise, Oscillators
and Algebraic Randomness' La Chapelle des Bois, France 1999-04-05 1999-04-10
April 5-10, 1999 [SPhT-T99/116
Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices
We employ KAM theory to rigorously investigate quasiperiodic dynamics in
cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and
superlattices. Toward this end, we apply a coherent structure ansatz to the
Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation
describing the spatial dynamics of the condensate. For shallow-well,
intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather
sets to prove that one obtains mostly quasiperiodic dynamics for condensate
wave functions of sufficiently large amplitude, where the minimal amplitude
depends on the experimentally adjustable BEC parameters. We show that this
threshold scales with the square root of the inverse of the two-body scattering
length, whereas the rotation number of tori above this threshold is
proportional to the amplitude. As a consequence, one obtains the same dynamical
picture for lattices of all depths, as an increase in depth essentially only
affects scaling in phase space. Our approach is applicable to periodic
superlattices with an arbitrary number of rationally dependent wave numbers.Comment: 29 pages, 6 figures (several with multiple parts; higher-quality
versions of some of them available at
http://www.its.caltech.edu/~mason/papers), to appear very soon in Journal of
Nonlinear Scienc
Best possible rates of distribution of dense lattice orbits in homogeneous spaces
The present paper establishes upper and lower bounds on the speed of
approximation in a wide range of natural Diophantine approximation problems.
The upper and lower bounds coincide in many cases, giving rise to optimal
results in Diophantine approximation which were inaccessible previously. Our
approach proceeds by establishing, more generally, upper and lower bounds for
the rate of distribution of dense orbits of a lattice subgroup in a
connected Lie (or algebraic) group , acting on suitable homogeneous spaces
. The upper bound is derived using a quantitative duality principle for
homogeneous spaces, reducing it to a rate of convergence in the mean ergodic
theorem for a family of averaging operators supported on and acting on
. In particular, the quality of the upper bound on the rate of
distribution we obtain is determined explicitly by the spectrum of in the
automorphic representation on . We show that the rate
is best possible when the representation in question is tempered, and show that
the latter condition holds in a wide range of examples
Rational points near planar curves and Diophantine approximation
In this paper, we establish asymptotic formulae with optimal errors for the
number of rational points that are close to a planar curve, which unify and
extend the results of Beresnevich-Dickinson-Velani and Vaughan-Velani.
Furthermore, we complete the Lebesgue theory of Diophantine approximation on
weakly non-degenerate planar curves that was initially developed by
Beresnevich-Zorin in the divergence case.Comment: 27 pages, corrected typos, to appear in Adv. Mat
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