26 research outputs found
On the averaging principle for one-frequency systems. Seminorm estimates for the error
We extend some previous results of our work [1] on the error of the averaging
method, in the one-frequency case. The new error estimates apply to any
separating family of seminorms on the space of the actions; they generalize our
previous estimates in terms of the Euclidean norm. For example, one can use the
new approach to get separate error estimates for each action coordinate. An
application to rigid body under damping is presented. In a companion paper [2],
the same method will be applied to the motion of a satellite around an oblate
planet.Comment: LaTeX, 23 pages, 4 figures. The final version published in Nonlinear
Dynamic
Statistics and geometry of cosmic voids
We introduce new statistical methods for the study of cosmic voids, focusing
on the statistics of largest size voids. We distinguish three different types
of distributions of voids, namely, Poisson-like, lognormal-like and Pareto-like
distributions. The last two distributions are connected with two types of
fractal geometry of the matter distribution. Scaling voids with Pareto
distribution appear in fractal distributions with box-counting dimension
smaller than three (its maximum value), whereas the lognormal void distribution
corresponds to multifractals with box-counting dimension equal to three.
Moreover, voids of the former type persist in the continuum limit, namely, as
the number density of observable objects grows, giving rise to lacunar
fractals, whereas voids of the latter type disappear in the continuum limit,
giving rise to non-lacunar (multi)fractals. We propose both lacunar and
non-lacunar multifractal models of the cosmic web structure of the Universe. A
non-lacunar multifractal model is supported by current galaxy surveys as well
as cosmological -body simulations. This model suggests, in particular, that
small dark matter halos and, arguably, faint galaxies are present in cosmic
voids.Comment: 39 pages, 8 EPS figures, supersedes arXiv:0802.038