234 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
An approximation for zero-balanced Appell function near
We suggest an approximation for the zero-balanced Appell hypergeometric
function near the singular point . Our approximation can be viewed
as a generalization of Ramanujan's approximation for zero-balanced
and is expressed in terms of . We find an error bound and prove some
basic properties of the suggested approximation which reproduce the similar
properties of the Appell function. Our approximation reduces to the
approximation of Carlson-Gustafson when the Appell function reduces to the
first incomplete elliptic integral.Comment: 10 page
On approximate solutions of the incompressible Euler and Navier-Stokes equations
We consider the incompressible Euler or Navier-Stokes (NS) equations on a
torus T^d in the functional setting of the Sobolev spaces H^n(T^d) of
divergence free, zero mean vector fields on T^d, for n > d/2+1. We present a
general theory of approximate solutions for the Euler/NS Cauchy problem; this
allows to infer a lower bound T_c on the time of existence of the exact
solution u analyzing a posteriori any approximate solution u_a, and also to
construct a function R_n such that || u(t) - u_a(t) ||_n <= R_n(t) for all t in
[0,T_c). Both T_c and R_n are determined solving suitable "control
inequalities", depending on the error of u_a; the fully quantitative
implementation of this scheme depends on some previous estimates of ours on the
Euler/NS quadratic nonlinearity [15][16]. To keep in touch with the existing
literature on the subject, our results are compared with a setting for
approximate Euler/NS solutions proposed in [3]. As a first application of the
present framework, we consider the Galerkin approximate solutions of the
Euler/NS Cauchy problem, with a specific initial datum considered in [2]: in
this case our methods allow, amongst else, to prove global existence for the NS
Cauchy problem when the viscosity is above an explicitly given bound.Comment: LaTex, 44 pages, 18 figure
Diffusion and wave behaviour in linear Voigt model
A boundary value problem related to a third- order parabolic equation with a
small parameter is analized. This equation models the one-dimensional evolution
of many dissipative media as viscoelastic fluids or solids, viscous gases,
superconducting materials, incompressible and electrically conducting fluids.
Moreover, the third-order parabolic operator regularizes various non linear
second order wave equations. In this paper, the hyperbolic and parabolic
behaviour of the solution is estimated by means of slow time and fast time. As
consequence, a rigorous asymptotic approximation for the solution is
established
Recognising the Suzuki groups in their natural representations
Under the assumption of a certain conjecture, for which there exists strong
experimental evidence, we produce an efficient algorithm for constructive
membership testing in the Suzuki groups Sz(q), where q = 2^{2m + 1} for some m
> 0, in their natural representations of degree 4. It is a Las Vegas algorithm
with running time O{log(q)} field operations, and a preprocessing step with
running time O{log(q) loglog(q)} field operations. The latter step needs an
oracle for the discrete logarithm problem in GF(q).
We also produce a recognition algorithm for Sz(q) = . This is a Las Vegas
algorithm with running time O{|X|^2} field operations.
Finally, we give a Las Vegas algorithm that, given ^h = Sz(q) for some h
in GL(4, q), finds some g such that ^g = Sz(q). The running time is O{log(q)
loglog(q) + |X|} field operations.
Implementations of the algorithms are available for the computer system
MAGMA
Free Energies of Isolated 5- and 7-fold Disclinations in Hexatic Membranes
We examine the shapes and energies of 5- and 7-fold disclinations in
low-temperature hexatic membranes. These defects buckle at different values of
the ratio of the bending rigidity, , to the hexatic stiffness constant,
, suggesting {\em two} distinct Kosterlitz-Thouless defect proliferation
temperatures. Seven-fold disclinations are studied in detail numerically for
arbitrary . We argue that thermal fluctuations always drive
into an ``unbuckled'' regime at long wavelengths, so that
disclinations should, in fact, proliferate at the {\em same} critical
temperature. We show analytically that both types of defects have power law
shapes with continuously variable exponents in the ``unbuckled'' regime.
Thermal fluctuations then lock in specific power laws at long wavelengths,
which we calculate for 5- and 7-fold defects at low temperatures.Comment: LaTeX format. 17 pages. To appear in Phys. Rev.
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
Mutation, selection, and ancestry in branching models: a variational approach
We consider the evolution of populations under the joint action of mutation
and differential reproduction, or selection. The population is modelled as a
finite-type Markov branching process in continuous time, and the associated
genealogical tree is viewed both in the forward and the backward direction of
time. The stationary type distribution of the reversed process, the so-called
ancestral distribution, turns out as a key for the study of mutation-selection
balance. This balance can be expressed in the form of a variational principle
that quantifies the respective roles of reproduction and mutation for any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth rate, as
given by the difference between the current mean reproduction rate, and an
asymptotic decay rate related to the mutation process; this tradeoff is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence evolution with
mutation coupled to reproduction but independent across sites, and a fitness
function that is invariant under permutation of sites. Here, the variational
principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure
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