650 research outputs found
Sums and differences of four k-th powers
We prove an upper bound for the number of representations of a positive
integer as the sum of four -th powers of integers of size at most ,
using a new version of the Determinant method developed by Heath-Brown, along
with recent results by Salberger on the density of integral points on affine
surfaces. More generally we consider representations by any integral diagonal
form. The upper bound has the form , whereas earlier
versions of the Determinant method would produce an exponent for of order
in this case. Furthermore, we prove that the number of
representations of a positive integer as a sum of four -th powers of
non-negative integers is at most for
, improving upon bounds by Wisdom.Comment: 18 pages. Mistake corrected in the statement of Theorem 1.2. To
appear in Monatsh. Mat
The role of chaotic resonances in the solar system
Our understanding of the Solar System has been revolutionized over the past
decade by the finding that the orbits of the planets are inherently chaotic. In
extreme cases, chaotic motions can change the relative positions of the planets
around stars, and even eject a planet from a system. Moreover, the spin axis of
a planet-Earth's spin axis regulates our seasons-may evolve chaotically, with
adverse effects on the climates of otherwise biologically interesting planets.
Some of the recently discovered extrasolar planetary systems contain multiple
planets, and it is likely that some of these are chaotic as well.Comment: 28 pages, 9 figure
Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions
We show that when time-reversible symplectic algorithms are used to solve
periodic motions, the energy error after one period is generally two orders
higher than that of the algorithm. By use of correctable algorithms, we show
that the phase error can also be eliminated two orders higher than that of the
integrator. The use of fourth order forward time step integrators can result in
sixth order accuracy for the phase error and eighth accuracy in the periodic
energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in
great details, and compare the effectiveness of some recent fourth order
algorithms.Comment: Submitted to Phys. Rev. E, 29 Page
Influence of the coorbital resonance on the rotation of the Trojan satellites of Saturn
The Cassini spacecraft collects high resolution images of the saturnian
satellites and reveals the surface of these new worlds. The shape and rotation
of the satellites can be determined from the Cassini Imaging Science Subsystem
data, employing limb coordinates and stereogrammetric control points. This is
the case for Epimetheus (Tiscareno et al. 2009) that opens elaboration of new
rotational models (Tiscareno et al. 2009; Noyelles 2010; Robutel et al. 2011).
Especially, Epimetheus is characterized by its horseshoe shape orbit and the
presence of the swap is essential to introduce explicitly into rotational
models. During its journey in the saturnian system, Cassini spacecraft
accumulates the observational data of the other satellites and it will be
possible to determine the rotational parameters of several of them. To prepare
these future observations, we built rotational models of the coorbital (also
called Trojan) satellites Telesto, Calypso, Helene, and Polydeuces, in addition
to Janus and Epimetheus. Indeed, Telesto and Calypso orbit around the L_4 and
L_5 Lagrange points of Saturn-Tethys while Helene and Polydeuces are coorbital
of Dione. The goal of this study is to understand how the departure from the
Keplerian motion induced by the perturbations of the coorbital body, influences
the rotation of these satellites. To this aim, we introduce explicitly the
perturbation in the rotational equations by using the formalism developed by
Erdi (1977) to represent the coorbital motions, and so we describe the
rotational motion of the coorbitals, Janus and Epimetheus included, in compact
form
Tidal friction in close-in satellites and exoplanets. The Darwin theory re-visited
This report is a review of Darwin's classical theory of bodily tides in which
we present the analytical expressions for the orbital and rotational evolution
of the bodies and for the energy dissipation rates due to their tidal
interaction. General formulas are given which do not depend on any assumption
linking the tidal lags to the frequencies of the corresponding tidal waves
(except that equal frequency harmonics are assumed to span equal lags).
Emphasis is given to the cases of companions having reached one of the two
possible final states: (1) the super-synchronous stationary rotation resulting
from the vanishing of the average tidal torque; (2) the capture into a 1:1
spin-orbit resonance (true synchronization). In these cases, the energy
dissipation is controlled by the tidal harmonic with period equal to the
orbital period (instead of the semi-diurnal tide) and the singularity due to
the vanishing of the geometric phase lag does not exist. It is also shown that
the true synchronization with non-zero eccentricity is only possible if an
extra torque exists opposite to the tidal torque. The theory is developed
assuming that this additional torque is produced by an equatorial permanent
asymmetry in the companion. The results are model-dependent and the theory is
developed only to the second degree in eccentricity and inclination
(obliquity). It can easily be extended to higher orders, but formal accuracy
will not be a real improvement as long as the physics of the processes leading
to tidal lags is not better known.Comment: 30 pages, 7 figures, corrected typo
Explicit Lie-Poisson integration and the Euler equations
We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson
integrators, preserving all Casimirs, can be constructed. The integrators are
extremely simple. Examples are the rigid body, a moment truncation, and a new,
fast algorithm for the sine-bracket truncation of the 2D Euler equations.Comment: 7 pages, compile with AMSTEX; 2 figures available from autho
Long-Term Stability of Horseshoe Orbits
Unlike Trojans, horseshoe coorbitals are not generally considered to be
long-term stable (Dermott and Murray, 1981; Murray and Dermott, 1999). As the
lifetime of Earth's and Venus's horseshoe coorbitals is expected to be about a
Gyr, we investigated the possible contribution of late-escaping inner planet
coorbitals to the lunar Late Heavy Bombardment. Contrary to analytical
estimates, we do not find many horseshoe objects escaping after first 100 Myr.
In order to understand this behaviour, we ran a second set of simulations
featuring idealized planets on circular orbits with a range of masses. We find
that horseshoe coorbitals are generally long lived (and potentially stable) for
systems with primary-to-secondary mass ratios larger than about 1200. This is
consistent with results of Laughlin and Chambers (2002) for equal-mass pairs or
coorbital planets and the instability of Jupiter's horseshoe companions (Stacey
and Connors, 2008). Horseshoe orbits at smaller mass ratios are unstable
because they must approach within 5 Hill radii of the secondary. In contrast,
tadpole orbits are more robust and can remain stable even when approaching
within 4 Hill radii of the secondary.Comment: Accepted for MNRA
Shape models and physical properties of asteroids
Despite the large amount of high quality data generated in recent space
encounters with asteroids, the majority of our knowledge about these objects
comes from ground based observations. Asteroids travelling in orbits that are
potentially hazardous for the Earth form an especially interesting group to be
studied. In order to predict their orbital evolution, it is necessary to
investigate their physical properties. This paper briefly describes the data
requirements and different techniques used to solve the lightcurve inversion
problem. Although photometry is the most abundant type of observational data,
models of asteroids can be obtained using various data types and techniques. We
describe the potential of radar imaging and stellar occultation timings to be
combined with disk-integrated photometry in order to reveal information about
physical properties of asteroids.Comment: From Assessment and Mitigation of Asteroid Impact Hazards boo
The interaction between typically developing students and peers with autism spectrum disorder in regular schools in Ghana : an exploration using the theory of planned behaviour
The purpose of this study is to assess the intention of typically developing peers towards learning in the classroom with students with Autism Spectrum Disorder (ASD). In developing countries, such as Ghana, the body of literature on the relationship between students with disabilities and typically developing peers has been sparsely studied. Using Ajzen’s theory of planned behaviour as a theoretical framework for this study, 516 typically developing students completed four scales representing belief constructs, attitudes, subjective norms, and perceived behavioural controls (self-efficacy), hypothesised to
predict behavioural intention. The data were subjected to a t-test, analysis of variance, and structural equation modelling. The modelling confirmed the combining ability of attitude, subjective norms, and perceived behavioural controls to predict intention. We conclude by revealing the need for policymakers to consider designing programmes aimed towards promoting social relationships between students with ASD and typically developing peers
Quantum Statistical Calculations and Symplectic Corrector Algorithms
The quantum partition function at finite temperature requires computing the
trace of the imaginary time propagator. For numerical and Monte Carlo
calculations, the propagator is usually split into its kinetic and potential
parts. A higher order splitting will result in a higher order convergent
algorithm. At imaginary time, the kinetic energy propagator is usually the
diffusion Greens function. Since diffusion cannot be simulated backward in
time, the splitting must maintain the positivity of all intermediate time
steps. However, since the trace is invariant under similarity transformations
of the propagator, one can use this freedom to "correct" the split propagator
to higher order. This use of similarity transforms classically give rises to
symplectic corrector algorithms. The split propagator is the symplectic kernel
and the similarity transformation is the corrector. This work proves a
generalization of the Sheng-Suzuki theorem: no positive time step propagators
with only kinetic and potential operators can be corrected beyond second order.
Second order forward propagators can have fourth order traces only with the
inclusion of an additional commutator. We give detailed derivations of four
forward correctable second order propagators and their minimal correctors.Comment: 9 pages, no figure, corrected typos, mostly missing right bracket
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