Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories

We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, that can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form, so as to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that is at basis of the analytic part of the Nekhoroshev's theorem, so as to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller.Comment: 31 pages, 2 figures. arXiv admin note: text overlap with arXiv:1010.260

Implementation of the Trigonometric LMS Algorithm using Original Cordic Rotation

The LMS algorithm is one of the most successful adaptive filtering algorithms. It uses the instantaneous value of the square of the error signal as an estimate of the mean-square error (MSE). The LMS algorithm changes (adapts) the filter tap weights so that the error signal is minimized in the mean square sense. In Trigonometric LMS (TLMS) and Hyperbolic LMS (HLMS), two new versions of LMS algorithms, same formulations are performed as in the LMS algorithm with the exception that filter tap weights are now expressed using trigonometric and hyperbolic formulations, in cases for TLMS and HLMS respectively. Hence appears the CORDIC algorithm as it can efficiently perform trigonometric, hyperbolic, linear and logarithmic functions. While hardware-efficient algorithms often exist, the dominance of the software systems has kept those algorithms out of the spotlight. Among these hardware- efficient algorithms, CORDIC is an iterative solution for trigonometric and other transcendental functions. Former researches worked on CORDIC algorithm to observe the convergence behavior of Trigonometric LMS (TLMS) algorithm and obtained a satisfactory result in the context of convergence performance of TLMS algorithm. But revious researches directly used the CORDIC block output in their simulation ignoring the internal step-by-step rotations of the CORDIC processor. This gives rise to a need for verification of the convergence performance of the TLMS algorithm to investigate if it actually performs satisfactorily if implemented with step-by-step CORDIC rotation. This research work has done this job. It focuses on the internal operations of the CORDIC hardware, implements the Trigonometric LMS (TLMS) and Hyperbolic LMS (HLMS) algorithms using actual CORDIC rotations. The obtained simulation results are highly satisfactory and also it shows that convergence behavior of HLMS is much better than TLMS.Comment: 12 pages, 5 figures, 1 table. Published in IJCNC; http://airccse.org/journal/cnc/0710ijcnc08.pdf, http://airccse.org/journal/ijc2010.htm

A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems

The inclinations of exoplanets detected via radial velocity method are essentially unknown. We aim to provide estimations of the ranges of mutual inclinations that are compatible with the long-term stability of the system. Focusing on the skeleton of an extrasolar system, i.e., considering only the two most massive planets, we study the Hamiltonian of the three-body problem after the reduction of the angular momentum. Such a Hamiltonian is expanded both in Poincar\'e canonical variables and in the small parameter $D_2$, which represents the normalised Angular Momentum Deficit. The value of the mutual inclination is deduced from $D_2$ and, thanks to the use of interval arithmetic, we are able to consider open sets of initial conditions instead of single values. Looking at the convergence radius of the Kolmogorov normal form, we develop a reverse KAM approach in order to estimate the ranges of mutual inclinations that are compatible with the long-term stability in a KAM sense. Our method is successfully applied to the extrasolar systems HD 141399, HD 143761 and HD 40307.Comment: 19 pages, 3 figure

Rotation method for accelerating multiple-spherical Bessel function integrals against a numerical source function

A common problem in cosmology is to integrate the product of two or more spherical Bessel functions (sBFs) with different configuration-space arguments against the power spectrum or its square, weighted by powers of wavenumber. Naively computing them scales as $N_{\rm g}^{p+1}$ with $p$ the number of configuration space arguments and $N_{\rm g}$ the grid size, and they cannot be done with Fast Fourier Transforms (FFTs). Here we show that by rewriting the sBFs as sums of products of sine and cosine and then using the product to sum identities, these integrals can then be performed using 1-D FFTs with $N_{\rm g} \log N_{\rm g}$ scaling. This "rotation" method has the potential to accelerate significantly a number of calculations in cosmology, such as perturbation theory predictions of loop integrals, higher order correlation functions, and analytic templates for correlation function covariance matrices. We implement this approach numerically both in a free-standing, publicly-available \textsc{Python} code and within the larger, publicly-available package \texttt{mcfit}. The rotation method evaluated with direct integrations already offers a factor of 6-10$\times$ speed-up over the naive approach in our test cases. Using FFTs, which the rotation method enables, then further improves this to a speed-up of $\sim$$1000-3000\times$ over the naive approach. The rotation method should be useful in light of upcoming large datasets such as DESI or LSST. In analysing these datasets recomputation of these integrals a substantial number of times, for instance to update perturbation theory predictions or covariance matrices as the input linear power spectrum is changed, will be one piece in a Monte Carlo Markov Chain cosmological parameter search: thus the overall savings from our method should be significant

Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples

In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sample points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Groechenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm---based on the existing generalized sampling framework---that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice---jittered, radial and spiral sampling schemes---and provide several examples illustrating the effectiveness of our approach when tested on these schemes

The SISCone jet algorithm optimised for low particle multiplicities

The SISCone jet algorithm is a seedless infrared-safe cone jet algorithm. There exists an implementation which is highly optimised for a large number of final state particles. However, in fixed-order perturbative calculations with a small number of final state particles, it turns out that the computer time needed for the jet clustering of this implementation is comparable to the computer time of the matrix elements. This article reports on an implementation of the SISCone algorithm optimised for low particle multiplicities.Comment: 16 pages, version to be publishe

Reconstruction of Binary Functions and Shapes from Incomplete Frequency Information

The characterization of a binary function by partial frequency information is considered. We show that it is possible to reconstruct binary signals from incomplete frequency measurements via the solution of a simple linear optimization problem. We further prove that if a binary function is spatially structured (e.g. a general black-white image or an indicator function of a shape), then it can be recovered from very few low frequency measurements in general. These results would lead to efficient methods of sensing, characterizing and recovering a binary signal or a shape as well as other applications like deconvolution of binary functions blurred by a low-pass filter. Numerical results are provided to demonstrate the theoretical arguments.Comment: IEEE Transactions on Information Theory, 201