On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions

In this paper, we complement recent results of Bronski and Johnson and of Johnson and Zumbrun concerning the modulational stability of spatially periodic traveling wave solutions of the generalized Korteweg-de Vries equation. In this previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to long wavelength perturbations. Here, we reproduce this result without reference to the Evans function by using direct Bloch-expansion methods and spectral perturbation analysis. This approach has the advantage of applying also in the more general multi-periodic setting where no conveniently computable Evans function is yet devised. In particular, we complement the picture of modulational stability described by Bronski and Johnson by analyzing the projectors onto the total eigenspace bifurcating from the origin in a neighborhood of the origin and zero Floquet parameter. We show the resulting linear system is equivalent, to leading order and up to conjugation, to the Whitham system and that, consequently, the characteristic polynomial of this system agrees (to leading order) with the linearized dispersion relation derived through Evans function calculation.Comment: 19 pages

A complex ray-tracing tool for high-frequency mean-field flow interaction effects in jets

This paper presents a complex ray-tracing tool for the calculation of high-frequency Green’s functions in 3D mean field jet flows. For a generic problem, the ray solution suffers from three main deficiencies: multiplicity of solutions, singularities at caustics, and the determining of complex solutions. The purpose of this paper is to generalize, combine and apply existing stationary media methods to moving media scenarios. Multiplicities are dealt with using an equivalent two-point boundary-value problem, whilst non-uniformities at caustics are corrected using diffraction catastrophes. Complex rays are found using a combination of imaginary perturbations, an assumption of caustic stability, and analytic continuation of the receiver curve. To demonstrate this method, the ray tool is compared against a high-frequency modal solution of Lilley’s equation for an off-axis point source. This solution is representative of high-frequency source positions in real jets and is rich in caustic structures. A full utilization of the ray tool is shown to provide excellent results<br/

Numerical Solution of Differential Equations by the Parker-Sochacki Method

A tutorial is presented which demonstrates the theory and usage of the Parker-Sochacki method of numerically solving systems of differential equations. Solutions are demonstrated for the case of projectile motion in air, and for the classical Newtonian N-body problem with mutual gravitational attraction.Comment: Added in July 2010: This tutorial has been posted since 1998 on a university web site, but has now been cited and praised in one or more refereed journals. I am therefore submitting it to the Cornell arXiv so that it may be read in response to its citations. See "Spiking neural network simulation: numerical integration with the Parker-Sochacki method:" J. Comput Neurosci, Robert D. Stewart & Wyeth Bair and http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2717378

Deflections in Magnet Fringe Fields

A transverse multipole expansion is derived, including the longitudinal components necessarily present in regions of varying magnetic field profile. It can be used for exact numerical orbit following through the fringe field regions of magnets whose end designs introduce no extraneous components, {\it i.e.} fields not required to be present by Maxwell's equations. Analytic evaluations of the deflections are obtained in various approximations. Mainly emphasized is a ``straight-line approximation'', in which particle orbits are treated as straight lines through the fringe field regions. This approximation leads to a readily-evaluated figure of merit, the ratio of r.m.s. end deflection to nominal body deflection, that can be used to determine whether or not a fringe field can be neglected. Deflections in ``critical'' cases (e.g. near intersection regions) are analysed in the same approximation.Comment: To be published in Physical Review

Sampling algebraic sets in local intrinsic coordinates

Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descriptions are respectively called extrinsic and intrinsic representations. While intrinsic representations lower the cost of the linear algebra operations, we observe worse condition numbers. In this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets. Local intrinsic coordinates also lead to a better stepsize control. We illustrate our results with Maple experiments and computations with PHCpack on some benchmark polynomial systems.Comment: 13 pages, 2 figures, 2 algorithms, 2 table

Fast Computing on Vehicle Dynamics Using Chebyshev Series Expansions.

This article focusses on faster computation techniques to integrate mechanical models in electronic advanced active safety applications. It shows the different techniques of approximation in series of functions and differential equations applied to vehicle dynamics. This allows the achievement of approximate polynomial and rational solutions with a very fast and efficient computation. Firstly, the whole theoretical basic principles related to the techniques used are presented: orthogonality of functions, function expansion in Chebyshev and Jacobi series, approximation through rational functions, the Minimax-Remez algorithm, orthogonal rational functions (ORF’s) and the perturbation of dynamic systems theory, that reduces the degree of the expansion polynomials used. As an application, it is shown the obtaining of approximate solutions to the longitudinal dynamics, vertical dynamics, steering geometry and a tyre model, all obtained through development in series of orthogonal functions with a computation much faster than those of its equivalents in the classic vehicle theory. These polynomial partially symbolic solutions present very low errors and very favourable analytical properties due to their simplicity, becoming ideal for real time computation as those required for the simulation of evasive manoeuvres prior a crash. This set of techniques had never been applied to vehicle dynamics before.pre-print748 K

Finite sample approximation results for principal component analysis: a matrix perturbation approach

Principal component analysis (PCA) is a standard tool for dimensional reduction of a set of $n$ observations (samples), each with $p$ variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size $n$, and those of the limiting population PCA as $n\to\infty$. As in machine learning, we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample PCA and population PCA under a spiked covariance model. In addition, we also consider the relation between finite sample PCA and the asymptotic results in the joint limit $p,n\to\infty$, with $p/n=c$. We present a matrix perturbation view of the "phase transition phenomenon," and a simple linear-algebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit. Moreover, our analysis also applies for finite $p,n$ where we show that although there is no sharp phase transition as in the infinite case, either as a function of noise level or as a function of sample size $n$, the eigenvector of sample PCA may exhibit a sharp "loss of tracking," suddenly losing its relation to the (true) eigenvector of the population PCA matrix. This occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise, whose eigenvector points in a random direction.Comment: Published in at http://dx.doi.org/10.1214/08-AOS618 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org