On the resolution power of Fourier extensions for oscillatory functions

Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. When constructed in a particular manner, Fourier extensions share many of the same features of a standard Fourier series. In particular, one can compute Fourier extensions which converge spectrally fast whenever the function is smooth, and exponentially fast if the function is analytic, much the same as the Fourier series of a smooth/analytic and periodic function. With this in mind, the purpose of this paper is to describe, analyze and explain the observation that Fourier extensions, much like classical Fourier series, also have excellent resolution properties for representing oscillatory functions. The resolution power, or required number of degrees of freedom per wavelength, depends on a user-controlled parameter and, as we show, it varies between 2 and \pi. The former value is optimal and is achieved by classical Fourier series for periodic functions, for example. The latter value is the resolution power of algebraic polynomial approximations. Thus, Fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation.Comment: Revised versio

The Commonality of Earthquake and Wind Analysis

Earthquakes and wind loadings constitute dynamic effects that often must be considered in the design of buildings and structures. The primary purpose of this research study was to investigate the common features of general dynamic analysis procedures employed for evaluating the effects of wind and earthquake excitation. Another major goal was to investigate and develop a basis for generating response spectra for wind loading, which in turn would permit the use of modal analysis techniques for wind analysis in a manner similar to that employed for earthquake engineering. In order to generate wind response spectra, the wind loading is divided into two parts, a mean load treated as a static component and a fluctuating load treated as a dynamic component. The spectral representation of the wind loading constitutes a simple procedure for estimating the forces associated with the dynamic component of the gusting wind. Several illustrative examples are presented demonstrating the commonality.National Science Foundation Grants ENV 75-08456 and ENV 77-0719

Knowledge discovery for friction stir welding via data driven approaches: Part 1 – correlation analyses of internal process variables and weld quality

For a comprehensive understanding towards Friction Stir Welding (FSW) which would lead to a unified approach that embodies materials other than aluminium, such as titanium and steel, it is crucial to identify the intricate correlations between the controllable process conditions, the observable internal process variables, and the characterisations of the post-weld materials. In Part I of this paper, multiple correlation analyses techniques have been developed to detect new and previously unknown correlations between the internal process variables and weld quality of aluminium alloy AA5083. Furthermore, a new exploitable weld quality indicator has, for the first time, been successfully extracted, which can provide an accurate and reliable indication of the ‘as-welded’ defects. All results relating to this work have been validated using real data obtained from a series of welding trials that utilised a new revolutionary sensory platform called ARTEMIS developed by TWI Ltd., the original inventors of the FSW process

New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS

Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. Two of them are nearly optimal. The first one, proposed in 1995, relies on recursive factorization of a polynomial, is quite involved, and has never been implemented. The second one, proposed in 2016, relies on subdivision iterations, was implemented in 2018, and promises to be practically competitive, although user's current choice for univariate polynomial root-finding is the package MPSolve, proposed in 2000, revised in 2014, and based on Ehrlich's functional iterations. By proposing and incorporating some novel techniques we significantly accelerate both subdivision and Ehrlich's iterations. Moreover our acceleration of the known subdivision root-finders is dramatic in the case of sparse input polynomials. Our techniques can be of some independent interest for the design and analysis of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table

The complexity of class polynomial computation via floating point approximations

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. It runs in time $O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon}) = O (h^{2 + \epsilon})$ for any $\epsilon > 0$, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $\log |D|$. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary-quadratic order and on a rigorously proven upper bound for the height of class polynomials