Fast Fourier Transform

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

: Scientific learning is an iterative process employing Criticism and Estimation. Correspondingly the formulated model factors into two complimentary parts - a predictive part allowing model criticism, and a Bayes posterior part allowing estimation. Implications for significance tests, the theory of precise measurement, and for ridge estimates are considered. Predictive checking functions for transformation, serial correlation, bad values, and their relation with Bayesian options are considered. Robustness is seen from a Bayesian viewpoint and examples are given. For the bad value problem a comparison with M estimators is made. (Author)

Sampling and Bayes' inference in scientific modelling and robustness

Correlation and regression are different, but not mutually exclusive, techniques. Roughly, regression is used for prediction (which does not extrapolate beyond the data used in the analysis) whereas correlation is used to determine the degree of association. There situations in which the x variable is not fixed or readily chosen by the experimenter, but instead is a random covariate to the y variable. This paper shows the relationships between the coefficient of determination, the multiple correlation coefficient, the covariance, the correlation coefficient and the coefficient of alienation, for the case of two related variables x and y. It discusses the uses of the correlation coefficient r, either as a way to infer correlation, or to test linearity. A number of graphical examples are provided as well as examples of actual chemical applications. The paper recommends the use of z Fisher transformation instead of r values because r is not normally distributed but z is (at least in approximation). For eithe...

The Correlation Coefficient: An Overview

We report on the neutrino mass measurement result from the first four-week science run of the Karlsruhe Tritium Neutrino experiment KATRIN in spring 2019. Beta-decay electrons from a high-purity gaseous molecular tritium source are energy analyzed by a high-resolution MAC-E filter. A fit of the integrated electron spectrum over a narrow interval around the kinematic end point at 18.57 keV gives an effective neutrino mass square value of (-1.0_{-1.1}^{+0.9})  eV^{2}. From this, we derive an upper limit of 1.1 eV (90% confidence level) on the absolute mass scale of neutrinos. This value coincides with the KATRIN sensitivity. It improves upon previous mass limits from kinematic measurements by almost a factor of 2 and provides model-independent input to cosmological studies of structure formation.

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Improved Upper Limit on the Neutrino Mass from a Direct Kinematic Method by KATRIN

A number of Joint Committees of the Bureau International des Poids et Mesures and other international organizations carry out particular tasks of common interest. The Joint Committee for Guides in Metrology (JCGM) has amongst its tasks the promotion of the 'Guide to the Expression of Uncertainty in Measurement' (GUM), the preparation of further documents for its broad application, and revision and promotion of the use of the 'International Vocabulary of Basic and General Terms in Metrology'. This paper summarizes the documents relating to the GUM planned by JCGM.

https://iopscience.iop.org/article/10.1088/0026-1394/43/4/S01/pdf

Evolution of the ‘Guide to the Expression of Uncertainty in Measurement’

The Comite international des poids et mesures (CIPM) has projected a major revision of the International System of Units (SI) in which all of the base units will be defined by fixing the values of fundamental constants of nature. In preparation for this we have carried out a new, low-uncertainty determination of the Boltzmann constant, kB, in terms of which the SI unit of temperature, the kelvin, can be re-defined. We have evaluated kB from exceptionally accurate measurements of the speed of sound in argon gas which can be related directly to the mean molecular kinetic energy, . Our new estimate is kB = 1.380 651 56 (98) × 10−23 J K−1 with a relative standard uncertainty uR = 0.71 × 10−6.

https://iopscience.iop.org/article/10.1088/0026-1394/50/4/354/pdf

A low-uncertainty measurement of the Boltzmann constant

This guide provides best practice in the evaluation of uncertainty within metrology, and in the support to this topic given by a treatment that is probabilistic. It is motivated by two principle considerations. The first is that although the primary guide on uncertainty evaluation, the 'Guide to the expression of uncertainty in measurement' (GUM), can be expected to be very widely applicable, the approach it predominantly endorses contains some limitations. The other is that on the basis of the authors' considerable contact with practitioners in the metrology community it is evident that important classes of problem are encountered that are subject to these limitations. A further consideration is that measurement models are encountered in practice that lie outside the scope of the model type (viz., a measurement function with a single (scalar) output quantity) that is the focus of the presentation given in the GUM. Central to consideration is the need to carry out uncertainty evaluation in as scientific a manner as economically possible. Although several approaches to uncertainty evaluation exist, the GUM has been very widely adopted (and is strongly supported by the authors of this guide). The emphasis of this guide is on making good use of the GUM, on aspects that yield greater generality, and especially on the provision in some cases of measurement uncertainties that are more objectively based and numerically more sustainable. The guide is also concerned with validating the current usage of the GUM in circumstances where there is doubt concerning its applicability. The relationship of this guide to the work being carried out by the Joint Committee on Guides in Metrology to prepare a new edition of the GUM and to provide supporting documents to the GUM is indicated.

Software Support for Metrology Best Practice Guide No. 6. Uncertainty evaluation.

The uncertainty associated with a value of some quantity is widely recognized throughout scientific disciplines as a quantitative measure of the reliability of that value. In addition, measurement uncertainty is increasingly seen as essential in quality assurance for industry. The Guide to the Expression of Uncertainty in Measurement (GUM) provides internationally agreed recommendations for the evaluation of uncertainties. This paper outlines the current situation of uncertainty evaluation in the context of international norms and arrangements. It describes the basic ideas and concepts that underlie the GUM and serves as a brief tutorial on methods for evaluating measurement uncertainty in a manner consistent with the GUM. It recommends an approach to evaluating measurement uncertainty based on the propagation of distributions using Monte Carlo simulation. An example is presented to illustrate Monte Carlo simulation.

Evaluation of measurement uncertainty based on the propagation of distributions using Monte Carlo simulation.

It is now fully appreciated that metrology will play an integral role in the successful development and commercialization of micro- and nanotechnology. To this end, the UK Government, through the National Measurement System, funded several groundbreaking projects in its 2002–2005 Programme for Length. This paper will briefly describe the background of the research, concentrating on the technical details of the projects. The Programme for Length normally only funds work into dimensional metrology but this funding cycle also funded work into low force metrology as this area is crucial to most mechanical probing techniques. The projects described include a traceable areal contacting instrument designed to calibrate areal transfer artefacts and hence offer traceability for industrial areal instruments, the production of the areal transfer artefacts, the development of Internet-based softgauges for profile parameters, a primary low force balance with a force resolution of 50 pN and the development of methods for measuring complex micro-scale structures. Amongst others, the projects involved collaboration with PTB, TNO, Taylor Hobson, AWE, Rubert & Co. and the Universities of Warwick, Huddersfield and Eindhoven.

Peter M. Harris

Papers

Evolution of the ‘Guide to the Expression of Uncertainty in Measurement’

A low-uncertainty measurement of the Boltzmann constant

Software Support for Metrology Best Practice Guide No. 6. Uncertainty evaluation.

Evaluation of measurement uncertainty based on the propagation of distributions using Monte Carlo simulation.

Recent advances in traceable nanoscale dimension and force metrology in the UK