THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

Publisher Summary This chapter discusses the mathematical theory of computation. Computation essentially explores how machines can be made to carry out intellectual processes. Any intellectual process that can be carried out mechanically can be performed by a general purpose digital computer. There are three established directions of mathematical research that are relevant to the science of computation—namely, numerical analysis, theory of computability, and theory of finite automata. The chapter explores what practical results can be expected from a suitable mathematical theory. Further, the chapter presents several descriptive formalisms with a few examples of their use and theories that enable to prove the equivalence of computations expressed in these formalisms. A few mathematical results about the properties of the formalisms are also presented.

A Basis for a Mathematical Theory of Computation

This paper presents an overview of singular perturbations and time scales (SPaTS) in control theory and applications during the period 1984-2001 (the last such overviews were provided by [231, 371]). Due to the limitations on space, this is in way intended to be an exhaustive survey on the topic.

Singular perturbations and time scales in control theory and applications: An overview

Triangle centers and central triangles , by Clark Kimberling. Pp. 295. $42.90. 1998. ISSN 0316 1282 ( Congressum Numerantium , Vol. 129, Winnipeg).

A comprehensive survey on impulse and Gaussian denoising filters for digital images

We propose a novel multichannel image restoration scheme based on Bayesian maximum a posteriori estimation. Restoration schemes for mono-channel images fail in the case of multichannel images since they do not take into account the cross channel relations present. We propose a robust M-estimator based Gibbs prior which uses not only spatial but also interchannel term for multichannel image data modelled as a Markov random field. Regularized total variation potential function is used as a robust M-estimator in each channel and a weighted inter-channel coupling term with Gaussian prior is used to capture the cross-correlations of the given image. We prove the well-posedness of the corresponding minimization in the bounded variation space. This variational Bayesian scheme reduces the channel mixup artifacts and preserves color edges when compared with other schemes on real and noisy multidimensional images.

Multichannel image restoration using combined channelinformation and robust M-estimator approach

Partial decoupling of slow and fast variables

A numerical method for singularly perturbed systems of linear two-point boundary-value problems using partial decoupling

It is generally believed that the Ritz vectors do not coincide with the refined Ritz vectors in the Arnoldi method for computing eigenvalues of matrices. We show that this coincidence is theoretically possible. We provide a necessary and sufficient condition for this coincidence to happen and give examples to illustrate the same. Using Lanczos polynomials, we give a polynomial characterization of refined Ritz vectors of symmetric matrices that is different from the one available in the literature.

On refined Ritz vectors and polynomial characterization

The methods used for extracting an approximate eigenpair are crucial in sparse iterative eigensolvers. Using least squares and line search techniques this paper devises a method for an approximate eigenpair extraction. Numerical comparison of the Jacobi–Davidson method using the suggested method of eigenpair extraction, Rayleigh–Ritz, and refined Ritz projections shows that the suggested method is a viable alternative.

Arindama Singh

Papers

Multichannel image restoration using combined channelinformation and robust M-estimator approach

Partial decoupling of slow and fast variables

A numerical method for singularly perturbed systems of linear two-point boundary-value problems using partial decoupling

On refined Ritz vectors and polynomial characterization

The Least squares and line search in extracting eigenpairs in Jacobi–Davidson method