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 revisit a problem faced by all programmers. Can one write a program that determines whether any given program is the shortest program? How does one prove that a given program is the shortest? After answering these questions, we discuss very briefly the Kolmogorov complexity of a string of zero and one, which leads to a barrier on any axiomatic system, known as Chaitin’s barrier.

On Proving a Program Shortest

In Chap. 1 we defined a vector space as an abstraction of the familiar Euclidean space. In doing so, we took into account only two aspects of the set of vectors in a plane, namely the vector addition and scalar multiplication. Now, we consider the third aspect, namely the angle between vectors.

Inner Product Spaces

In this paper, a modified version of Newton's Iteration (also called Quasilinearization) for nonlinear systems of ordinary differential equations is considered. Under certain conditions, a simple proof of quadratic convergence of the iteration is derived.

A modified version of newton's iteration

In this note, the thickness of the boundary layers for linear, homogeneous, singularly perturbed first-order ordinary differential systems is considered. Inequalities concerning this thickness are derived. It is also remarked that the present analysis can be carried over to the nonhomogeneous case.

Estimate of boundary layer thickness for linear, singularly perturbed two-point boundary-value problems

A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. We define a related notion of a true-pair of a linear operator, and then show that each linear operator on a finite dimensional nonzero real vector space has a true-pair. This is usually proved by using the Fundamental theorem of algebra and Cayley-Hamilton theorem. We construct an inductive proof of this fact without using these theorems. From this we deduce that a polynomial with real coefficients can be written as a product of linear factors and quadratic factors with negative discriminant. It thus gives a proof of the latter fact about polynomials with real coefficients, which does not use complex numbers.

Arindama Singh

Papers

On Proving a Program Shortest

Inner Product Spaces

A modified version of newton's iteration

Estimate of boundary layer thickness for linear, singularly perturbed two-point boundary-value problems

True-pairs of Real Linear Operators and Factorization of Real Polynomials