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

An algorithm based on consensus method to compute the set of prime implicates of a quantifier free first order formula X was presented in an earlier work. In this paper the notion of prime implicates is extended to theory prime implicates in the first order case. We provide an algorithm to compute the theory prime implicates of a knowledge base X with respect to another knowledge base Y where both X and Y are assumed to be unquantified first order formulas. The partial correctness of the algorithm is proved.

An algorithm for computing theory prime implicates in first order logic

A method is proposed to solve fixed end-point, linear optimal control problems with quadratic cost and singularly perturbed state. After translating the problem into a two-point boundary value problem, we choose two points t1, t2 ϵ [t0, tf] and let τ = (t-t0)/ϵ and σ = (tf-t)/ϵ. The τ-scaled, original and σ-scaled boundary value problems are then solved on the intervals [t0, t1], [t1, t2] and [t2, tf] respectively. A test example is solved to illustrate the method.

A Boundary value technique for solving singularly perturbed, fixed end-point optimal control problems

In this paper we study how Lavrentiev regularization can be used in the context of learning theory, especially in regularization networks that are closely related to support vector machines. We briefly discuss formulations of learning from examples in the context of ill-posed inverse problem and regularization. We then study the interplay between the Lavrentiev regularization of the concerned continuous and discretized ill-posed inverse problems. As the main result of this paper, we give an improved probabilistic bound for the regularization networks or least square algorithms, where we can afford to choose the regularization parameter in a larger interval.

An error analysis of lavrentiev regularization in learning theory

Showalter regularization of a singularly perturbed PDE

We present a ball game that can be continued as long as we wish. It looks as though the game would never end. But by applying a result on trees, we show that the game nonetheless ends in some finite number of moves. We then point out some deep results on the natural number system connected with the game.

Arindama Singh

Papers

An algorithm for computing theory prime implicates in first order logic

A Boundary value technique for solving singularly perturbed, fixed end-point optimal control problems

An error analysis of lavrentiev regularization in learning theory

Showalter regularization of a singularly perturbed PDE

From a ball game to incompleteness