Arindama Singh

Bio: Arindama Singh is an academic researcher from Indian Institute of Technology Madras. The author has contributed to research in topics: Image restoration & Singular perturbation. The author has an hindex of 8, co-authored 37 publications receiving 202 citations. Previous affiliations of Arindama Singh include University UCINF & Indian Institute of Technology Kanpur.

Cantor’s Little Theorem

Abstract: This article discusses two theorems of Georg Cantor: Cantor’s Little Theorem and Cantor’s Diagonal Theorem. The results are obtained by generalizing the method of proof of the well known Cantor’s theorem about the cardinalities of a set and its power set. As an application of these, Godel’s first incompleteness theorem is proved. Hints are given as to how to derive other deeper results including the existence of Parikh’s sentence.

Boundary-value technique to solve linear state regulator problems

Abstract: In this paper, a method is proposed to solve free endpoint optimal control problems with linear state and quadratic cost. After translating the problem into a two-point boundary-value problem (TPBVP) that arises from the optimization of the Hamiltonian, two pointst 1 andt 2,t 1

Construction and Categoricity of the Real Number System Using Decimals

Abstract: In this expository article, the real numbers are defined as infinite decimals. After defining an ordering relation and the arithmetic operations, it is shown that the set of real numbers is a complete ordered field. It is further shown that any complete ordered field is isomorphic to the constructed set of real numbers.

A numerical method for solving singularly perturbed systems of nonlinear two-point boundary-value problems

Abstract: In this paper, a numerical method is proposed to solve singularly perturbed systems of nonlinear two-point boundary-value problems. First, Newton's iteration is used to linearize such problems, reducing these to a sequence of linear singularly perturbed two-point boundary-value problems. Then,a difference scheme is applied to solve the linear systems. The difference scheme is accurate up to O(h 2). Test examples are included to demonstrate the efficiency of the method.

Theory of Functions

Abstract: 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.

A Basis for a Mathematical Theory of Computation

Abstract: 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.

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

Abstract: 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.