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

A hybrid convex variational model for image restoration

We study an inhomogeneous partial differential equation which includes a separate edge detection part to control smoothing in and around possible discontinuities, under the framework of anisotropic diffusion. By incorporating edges found at multiple scales via an adaptive edge detector-based indicator function, the proposed scheme removes noise while respecting salient boundaries. We create a smooth transition region around probable edges found and reduce the diffusion rate near it by a gradient-based diffusion coefficient. In contrast to the previous anisotropic diffusion schemes, we prove the well-posedness of our scheme in the space of bounded variation. The proposed scheme is general in the sense that it can be used with any of the existing diffusion equations. Numerical simulations on noisy images show the advantages of our scheme when compared to other related schemes.

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Well-Posed Inhomogeneous Nonlinear Diffusion Scheme for Digital Image Denoising

A novel way to denoise multispectral images is proposed via an anisotropic diffusion based partial differential equation (PDE). A coupling term is added to the divergence term and it facilitates the modelling of interchannel relations in multidimensional image data. A total variation function is used to model the intrachannel smoothing and gives a piecewise smooth result with edge preservation. The coupling term uses weights computed from different bands of the input image and balances the interchannel information in the diffusion process. It aligns edges from different channels and stops the diffusion transfer using the weights. Well-posedness of the PDE is proved in the space of bounded variation functions. Comparison with the previous approaches is provided to demonstrate the advantages of the proposed scheme. The simulation results show that the proposed scheme effectively removes noise and preserves the main features of multispectral image data by taking channel coupling into consideration.

Multispectral image denoising by well-posed anisotropic diffusion scheme with channel coupling

Anisotropic partial differential equation (PDE)-based image restoration schemes employ a local edge indicator function typically based on gradients. In this paper, an alternative pixel-wise adaptive diffusion scheme is proposed. It uses a spatial function giving better edge information to the diffusion process. It avoids the over-locality problem of gradient-based schemes and preserves discontinuities coherently. The scheme satisfies scale space axioms for a multiscale diffusion scheme; and it uses a well-posed regularized total variation (TV) scheme along with Perona-Malik type functions. Median-based weight function is used to handle the impulse noise case. Numerical results show promise of such an adaptive approach on real noisy images.

An adaptive diffusion scheme for image restoration and selective smoothing

This paper extends the notion of prime implicates to first order logic formulas without equality which are assumed to be in Skolem Conjunctive Normal Form. Using the extended notions of consensus and subsumption it is shown that the consensus-subsumption algorithm for computing prime implicates well known for propositional formulas can be conditionally lifted to first order formulas.

Arindama Singh

Papers

A hybrid convex variational model for image restoration

Well-Posed Inhomogeneous Nonlinear Diffusion Scheme for Digital Image Denoising

Multispectral image denoising by well-posed anisotropic diffusion scheme with channel coupling

An adaptive diffusion scheme for image restoration and selective smoothing

Prime Implicates of First Order Formulas.