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In this article we study the (Cartesian) Products of sets. As a prelude to it, we examine various definitions offered by different mathematicians to the concept of an ordered pair. The subtle difference between the notions of ordered products of sets and unordered products of sets is highlighted.
This section of Resonance presents thought-provoking questions, and discusses answers a few months later. Readers are invited to send new questions, solutions to old ones and comments, to ‘Think It Over’, Resonance, Indian Academy of Sciences, Bangalore 560 080. Items illustrating ideas and concepts will generally be chosen.
Against the background of the development of physics, and in particular of mechanics, over the centuries since Galileo and Newton, we describe the life and work of William Rowan Hamilton in the 19th century. The depth of his ideas which brought together the understanding of ray optics and classical mechanics, and the remarkable ways in which his work paved the way to the construction of quantum mechanics...
Building on work by Fermat and Huygens, Hamilton transformed the study of geometrical optics in his very first paper, presented when still in his teens. His ‘characteristic function’ was an analytical way to describe wavefronts, and in his hands a powerful tool to look at families of rays rather than isolated ones. His prediction of internal and external conical refraction in some crystals and its...
Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three-dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated...
Wetlands are repositories of unique biodiversity. Wetland organisms are well adapted to their habitat, lying at the interface of aquatic and terrestrial environments. In order to understand their adaptations in a better way, it is essential to grasp the basic properties of the medium in which various organisms live. This is attempted here by first examining the properties of the two contrasting environments,...
We consider the quantum-mechanical non-relativistic hydrogen atom. We show that for bound states with size much larger than the Bohr radius, one can construct a wave packet that is localized in space corresponding to a classical particle moving in a circular orbit.
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