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Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of simplicity was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E - However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).","brand":"WoB","offers":[{"title":"GB \/ NEW \/ INGRAM","offer_id":52089167708433,"sku":"NLS9783540177586","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9783540177586.jpg?v=1785858269"},{"product_id":"classical-groups-and-k-theory-book-alexander-j-hahn-9783642057373","title":"The Classical Groups and K-Theory","description":"It is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. 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Much of the underlying mathematical story is presented alongside the astonishing images and extensive data that NASA’s \u003ci\u003eVoyager, NEAR-Shoemaker, Cassini, \u003c\/i\u003eand \u003ci\u003eJuno \u003c\/i\u003emissions have sent back to us. \u003c\/p\u003e\n\n\u003cp\u003eFirst and second year college students in mathematics, engineering, or science, and those seeking an enriching independent study, will experience the mathematical language and methods of single variable calculus within their application to relevant conceptual and strategic aspects of the navigation of a spacecraft. The reader is expected to have taken one or two semesters of the basic calculus of derivatives, integrals, and the role that limits play. Additional prerequisites include knowledge of coordinate plane geometry, basic trigonometry, functions and graphs, including trig, inverse, exponential, and log functions.\u003c\/p\u003e\n\n\u003cp\u003eThe discussions begin with the rich history of humanity’s efforts to understand the universe from the Greeks, to Newton and the Scientific Revolution, to Hubble and galaxies, to NASA and the space missions.   The calculus of polar functions that plays a central mathematical role is presented in a self-contained way in complete detail. Each of the six chapters is followed by an extensive problem set that deals with and also expands on the concerns of the chapter. 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