BOOKS  
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Books

Books in English Edited books Books in Danish
Selected Books Shadows of the Circle: From Conic Sections to Planetary Motion(2^{nd} ed.) The ancient Greeks were the first to seriously ask for scientific explanations of the panorama of the heavens based on mathematical ideas. Ever since, mathematics has played a major role for human perception and description of the outside physical world, and in a larger perspective for comprehending the universe. This second edition pays tribute to this line of thought and takes the reader on a journey in the mathematical universe from conic sections to mathematical modelling of planetary systems. In the second edition, the four chapters in the first edition on conic sections (two chapters), isoperimetric problems for plane figures, and nonEuclidean geometry, are treated in four revised chapters with many new exercises added. In three new chapters, the reader is taken through mathematics in curves, mathematics in a Nautilus shell, and mathematics in the panorama of the heavens. In all chapters of the book, the circle plays a prominent role. This book is addressed to undergraduate and graduate students as well as researchers interested in the geometry of conic sections, including the historical background and mathematical methods used. It features selected important results, and proofs that not only proves but also 'explains' the results.
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Many applied mathematical disciplines, such as dynamical systems and optimization theory as well as classical mathematical disciplines like differential geometry and the theory of Lie groups, have a common foundation in general topology and multivariate calculus in normed vector spaces. In this book, students from both pure and applied subjects are offered an opportunity to work seriously with fundamental notions from mathematical analysis that are important not only from a mathematical point of view but also occur frequently in the theoretical parts of, for example, the engineering sciences. The book provides complete proofs of the basic results from topology and differentiability of mappings in normed vector spaces. It is a useful resource for students and researchers in mathematics and the many sciences that depend on fundamental techniques from mathematical analysis. In this second edition, the notions of compactness and sequentially compactness are developed with independent proofs for the main results. Thereby the material on compactness is apt for direct applications also in functional analysis, where the notion of sequentially compactness prevails. This edition also covers a new section on partial derivatives, and new material has been incorporated to make a more complete account of higher order derivatives in Banach spaces, including full proofs for symmetry of higher order derivatives and Taylor's formula. The exercise material has been reorganized from a collection of problem sets at the end of the book to a section at the end of each chapter with further results. Readers will find numerous new exercises at different levels of difficulty for practice.
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This book presents basic elements of the theory of Hilbert spaces and operators on Hilbert spaces, culminating in a proof of the spectral theorem for compact, selfadjoint operators on separable Hilbert spaces. It exhibits a construction of the space of p^{th} power Lebesgue integrable functions by a completion procedure with respect to a suitable norm in a space of continuous functions, including proofs of the basic inequalities of Hölder and Minkowski. The L^{p}spaces thereby emerges in direct analogy with a construction of the real numbers from the rational numbers. This allows grasping the main ideas more rapidly. Other important Banach spaces arising from function spaces and sequence spaces are also treated. In this second edition, I have expanded the material on normed vector spaces and their operators presented in Chapter 1 to include proofs of the Open Mapping Theorem, the Closed Graph Theorem and the Hahn Banach Theorem. The material on operators between normed vector spaces is further expanded in a new Chapter 6, which presents the basic elements of the theory of Fredholm operators on general Banach spaces, not only on Hilbert spaces. This requires that we develop the theory of dual operators between Banach spaces to replace the use of adjoint operators between Hilbert spaces. With the addition of the new material on normed vector spaces and their operators, the book can serve as a general introduction to functional analysis viewed as a theory of infinite dimensional linear spaces and linear operators acting on them.
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Denne bog henvender sig til alle der ønsker et indblik i matematikkens mange bidrag til kultur, videnskab og samfund. Bogen er derfor tilrettelagt som en række små essays, hvor der tilstræbes en mere beskrivende og fortællende fremstilling end normalt i bøger med et matematisk indhold. Emnerne i bogen er mangfoldige og strækker sig fra beskrivelse af geometriske former og fænomener i omverdenen til diskussion af nogle af uendelighedsbegrebets mange aspekter. Store dele af bogen kræver ikke specielle matematiske forkundskaber, men nok en vis optagethed af faget. Bogen giver i glimt smagsprøver på matematikkens abstrakte idé verden, herunder et indblik i nogle af matematikkens nyeste landvindinger. 