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Differential geometry. Global analysis --- Curves.

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Curves on surfaces. --- Flows (Differentiable dynamical systems). --- Hamiltonian systems.

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During the last five years, after the first meeting on "Quaternionic Structures in Mathematics and Physics", interest in quaternionic geometry and its applications has continued to increase. Progress has been made in constructing new classes of manifolds with quaternionic structures (quaternionic Kähler, hyper-Kähler, hyper-complex, etc.), studying the differential geometry of special classes of such manifolds and their submanifolds, understanding relations between the quaternionic structure and other differential-geometric structures, and also in physical applications of quaternionic geometry

Geometry, Differential --- Complex manifolds --- Quaternions --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis --- Analytic spaces --- Manifolds (Mathematics)

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This volume takes a new look at one of the greatest works of Hellenistic mathematics, Apollonius of Perga's Conica . It provides a long overdue alternative to H.G. Zeuthen's Die Lehre von den Kegelschnitten im Altertum . The central part of the volume contains a historically sensitive analysis and interpretation of the entire Conica , both from the standpoint of its individual books and of the text as a whole. Particular attention is given to Books V-VII, which have had scant treatment until now. Two chapters in the volume concern histioriographic issues connected with the Conica in paricular and Greek mathematics in general. Although the volume is intended primarily for historians of ancient mathematics, its approach is fresh and engaging enough to be of interest also to historians, philosophers, linguists, and open-minded mathematicians.

Mathematics, Greek --- Conic sections --- Mathématiques grecques --- Coniques --- Apollonius, --- Mathematics, Greek. --- Greek mathematics --- Geometry --- Curves, Plane --- Geometry, Plane --- Mathematics --- Ellipse --- Geometry, Analytic --- Parabola --- Apollonius of Perga --- Mathématiques grecques

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Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves over finite fields with many rational points (relative to the genus). Recently, the authors discovered another important application of such curves, namely to the construction of low-discrepancy sequences. These sequences are needed for numerical methods in areas as diverse as computational physics and mathematical finance. This has given additional impetus to the theory of, and the search for, algebraic curves over finite fields with many rational points. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences.

Curves, Algebraic. --- Finite fields (Algebra) --- Rational points (Geometry) --- Coding theory. --- Data compression (Telecommunication) --- Digital electronics --- Information theory --- Machine theory --- Signal theory (Telecommunication) --- Computer programming --- Points, Rational (Geometry) --- Arithmetical algebraic geometry --- Modular fields (Algebra) --- Algebra, Abstract --- Algebraic fields --- Galois theory --- Modules (Algebra) --- Algebraic curves --- Algebraic varieties