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" General Theory of Irregular Curves "
by A.D. Alexandrov, Yu. G. Reshetnyak.
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BL
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772916
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b592910
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| Main Entry
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by A.D. Alexandrov, Yu. G. Reshetnyak.
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| Title & Author
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General Theory of Irregular Curves\ by A.D. Alexandrov, Yu. G. Reshetnyak.
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| Publication Statement
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Dordrecht : Springer Netherlands, 1989
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| Series Statement
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Mathematics and Its Applications, Soviet Series, 29.
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(x, 288 pages)
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| ISBN
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9400925913
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: 9789400925915
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| Contents
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I: General Notion of a Curve --; 1.1. Definition of a Curve --; 1.2. Normal Parametrization of a Curve --; 1.3. Chains on a Curve and the Notion of an Inscribed Polygonal Line --; 1.4. Distance Between Curves and Curve Convergence --; 1.5. On a Non-Parametric Definition of the Notion of a Curve --; II: Length of a Curve --; 2.1. Definition of a Curve Length and its Basic Properties --; 2.2. Rectifiable Curves in Euclidean Spaces --; 2.3. Rectifiable Curves in Lipshitz Manifolds --; III: Tangent and the Class of One-Sidedly Smooth Curves --; 3.1. Definition and Basic Properties of One-Sidedly Smooth Curves --; 3.2. Projection Criterion of the Existence of a Tangent in the Strong Sense --; 3.3. Characterizing One-Sidedly Smooth Curves with Contingencies --; 3.4. One-Sidedly Smooth Functions --; 3.5. Notion of c-Correspondence. Indicatrix of Tangents of a Curve --; 3.6. One-Sidedly Smooth Curves in Differentiable Manifolds --; IV: Some Facts of Integral Geometry --; 4.1. Manifold Gnk of k-Dimensional Directions in Vn --; 4.2. Imbedding of Gnk into a Euclidean Space --; 4.3. Existence of Invariant Measure of Gnk --; 4.4. Invariant Measure in Gnk and Integral. Uniqueness of an Invariant Measure --; 4.5. Some Relations for Integrals Relative to the Invariant Measure in Gnk --; 4.6. Some Specific Subsets of Gnk --; 4.7. Length of a Spherical Curve as an Integral of the Function Equal to the Number of Intersection Points --; 4.8. Length of a Curve as an Integral of Lengths of its Projections --; 4.9. Generalization of Theorems on the Mean Number of the Points of Intersection and Other Problems --; V: Turn or Integral Curvature of a Curve --; 5.1. Definition of a Turn. Basic Properties of Curves of a Finite Turn --; 5.2. Definition of a Turn of a Curve by Contingencies --; 5.3. Turn of a Regular Curve --; 5.4. Analytical Criterion of Finiteness of a Curve Turn --; 5.5. Basic Integra-Geometrical Theorem on a Curve Turn --; 5.6. Some Estimates and Theorems on a Limiting Transition --; 5.7. Turn of a Curve as a Limit of the Sum of Angles Between the Secants --; 5.8. Exact Estimates of the Length of a Curve --; 5.9. Convergence with a Turn --; 5.10 Turn of a Plane Curve --; VI: Theory of a Turn on an n-Dimensional Sphere --; 6.1. Auxiliary Results --; 6.2. Integro-Geometrical Theorem on Angles and its Corrolaries --; 6.3. Definition and Basic Properties of Spherical Curves of a Finite Geodesic Turn --; 6.4. Definition of a Geodesic Turn by Means of Tangents --; 6.5. Curves on a Two-Dimensional Sphere --; VII: Osculating Planes and Class of Curves with an Osculating Plane in the Strong Sense --; 7.1. Notion of an Osculating Plane --; 7.2. Osculating Plane of a Plane Curve --; 7.3. Properties of Curves with an Osculating Plane in the Strong Sense --; VIII: Torsion of a Curve in a Three-Dimensional Euclidean Space --; 8.1. Torsion of a Plane Curve --; 8.2. Curves of a Finite Complete Torsion --; 8.3. Complete Two-Dimensional Indicatrix of a Curve of a Finite Complete Torsion --; 8.4. Continuity and Additivity of Absolute Torsion --; 8.5. Definition of an Absolute Torsion Through Triple Chains and Paratingences --; 8.6. Right-Hand and Left-Hand Indices of a Point. Complete Torsion of a Curve --; IX: Frenet Formulas and Theorems on Natural Parametrization --; 9.1. Frenet Formulas --; 9.2. Theorems on Natural Parametrization --; X: Some Additional Remarks --; References.
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| Abstract
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One service mathematics has rendered the "Et moi ... si j'a\'ait su comment en revenir, human race. It has put common sense back je n'y scrais point alit: Jules Verne where it belongs, on the topmost shelf next to the dusty canister labc\led 'discarded non The series is divergent; therefore we may be sense'. Eric T. 8c\l able to do something with it. O. Hcaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
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| Subject
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Geometry.
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Global analysis (Mathematics)
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Mathematics.
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| Added Entry
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A D Alexandrov
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Yu G Reshetnyak
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