Table of contents

Contents Introduction Chapter I. The geometry of curves on §1. The elementary geometry of smooth curves and wavefronts §2. Contact manifolds, their Legendrian submanifolds and their fronts §3. Dual curves and derivative curves of fronts §4. The caustic and the derivatives of fronts Chapter II. Quaternions and the triality theorem §5. Quaternions and the standard contact structures on the sphere §6. Quaternions and contact elements of the sphere §7. The action of quaternions on the contact elements of the sphere §8. The action of right shifts on left-invariant fields §9. The duality of j-fronts and k-fronts of i-Legendrian curves Chapter III. Quaternions and curvatures §10. The spherical radii of curvature of fronts §11. Quaternions and caustics §12. The geodesic curvature of the derivative curve §13. The derivative of a small curve and the derivative of curvature of the curve Chapter IV. The characteristic chain and spherical indices of a hypersurface §14. The characteristic 2-chain §15. The indices of hypersurfaces on a sphere §16. Indices as linking coefficients §17. The indices of hypersurfaces on a sphere as intersection indices §18. Proofs of the index theorems §19. The indices of fronts of Legendrian submanifolds on an evendimensional sphere Chapter V. Exact Lagrangian curves on a sphere and their Maslov indices §20. Exact Lagrangian curves and their Legendrian lifts §21. The integral of a horizontal form as the area of the characteristic chain §22. A horizontal contact form as a Levi-Civita connection and a generalized Gauss-Bonnet formula §23. Proof of the formula for the Maslov index §24. The area-length duality §25. The parities of fronts and caustics Chapter VI. The Bennequin invariant and the spherical invariant §26. The spherical invariant §27. The topological meaning of the invariant Chapter VII. Pseudo-functions §28. The quasi-functions of Chekanov §29. From quasi-functions on the cylinder to pseudo-functions on the sphere, and conversely §30. Conjectures concerning pseudo-functions §31. Space curves and Sturm's theorem

Bibliography

Contents Introduction Chapter I. Infinitesimal projective transformations §1. Infinitesimal affine transformations §2. Infinitesimal conformal transformations §3. Infinitesimal projective transformations §4. Groups of projective transformations Chapter II. Projective and affine transformations determined by concircular vector fields §1. Conformal and projective Lie algebras determined by concircular vector fields §2. Projective Lie algebras determined by concurrent and parallel vector fields §3. Lie algebras of conformal and projective motions in spaces of constant curvature Chapter III. Lorentz manifolds admitting projective and affine motions §1. The skew-normal frame §2. Integration of the Eisenhart equations, h-spaces §3. The properties of h-spaces. Ordinary h-spaces and K-spaces §4. Non-homothetic projective motions in ordinary h-spaces Chapter IV. Classification of Lorentz manifolds with respect to the Lie algebras of projective and affine motions §1. Lie algebras of projective motions of ordinary h-spaces. The general case §2. Lie algebras of projective motions of ordinary h-spaces of type [I(I...I)] §3. Lie algebras of projective motions of ordinary h-spaces of type [3] §4. On K-spaces and the spaces V(K) §5. Lie algebras of projective motions of spaces V(0) with Lorentz signature §6. Lie algebras of projective motions of spaces V(K) with Lorentz signature §7. Lie algebras of affine motions of Lorentz manifolds Conclusion

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Contents Introduction. Description of main results Problem I. Justification of the second element of thermodynamics on the basis of classical mechanics Problem II. Justification of the Fermi acceleration mechanism Part 1. Rigorous justification of the second element of thermodynamics from mechanics for the gas consisting of finitely many particles, and the ergodic theory Chapter I. One-dimensional gas §1. Introducing a measure on the phase space for the distribution function and entropy §2. Newtonian model §3. Relativistic model Chapter II. Three-dimensional gas §1. Introducing a measure on the phase space for the distribution function and entropy §2. Newtonian model of a three-dimensional gas §3. Relativistic model of a three-dimensional gas Part 2. Acceleration mechanisms in Newtonian and relativistic mechanics, and the stability problem for non-autonomous dynamical systems Chapter I. The Fermi-Ulam model §1. On the existence of invariant curves under ring mappings close to degenerate ones §2. On the Ulam problem Chapter II. Relativistic analogue of the Fermi-Ulam model Chapter III. Relativistic analogue of the generalization of the Fermi-Ulam model Chapter IV. An accelerator model with one wall in the gravity field §1. Existence of a continuum of accelerated trajectories, and the Anosov systems §2. Existence of a residual set of accelerated trajectories, and generalization of the Birkhoff problem to the nonautonomous case Chapter V. The non-autonomous Birkhoff problem, the nonautonomous function-theoretic centre problem, and their applications §1. The non-autonomous Birkhoff problem and its generalization §2. The non-autonomous function-theoretic centre problem, and stability and normality of some families of analytic functions

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