Showing papers in "Russian Mathematical Surveys in 1977"
TL;DR: In this article, the authors define the ergodicity of a diffeomorphism with non-zero exponents on a set of positive measure and the Bernoullian property of geodesic flows on closed Riemannian manifolds.
Abstract: CONTENTSPart I § 1. Introduction § 2. Prerequisites from ergodic theory § 3. Basic properties of the characteristic exponents of dynamical systems § 4. Properties of local stable manifoldsPart II § 5. The entropy of smooth dynamical systems § 6. "Measurable foliations". Description of the π-partition § 7. Ergodicity of a diffeomorphism with non-zero exponents on a set of positive measure. The K-property § 8. The Bernoullian property § 9. Flows § 10. Geodesic flows on closed Riemannian manifolds without focal pointsReferences
1,393 citations
TL;DR: The main ideas of the proof of the exponential estimate were discussed in this paper, including steepness conditions and forbidden motions of the discs of fast drift on the steepness of the unperturbed Hamiltonian.
Abstract: CONTENTS § 1 Introduction § 2 Unsolved problems Conjectures Generalizations § 3 The main ideas of the proof of the exponential estimate § 4 Steepness conditions Precise statement of the main theorem § 5 Forbidden motions § 6 Resonances Resonance zones and blocks § 7 Dependence of the diameters of the discs of fast drift on the steepness of the unperturbed Hamiltonian § 8 Condition for the non-overlapping of resonances § 9 Traps in frequency systems Completion of the proof of the main theorem § 10 Statement of the lemma on the elimination of non-resonance harmonics, and of the technical lemmas used in the proof of the main theorem § 11 Remarks on the proof of the main theorem § 12 Application of the main theorem to the many-body problem References
726 citations
TL;DR: The problem of multi-dimensional -algebraic operators is studied in this article, where the Hamiltonian formalism in equations of Lax and Novikov types is considered.
Abstract: CONTENTSIntroduction § 1. The Akhiezer function and the Zakharov-Shabat equations § 2. Commutative rings of differential operators § 3. The two-dimensional Schrodinger operator and the algebras associated with it § 4. The problem of multi-dimensional -algebraic operators Appendix 1. The Hamiltonian formalism in equations of Lax and Novikov types Appendix 2. Elliptic and rational solutions of the K-dV equations and systems of many particles Concluding Remarks References
508 citations
TL;DR: In this article, Solomyak proposed interpolation of bilinear maps from function spaces into operator spaces, and proved the main results on estimates of singular numbers based on the method of piecewise-polynomial approximation.
Abstract: ContentsIntroduction § 1. Operator spaces and function spaces § 2. Estimates of singular numbers based on the method of piecewise-polynomial approximation § 3. Interpolation methods § 4. General results on estimates (statements) § 5. Proof of the main results on estimates § 6. Operators with a difference kernel § 7. Tests for nuclearity § 8. Multipliers in classes of kernels of integral operators § 9. Analytical results for the multiplier problem § 10. Operators with a homogeneous difference kernel § 11. Some special casesAppendix. M. Z. Solomyak, The Interpolation of bilinear maps from function spaces into operator spacesReferences
196 citations
TL;DR: The degree of a Fredholm map is a measure of the degree of the map as discussed by the authors, and the degree is a function of the distance from the map to the ground plane of the graph.
Abstract: ContentsIntroduction § 1. Banach manifolds and their maps § 2. The degree of a Fredholm map § 3. Equivariant Fredholm maps § 4. Solubility of equations with Fredholm operators § 5. Some applications to existence theorems for differential equations § 6. Complete and local invertibility of proper maps § 7. Intersection indicesReferences
97 citations
TL;DR: In this article, the authors apply sublinear operators to the study of semigroups and apply superlinear point-set mappings to the problem of point set mappings in semigroup analysis.
Abstract: ContentsIntroduction ??1. Sublinear operators ??2. Application of sublinear operators to the study of semigroups ??3. Superlinear point-set mappingsReferences
95 citations
TL;DR: The Lewy equation on the boundary of a strictly convex domain has been studied in this article, where the main formula for global solutions of the -equation and a criterion for the local solubility of the Lewy equations are discussed.
Abstract: ContentsIntroductionChapter I. The Lewy equation on the boundary of a strictly convex domain ??1. The representation of a -closed form on as a difference between -exact forms and in the domains and ??2. A preliminary formula for the solutions of the equation on ??3. The main formula for global solutions of the -equation ??4. A criterion for the local solubility of the Lewy equation ??5. Local solutions for the equations of J.?J.?Kohn ??6. Estimates for the solutions of the -equation ??7. Formulae and estimates for solutions of the equation in a strictly convex domainChapter II. Analysis on a pseudoconvex manifold ??8. Solutions with uniform bounds for the -equation on a strictly pseudoconvex manifold ??9. The Levi problem and the Hartogs-Bochner effect on a strictly pseudoconvex manifold ??10. The global solubility of the Lewy equation on the boundary of a strictly pseudoconvex manifold and approximation of -closed formsReferences
90 citations
TL;DR: Gabrielov and Varchenko as discussed by the authors proposed a conjecture on the signature of the quadratic form of a quasihomogeneous singularity, which is a conjecture based on the inertia index of the singularity.
Abstract: ContentsIntroductionSome definitions § 1. Picard-Lefschetz theory § 2. The monodromy group and variation operator of a singularity § 3. The intersection matrices of specific singularities § 4. Intersection matrices of singularities of functions of two variablesAppendix 1. The connection between the modality of a singularity and the inertia index of its quadratic formAppendix 2. V. I. Arnol'd, A conjecture on the signature of the quadratic form of a quasihomogeneous singularityAppendix 3. A. M. Gabrielov, Monodromy groups and bordering of singularitiesAppendix 4. A. N. Varchenko, The characteristic polynomial of the monodromy operator and the Newton diagram of a singularityReferences
83 citations
77 citations
TL;DR: The Hamiltonian form of mechanics with friction, nonholonomic mechanics, invariant mechanics, the theory of refraction and impact, and invariant Hamiltonians of mechanical type are discussed in this article.
Abstract: ContentsIntroduction ? 0. Definitions, notation, and basic facts ? 1. The universal 1-form. Lagrangian manifolds ? 2. Fields and 1-forms on ? ? 3. Poisson brackets ? 4. Symmetries ? 5. The structure of Hamiltonian systems with a given degree of symmetry ? 6. Completely integrable systems ? 7. The theorems of Darboux and Weinstein ? 8. Invariant Hamilton-Jacobi theory ? 9. Hamiltonians of mechanical type ? 10. Mechanical systems. ExamplesAppendix I. BifurcationsAppendix II. A.?V.?Bocharov and A.?M.?Vinogradov, The Hamiltonian form of mechanics with friction, non-holonomic mechanics, invariant mechanics, the theory of refraction and impactReferences
65 citations
TL;DR: In this article, the authors introduce self-adjoint bi-extensions of closed Hermitian operators and quasi-Hermitian Bi-Involutively self-involutive bi-Inventive Self-Adjoint Bi-Extensions of Semi-Bounded Hermitians.
Abstract: CONTENTSIntroduction Chapter I. Operators on rigged Hilbert spaces Chapter II. Self-adjoint bi-extensions of closed Hermitian operators Chapter III. Self-adjoint bi-extensions of semibounded operators Chapter IV. Bi-involutively self-adjoint bi-extensions of J-Hermitian operators Chapter V. Quasi-Hermitian bi-extensions of closed Hermitian operators Chapter VI. Extended resolvents of Hermitian and quasi-Hermitian operators Chapter VII. Generalized operator colligations and their characteristic operator-functions Chapter VIII. Representation of Hermitian operators with improper reference subspace. The resolvent matrix Comments and guide to the literature References
TL;DR: A survey of results and problems in this area and connections with other topics can be found in this article, where a group representation is defined as a pair (A, Γ) where A is a module over a commutative ring K with identity and Γ is a group that acts on A.
Abstract: A group representation is a pair (A,Γ), where A is a module over a commutative ring K with identity and Γ is a group that acts on A. In the category of group representations over a fixed K, both A and Γ are variable. We study varieties in this category. This paper is a survey of results and problems in this area and connections with other topics.
TL;DR: This chapter discusses the space of random variables and random functions, and the tensor exponent of a contractive operator in Gaussian random functions and their polynomial hulls.
Abstract: ContentsIntroduction § 1. The space of random variables and random functions § 2. Polynomials in a random function § 3. Linear random functions and polynomials in such functions § 4. Gaussian random functions and their polynomial hulls § 5. The Ito-Wick mapping § 6. The tensor exponent of a contractive operatorReferences
TL;DR: In this article, the spectral decomposition of second-order elliptic operators is studied and an estimate of the remainder term of the spectral function in the L2 metric and its consequences is given.
Abstract: ContentsChapter III. A study, based on the mean-value formula, of spectral decompositions for second-order elliptic operators § 1. Basic notation and definitions § 2. Properties of fundamental functions § 3. An estimate of the remainder term of the spectral function in the L2 metric and its consequences § 4. A uniform estimate of the remainder term of the spectral function § 5. The general case of second-order elliptic operatorsReferences
TL;DR: In this paper, the behaviour of curves in spaces of non-negative curvature has been studied and a survey of results can be found in Section 5.1.1 and Section 6.2.
Abstract: CONTENTS § 1. IntroductionChapter I. Survey of results § 2. Closed spaces of non-negative curvature § 3. Open spaces of non-negative curvature § 4. Convex sets. Structure in the small § 5. Convex sets. Structure in the largeChapter II. Proofs § 6. Basic constructions § 7. Proof of the theorems in § 4 § 8. Proof of the theorems in § 5.2 § 9. Proof of Theorems 5.3 and 3.1.3 § 10. Proof of Theorem 5.4 § 11. Auxiliary propositions for § 10Chapter III. Appendix § 12. On the behaviour of curves in spaces of non-negative curvatureReferences
TL;DR: In this article, the singularities of integrals with oscillating kernels with general position were studied and the fundamental solution of a strictly hyperbolic equation was shown to have asymptotic behavior.
Abstract: CONTENTSIntroduction § 1. Singularities of integrals with oscillating kernels in the case of general position § 2. Singularities of type Am and Dm § 3. Singularities of the fundamental solution of a strictly hyperbolic equation § 4. Asymptotic behaviour of the solution of the mixed problem References
TL;DR: In this paper, the Riesz representation theorem for compact and locally compact spaces is used to define a functional representation by measures on compactifications and an integral representation of functionals on spaces of continuous functions of bounded support.
Abstract: CONTENTSIntroduction Chapter I. Preliminary information from the compactification theory of topological spaces § 1. The concept of a topological compactification § 2. Some methods of obtaining compactifications § 3. Some special compactifications Chapter II. The compactification theory of topological spaces and the integral representation of linear functionals § 1. The Riesz representation theorem for compact and locally compact spaces § 2. A functional representation by measures on compactifications § 3. An integral representation of functionals on the space of all continuous functions § 4. An integral representation of functionals on spaces of continuous functions of bounded support § 5. An integral representation of functionals on spaces without a topology § 6. The problem of extending measures with values in vector lattices Chapter III. Invariant Banach limits § 1. The classical statement of the problem § 2. The problem in a more general topological setting § 3. The support of the representative measure of an invariant permanent functional Chapter IV. Summability theory § 1. The classical statement of the problem § 2. The Stone-Cech compactification and multiplicative summability methods § 3. The Stone-Cech compactification and compatibility criteria § 4. The Stone-Cech compactification of non-discrete countable spaces and summability § 5. Summability problems in the more general setting of compactification theory and measure theory References
TL;DR: In this paper, the main theorem and the existence theorem are proved for Abelian spaces and simply-connected spaces, and a comparison of localizations is given. But the main and existence theorem is not applicable to the case of simply connected spaces.
Abstract: CONTENTSIntroduction § 1. The general concept of localization § 2. Localization of Abelian groups § 3. Localization of Abelian spaces. The main theorem § 4. The existence theorem. Special cases § 5. Fibrations § 6. Π-decompositions § 7. Proof of the main theorem and the existence theorem § 8. Localization of simply-connected spaces § 9. Comparison of localizations. Rationalization § 10. Nilpotent spaces § 11. Disintegrated Π-decompositions § 12. Localization of nilpotent spaces and groups References