Russian Mathematical Surveys 

About: Russian Mathematical Surveys is an academic journal published by IOP Publishing. The journal publishes majorly in the area(s): Obituary & Boundary value problem. It has an ISSN identifier of 0036-0279. Over the lifetime, 4534 publications have been published receiving 92745 citations. The journal is also known as: Uspekhi Matematicheskikh Nauk.

Characteristic lyapunov exponents and smooth ergodic theory

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

Quantum computations: algorithms and error correction

Abstract: Contents §0. Introduction §1. Abelian problem on the stabilizer §2. Classical models of computations2.1. Boolean schemes and sequences of operations2.2. Reversible computations §3. Quantum formalism3.1. Basic notions and notation3.2. Transformations of mixed states3.3. Accuracy §4. Quantum models of computations4.1. Definitions and basic properties4.2. Construction of various operators from the elements of a basis4.3. Generalized quantum control and universal schemes §5. Measurement operators §6. Polynomial quantum algorithm for the stabilizer problem §7. Computations with perturbations: the choice of a model §8. Quantum codes (definitions and general properties)8.1. Basic notions and ideas8.2. One-to-one codes8.3. Many-to-one codes §9. Symplectic (additive) codes9.1. Algebraic preparation9.2. The basic construction9.3. Error correction procedure9.4. Torus codes §10. Error correction in the computation process: general principles10.1. Definitions and results10.2. Proofs §11. Error correction: concrete procedures11.1. The symplecto-classical case11.2. The case of a complete basis Bibliography

The geometry of toric varieties

Abstract: ContentsIntroductionChapter I Affine toric varieties § 1 Cones, lattices, and semigroups § 2 The definition of an affine toric variety § 3 Properties of toric varieties § 4 Differential forms on toric varietiesChapter II General toric varieties § 5 Fans and their associated toric varieties § 6 Linear systems § 7 The cohomology of invertible sheaves § 8 Resolution of singularities § 9 The fundamental groupChapter III Intersection theory § 10 The Chow ring § 11 The Riemann-Roch theorem § 12 Complex cohomologyChapter IV The analytic theory § 13 Toroidal varieties § 14 Quasi-smooth varieties § 15 Differential forms with logarithmic polesAppendix 1 Depth and local cohomologyAppendix 2 The exterior algebraAppendix 3 DifferentialsReferences

Small denominators and problems of stability of motion in classical and celestial mechanics

Abstract: CONTENTSIntroduction § 1. Results § 2. Preliminary results from mechanics § 3. Preliminary results from mathematics § 4. The simplest problem of stability § 5. Contents of the paperChapter I. Theory of perturbations § 1. Integrable and non-integrable problems of dynamics § 2. The classical theory of perturbations § 3. Small denominators § 4. Newton's method § 5. Proper degeneracy § 6. Remark 1 § 7. Remark 2 § 8. Application to the problem of proper degeneracy § 9. Limiting degeneracy. Birkhoff's transformation § 10. Stability of positions of equilibrium of Hamiltonian systemsChapter II. Adiabatic invariants § 1. The concept of an adiabatic invariant § 2. Perpetual adiabatic invariance of action with a slow periodic variation of the Hamiltonian § 3. Adiabatic invariants of conservative systems § 4. Magnetic traps § 5. The many-dimensional caseChapter III. The stability of planetary motions § 1. Picture of the motion § 2. Jacobi, Delaunay and Poincare variables §3. Birkhoff's transformation § 4. Calculation of the asymptotic behaviour of the coefficients in the expansion of § 5. The many-body problemChapter IV. The fundamental theorem § 1. Fundamental theorem § 2. Inductive theorem § 3. Inductive lemma § 4. Fundamental lemma § 5. Lemma on averaging over rapid variables § 6. Proof of the fundamental lemma § 7. Proof of the inductive lemma § 8. Proof of the inductive theorem § 9. Lemma on the non-degeneracy of diffeomorphisms § 10. Averaging over rapid variables § 11. Polar coordinates § 12. The applicability of the inductive theorem § 13. Passage to the limit § 14. Proof of the fundamental theoremChapter V. Technical lemmas § 1. Domains of type D § 2. Arithmetic lemmas § 3. Analytic lemmas § 4. Geometric lemmas § 5. Convergence lemmas § 6. NotationChapter VI. Appendix § 1. Integrable systems § 2. Unsolved problems § 3. Neighbourhood of an invariant manifold §4. Intermixing § 5. Smoothing techniquesReferences

Gibbs measures in ergodic theory

Abstract: In this paper we introduce the concept of a Gibbs measure, which generalizes the concept of an equilibrium Gibbs distribution in statistical physics. The new concept is important in the study of Anosov dynamical systems. By means of this concept we construct a wide class of invariant measures for dynamical systems of this kind and investigate the problem of the existence of an invariant measure consistent with a smooth structure on the manifold; we also study the behaviour under small random excitations as . The cases of discrete time and continuous time are treated separately.