Showing papers in "Russian Mathematical Surveys in 1998"

Quantum coding theorems

Abstract: ContentsI. IntroductionII. General considerations § 1. Quantum communication channel § 2. Entropy bound and channel capacity § 3. Formulation of the quantum coding theorem. Weak conversionIII. Proof of the direct statement of the coding theorem § 1. Channels with pure signal states § 2. Reliability function § 3. Quantum binary channel § 4. Case of arbitrary states with bounded entropyIV. c-q channels with input constraints § 1. Coding theorem § 2. Gauss channel with one degree of freedom § 3. Classical signal on quantum background noise Bibliography

Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems

Abstract: ContentsIntroduction Chapter I. Differential geometry of symplectic structures on loop spaces of smooth manifolds § 1.1. Symplectic and Poisson structures on loop spaces of smooth manifolds. Basic definitions § 1.2. Homogeneous symplectic structures of the first order on loop spaces of pseudo-Riemannian manifolds and two-dimensional non-linear sigma-models with torsion § 1.3. Symplectic and Poisson structures of degenerate Lagrangian systems § 1.4. Homogeneous symplectic structures of the second order on loop spaces of almost symplectic and symplectic manifolds and symplectic connections § 1.5. Complexes of homogeneous forms on loop spaces of smooth manifolds and their cohomology groupsChapter II. Local and non-local Poisson structures of differential-geometric type § 2.1. Riemannian geometry of multidimensional local Poisson structures of hydrodynamic type § 2.2. Hamiltonian systems of hydrodynamic type and metrics of constant Riemannian curvature § 2.3. Non-homogeneous Hamiltonian systems of hydrodynamic type § 2.4. Killing-Poisson bivectors on spaces of constant Riemannian curvature and bi-Hamiltonian structure of the generalized Heisenberg magnets § 2.5. Homogeneous Poisson structures of differential-geometric typeChapter III. The equations of associativity in two-dimensional topological field theory and non-diagonalizable integrable systems of hydrodynamic type § 3.1. Equations of associativity as non-diagonalizable integrable homogeneous systems of hydrodynamic type § 3.2. Poisson and symplectic structures of the equations of associativity § 3.3. Theorem on a canonical Hamiltonian representation of the restriction of an arbitrary evolution system to the set of stationary points of its non-degenerate integral and its applications to the equations of associativity and systems of hydrodynamic type Bibliography

Thermodynamic formalism for countable symbolic Markov chains

Abstract: ContentsIntroductionChapter I. Variational principles and a classification of functions defined on a Markov set §1. Main definitions. The variational principle for topological pressure §2. Generating functions. Non-negative matrices §3. Criteria for stable positiveness §4. Equilibrium and Gibbs measures. Gibbs' variational principleChapter II. Two classes of invariant measures related to symbolic Markov chains. The method of subdifferentials §5. The subdifferential of topological pressure §6. Asymptotically equilibrium measures. Convergence of equilibrium measures corresponding to finite subchains of a countable symbolic Markov chain §7. Asymptotics of discrete invariant measuresChapter III. Dynamic zeta functions related to symbolic Markov chains §8. The factorization theorem and some corollaries §9. Meromorphic continuation of zeta functions and the behaviour of discrete invariant measures §10. The zeta function and spectral properties of non-negativematricesChapter IV. Some special classes of symbolic Markov chains §11. Symbolic Markov chains with a regular set of periodic orbits §12. Symbolic Markov chains with finite cycle-passage domain Bibliography

On teaching mathematics

Abstract: Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense. In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy’s warning that ugly mathematics has no permanent place under the Sun). Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was the universal hate towards mathematicians both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users. The ugly building, built by undereducated mathematicians who were exhausted by their inferiority complex and who were unable to make themselves familiar with physics, reminds one of the rigorous axiomatic theory of odd numbers. Obviously, it is possible to create such a theory and make pupils admire the perfection and internal consistency of the resulting structure (in which, for example, the sum of an odd number of terms and the product of any number of factors are defined). From this sectarian point of view, even numbers could either be declared a heresy or, with passage of time, be introduced into the theory supplemented with a few “ideal” objects (in order to comply with the needs of physics and the real world). Unfortunately, it was an ugly twisted construction of mathematics like the one above which predominated in the teaching of mathematics for decades. Having originated in France, this pervertedness quickly spread to teaching

Hyperidentities in algebras and varieties

Abstract: ContentsIntroduction ??1. The concept of hyperidentity. Hyperidentities of binary representations1.1. The concept of hyperidentity: examples1.2. Binary Cayley theorems for semigroups, idempotent and commutative semigroups, groups and the multiplicative groups of fields1.3. Stochastic algebras. The concept of coidentity ??2. The category of algebras of the same arithmetic type2.1. Homomorphisms and automorphisms of algebras of the same arithmetic type2.2. Hypervarieties of algebras ??3. The category of systems of the same arithmetic type. Nonclassical semantics in second order language ??4. Non-trivial associative and distributive hyperidentities in algebras that are invertible or close to being invertible Bibliography

Theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems

Abstract: ContentsIntroduction § 1. The concept of a set of determining functionals § 2. The completeness defect of a set of functionals § 3. Estimates for the completeness defect in Sobolev spaces § 4. Determining functionals for semilinear parabolic equations § 5. Determining functionals for equations of second order in time § 6. On boundary determining functionals Bibliography

Toric manifolds and complex cobordisms

Abstract: Algebraic topology of manifolds defined by simple polytopes V M Bukhshtaber and T E Panov Torus actions, combinatorial topology, and homological algebra V M Buchstaber and T E Panov Hirzebruch genera of manifolds with torus action T E Panov The index of an equivariant vector field and addition theorems for Pontryagin classes V M Bukhshtaber and K E Feldman Embedding of compacta, stable homotopy groups of spheres, and singularity theory P M Akhmet'ev Complex cobordism and formal groups Viktor M Buchstaber Massey Products and the Bukhshtaber spectral sequence I V Artel'nykh Toric manifolds and complex cobordisms

The universal Urysohn space, Gromov metric triples and random metrics on the natural numbers

Abstract: ContentsIntroduction1. The universal Urysohn space2. Gromov metric triples3. Random metrics on the natural numbers and the Urysohn space Bibliography

Serre's quantum problem

Abstract: Contents ??1. Introduction ??2. Skew polynomials ??3. Crossed products ??4. The functors K1 and K2 ??5. Projective modules over quantum polynomials ??6. Concluding remarks Bibliography

Vector-valued Hausdorff-Young inequality and applications

Abstract: Contents §1. Introduction §2. General definitions and auxiliary results §3. Interpolation spaces §4. Interpolation and the Fourier type of Banach spaces §5. The Rademacher type and cotype §6. The Fourier type of Banach spaces with respect to groups §7. Types and cotypes of Banach spaces with respect to general orthonormal systems §8. Relations between types (cotypes) with respect to the Rademacher system and other orthonormal systems §9. Theorems of Hardy and Paley for vector-valued functions §10. Estimates for Fourier transforms of smooth functions Bibliography

Controlled random sequences: methods of convex analysis and problems with functional constraints

Abstract: ContentsIntroduction § 1. Controlled random sequences: main definitions and traditional approaches § 1.1. Description of the mathematical model § 1.2. Models with integral functionals § 1.3. Homogeneous Markov decision processes with average cost criteria § 2. Application of methods of convex analysis § 2.1. Properties of the space § 2.2. Existence of optimal policies § 2.3. Sufficiency of selectors § 2.4. Preliminary results. The notion of an occupation measure § 2.5. Markov decision processes with total cost criteria and occupation measures § 2.6. Discounted costs and the corresponding occupation measures § 2.7. Average costs and ergodic occupation measures § 3. Problems with functional constraints § 3.1. General results § 3.2. Preliminary conclusions § 3.3. Markov decision processes with total cost criteria § 3.4. Homogeneous Markov decision processes with discounting § 3.5. Homogeneous Markov decision processes with average cost criteria § 3.6. Other constrained problems, related topics, and future prospectsConclusionAppendix. Elements of convex analysis and measure theory Bibliography

General gamma functions, exponentials, and hypergeometric functions

Abstract: Contents § 0. Introduction § 1. General gamma functions § 2. Series of exponential type § 3. The operators of u-differentiation and u-exponentials § 4. Many-dimensional analogues § 5. Series of hypergeometric type § 6. General u-hypergeometric systems Bibliography

Arithmetic of dimension theory

Abstract: ContentsIntroductionChapter I. Algebra § 1. Tensor classification of Abelian groups § 2. Kunneth algebras of polygroups § 3. The code latticeChapter II. Topology § 4. Theory of extensional dimension § 5. Criterion for extensivity § 6. Dimension and connectivityChapter III. Geometry § 7. Theorems on stable intersections § 8. Upper bounds for the dimension of an intersection § 9. Lower bounds for the dimension of an intersection Bibliography

On compatible potential deformations of Frobenius algebras and associativity equations

Abstract: Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems O I Mokhov Non-local Hamiltonian operators of hydrodynamic type with flat metrics, and the associativity equations O I Mokhov Singular spectral curves in finite-gap integration Iskander A Taimanov Quantum deformations of associative algebras and integrable systems B G Konopelchenko Classical r-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics Baej M Szablikowski On compatible Poisson structures of hydrodynamic type O I Mokhov Discrete integrable systems and deformations of associative algebras B G Konopelchenko On compatible potential deformations of Frobenius algebras and associativity equations

On a theorem of Poincare

Abstract: where w is any member of Ω. One says that Ω is contractible to a point iff there is a mapping H carrying Ω̂ to Ω and there is a member ω of Ω such that: (H · J0)(w) = H(0, w) = ω, (H · J1)(w) = H(1, w) = w where w is any member of Ω. One refers to H as a contraction mapping for Ω with contraction constant ω. 3◦ Now let ν be a k+1 form on Ω̂. Let K(ν) be the k form on Ω defined by the following rules. To make the rules legible, we adopt the following notation. Let A be any subset of the set {1, 2, 3, . . . , n} having members. We may display the members of A in order as follows: A : j1 < j2 < · · · j Let dw denote the form on R defined as follows: dw = dw1dw2 · · · dw