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 Doklady MathematicsJournal Prestige (SJR): 0.427 Number of Followers: 0      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1531-8362 - ISSN (Online) 1064-5624 Published by Springer-Verlag  [2574 journals]
• On Bilinear Hardy–Steklov Operators
• Abstract: Sharp estimates of the best constants in bilinear weighted inequalities with Hardy–Steklov integral operators are presented.
PubDate: 2018-11-01

• On Sets of Measurable Operators Convex and Closed in Topology of
Convergence in Measure
• Abstract: For a von Neumann algebra with a faithful normal semifinite trace, the properties of operator “intervals” of three types for operators measurable with respect to the trace are investigated. The first two operator intervals are convex and closed in the topology of convergence in measure, while the third operator interval is convex for all nonnegative operators if and only if the von Neumann algebra is Abelian. A sufficient condition for the operator intervals of the second and third types not to be compact in the topology of convergence in measure is found. For the algebra of all linear bounded operators in a Hilbert space, the operator intervals of the second and third types cannot be compact in the norm topology. A nonnegative operator is compact if and only if its operator interval of the first type is compact in the norm topology. New operator inequalities are proved. Applications to Schatten–von Neumann ideals are obtained. Two examples are considered.
PubDate: 2018-11-01

• Two-Dimensional (Two-Speed) Mode for a Nonstrictly Hyperbolic Problem
• Abstract: All types of solutions to the Riemann problem for a system of conservation laws that is nonstrictly hyperbolic in the sense of Petrovskii, but not in the sense of Friedrichs are described.
PubDate: 2018-11-01

• Axiomatizing Provable n -Provability
• Abstract: The set of all formulas whose n-provability in a given arithmetical theory S is provable in another arithmetical theory T is a recursively enumerable extension of S. We prove that such extensions can be naturally axiomatized in terms of transfinite progressions of iterated local reflection schemata over S. Specifically, the set of all provably 1-provable sentences in Peano arithmetic PA can be axiomatized by an ε0-times iterated local reflection schema over PA. The resulting characterizations provide additional information on the proof-theoretic strength of these theories and on the complexity of their axiomatization.
PubDate: 2018-11-01

• Generalized Maximum Principle in Optimal Control
• Abstract: The concept of a local infimum for an optimal control problem is introduced, and necessary conditions for it are formulated in the form of a family of “maximum principles.” If the infimum coincides with a strong minimum, then this family contains the classical Pontryagin maximum principle. Examples are given to show that the obtained necessary conditions strengthen and generalize previously known results.
PubDate: 2018-11-01

• Arithmetic Properties of Generalized Hypergeometric F -Series
• Abstract: A generalization of the Siegel–Shidlovskii method in the theory of transcendental numbers is used to prove the infinite algebraic independence of elements (generated by generalized hypergeometric series) of direct products of fields $$\mathbb{K}_v$$ , which are completions of an algebraic number field $$\mathbb{K}$$ of finite degree over the field of rational numbers with respect to valuations v of $$\mathbb{K}$$ extending p-adic valuations of the field ℚ over all primes p, except for a finite number of them.
PubDate: 2018-11-01

• Dualization Problem over the Product of Chains: Asymptotic Estimates for
the Number of Solutions
• Abstract: A key intractable problem in logical data analysis, namely, dualization over the product of partial orders, is considered. The important special case where each order is a chain is studied. If the cardinality of each chain is equal to two, then the considered problem is to construct a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form, which is equivalent to the enumeration of irreducible coverings of a Boolean matrix. The asymptotics of the typical number of irreducible coverings is known in the case where the number of rows in the Boolean matrix has a lower order of growth than the number of columns. In this paper, a similar result is obtained for dualization over the product of chains when the cardinality of each chain is higher than two. Deriving such asymptotic estimates is a technically complicated task, and they are required, in particular, for proving the existence of asymptotically optimal algorithms for the problem of monotone dualization and its generalizations.
PubDate: 2018-11-01

• A New Discrete Analogue of Pontryagin’s Maximum Principle
• Abstract: By introducing the concept of a γ-convex set, a new discrete analogue of Pontryagin’s maximum principle is obtained. By generalizing the concept of the relative interior of a set, an equality-type optimality condition is proved, which is called by the authors the Pontryagin equation.
PubDate: 2018-11-01

• Approximations in the Problem of Level Crossing by a Compound Renewal
Process
• Abstract: The classical problem of level crossing by a compound renewal process is considered, which has been extensively studied and has various applications. For the distribution of the first level crossing time, a new approximation is proposed, which is valid under minimal conditions and is obtained by applying a new method. It has a number of advantages over previously known approximations.
PubDate: 2018-11-01

• Blow-up of Solutions of the Cauchy Problem for a Nonlinear
Schrödinger Evolution Equation
• Abstract: The solution of the Cauchy problem for a nonlinear Schrödinger evolution equation with certain initial data is proved to blow up in a finite time, which is estimated from above. Additionally, lower bounds for the blow-up rate are obtained in some norms.
PubDate: 2018-11-01

• Uniqueness of Solution to the First Initial Boundary Value Problem for
Parabolic Systems on the Plane in a Model Case
• Abstract: The first initial boundary value problem for a one-dimensional (in x) Petrovskii parabolic second-order system with constant coefficients in a semibounded (in x) domain with a nonsmooth lateral boundary is proved to have a unique classical solution in certain Hölder classes.
PubDate: 2018-11-01

• A Disproof the Le Bars Conjecture about the Zero–One Law for Existential
• Abstract: The Le Bars conjecture (2001) states that the binomial random graph G(n, $$\frac{1}{2}$$ ) obeys the zero–one law for existential monadic sentences with two first-order variables. This conjecture is disproved. Moreover, it is proved that there exists an existential monadic sentence with a single monadic variable and two first-order variables whose truth probability does not converge.
PubDate: 2018-11-01

• Narrow Positively Graded Lie Algebras
• Abstract: The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n1 and n2 from our list, that are positive parts of the affine Kac–Moody algebras A1(1) and A2(2), respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2].
PubDate: 2018-11-01

• Types of Root Systems in Number Fields
• Abstract: We classify the types of root systems R in the rings of integers of number fields K such that the Weyl group W(R) lies in the group generated by Aut(K) and multiplications by the elements of K*.
PubDate: 2018-11-01

• Modeling of Phase Transitions in Weakly Bound Molecular Clusters
• Abstract: The method of molecular dynamics was used for modeling the isomerization of a hydrogen bonding network in small water clusters (hexamer and octamer). The collective modes of the particles moving in the clusters were determined by applying principal component analysis. An entropy criterion for phase transitions in water clusters was suggested. This criterion can be used to study phase transitions in weakly bound atomic and molecular clusters.
PubDate: 2018-11-01

• Soft Randomized Machine Learning
• Abstract: A new method for entropy-randomized machine learning is proposed based on empirical risk minimization instead of the exact fulfillment of empirical balance conditions. The corresponding machine learning algorithm is shown to generate a family of exponential distributions, and their structure is found.
PubDate: 2018-11-01

• On First Integrals of Linear Hamiltonian Systems
• Abstract: In the paper, Lax pairs for linear Hamiltonian systems of differential equations are found. Besides, first integrals of the system which are obtained from the Lax pairs are investigated.
PubDate: 2018-11-01

• Application of the Variational Method for Solving Inverse Problems of
Optimal Control
• Abstract: For optimal control problems, a new approach based on the search for an extremum of a special functional is proposed. The differential problem is reformulated as an ill-posed variational inverse problem. Taking into account ill-posedness leads to a stable numerical minimization procedure. The method developed has a high degree of generality, since it allows one to find special controls. Several examples of interest concerning the solution of classical optimal control problems are considered.
PubDate: 2018-11-01

• The Spectrum and Trace Formula for Bounded Perturbations of Differential
Operators
• Abstract: Spectrum properties and a method for deriving a regularized trace formula for perturbations of operators with discrete spectra in a separable Hilbert space are studied. A trace formula for a local perturbation of a two-dimensional harmonic oscillator in a strip is obtained based on this method.
PubDate: 2018-11-01

• On Unordered Multiplicative Partitions
• Abstract: In this paper, we find new identities and asymptotic formulas in the problem of multiplicative partitions, which is an analogue of the multidimensional problem of Dirichlet divisors with the condition of disorder of the factors.
PubDate: 2018-11-01

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