On M. G. Krein's contribution to the moment problem

This is a survey of M. G. Krein' s papers devoted to the moment problem.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 46, Nos. 1–2, pp. 63–75, January–February, 1994.     

M. G. Krein's ideas in the theory of orthogonal polynomials

This is a survey of M. G. Krein's ideas in the theory of orthogonal polynomials.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 76–86, January–February, 1994.     

Works on functional equations. II

This is the second part of the bibliography published in Aequationes Math.1, 152–191 (1968). For the classification and other explanations and remarks see pp. 152–155 and 190–191 of the first part,loc. cit.     

On M. G. Krein's papers in the theory of spaces with an indefinite metric

This is a survey of the papers by M. G. Krein (and his disciples) devoted to the theory of operators in spaces with an indefinite metric and its applications.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 5–17, January–February, 1994.     

Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary-value problems for differential equations. The further development of these results is also considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 55–62, January–February, 1994.     

Note on: N.E. Aguilera, M.S. Escalante, G.L. Nasini, “The disjunctive procedure and blocker duality”

Aguilera et al. [Discrete Appl. Math. 121 (2002) 1–13] give a generalization of a theorem of Lehman through an extension of the disjunctive procedure defined by Balas, Ceria and Cornuéjols. This generalization can be formulated as(A) For every clutter , the disjunctive index of its set covering polyhedron coincides with the disjunctive index of the set covering polyhedron of its blocker, .In Aguilera et al. [Discrete Appl. Math. 121 (2002) 1–3], (A) is indeed a corollary of the stronger result(B) .Motivated by the work of Gerards et al. [Math. Oper. Res. 28 (2003) 884–885] we propose a simpler proof of (B) as well as an alternative proof of (A), independent of (B). Both of them are based on the relationship between the “disjunctive relaxations” obtained by and the set covering polyhedra associated with some particular minors of .     

On M. G. Krein's work in the theory of linear periodic Hamiltonian systems

We present basic ideas of the theory of linear periodic Hamiltonian systems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 128–144, January–February, 1994.     

Generalized z-Distributions and Related Stochastic Processes

The class of generalized z–distributions is defined and their properties are investigated. Ornstein–Uhlenbeck–type and self–similar generalized z–processes are constructed and described. Esscher transforms of the generalized z–processes and the mixed generalized z–processes are characterized. Finally, construction and some properties of generalized z–diffusions are also discussed.     

A New Way to Represent Links. One-Dimensional Formalism and Untangling Technology

An alternative link representation different from planar diagrams is discussed. Isotopy classes of unordered nonoriented links are realized as central elements of a monoid presented explicitly by a finite number of generators and relations. The group presented by two generators and three relations [[a,b],a ² ba ^–2]=[[a,b],b ² ab ^–2]=[[a,b],[a ^–1,b ^–1]]=1, where [x,y]=xyx ^–1 y ^–1, is proved to have a commutator subgroup isomorphic to the braid group on infinitely many strands. A new partial algorithm for unknot recognition is constructed. Experiments show that the algorithm allows the untangling of unknots whose planar diagram has hundreds of crossings. Here 'untangling' means 'finding an isotopy to the circle'.     

Problems of adhesive contact. A review

Questions of importance to the understanding of the adhesion mechanism are examined. Special attention is given to the relief of the substrate surface and processes associated with the penetration of the adhesive into surface depressions. The possibility of diffusion of macromolecules across the adhesive-substrate interface is considered; the importance of a little-studied factor — the diffusion of low-molecular substances — is demonstrated.V. I. Lenin All-Union Electrical Engineering Institute, Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 840–849, September–October, 1969.     

Light absorption and laser-crack development. 2

Conclusions 1. The gray color of a laser crack in PMMA (black in polystyrene) is caused by carbon-black particles that adhere to its walls.2. Further experimentation under an electron microscope is necessary to determine the size of the carbon-black particles. Making use of these new data and the results described here, we can define their absorption index more precisely.3. The absorption index of the gaseous medium between the walls of the crack in a layer 30–100 µ thick is 20 cm^–1; this is 1000 times smaller than that of graphite, but approximately 1000 times greater than that of the ordinary block polymer.4. As a result of photochemical and photothermal reactions, the laser source creates a gray (PMMA) or black (polystyrene) absorbent layer located in the crack and on its walls. Thus, the energy of the laser beam is absorbed due to "self-inductive" opacity. This leads to an increase in the mass of the "gas wedge," to a pressure increase, and ultimately, to crack propagation.For communication 1, see [1].Institute of the Problems of Mechanics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 409–412, May–June, 1977.     

Generalized semilattices and m-degrees of index sets. II

Translated from Algebra i Logika, Vol. 30, No. 2, pp. 168–180, March–April, 1991.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. IV

In this paper, we apply the theory developed in parts I-III [Ukr. Math. Zh.,46, No. 9, 1171–1188; No. 11, 1509–1526; No. 12, 1627–1646 (1994)] to some classes of problems. We consider linear systems in zero approximation and investigate the problem of invariance of integral manifolds under perturbations. Unlike nonlinear systems, linear ones have centralized systems, which are always decomposable. Moreover, restrictions connected with the impossibility of diagonalization of the coefficient matrix in zero approximation are removed. In conclusion, we apply the method of local asymptotic decomposition to some mechanical problems.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1044–1068, August, 1995.This research was partially supported by the International Science Foundation, grant No. UB2000.     

A question of L. A. Shemetkov

Translated fromAlgebra i Logika, Vol. 31, No. 6, pp. 624–636, November–December, 1992.     

To the N. N. Luzin theorem

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 1, pp. 212–215, January–February, 1992.     

Least plurisuperharmonic majorant and multipliers of entire functions. I

Ufa. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 1, pp. 173–178, January–February, 1992.     

Thermal deformation of unidirectional hybrid composites. 2

Conclusions A variant of the solution of the problem of the thermorheologically complex temperature strain of a hybrid composite containing viscoelastic thermorheologically simple components with differing functions for temperature-time reduction in addition to elastic components, is proposed. An experimental study is conducted on unidirectional specimens of organic- and glass-fiber-reinforced plastic, organic- and carbon-fiber-reinforced plastic, and carbon- and glass-fiber-reinforced plastic at a constant rate of temperature change in the 20–150 °C range. Satisfactory correspondence is obtained between predicted and experimental data.For previous communication, see [1].Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 969–979, November–December, 1989.     

Toughness and the existence ofk-factors. II

In a paper with the same title [3], we proved Chvátal's conjecture thatk-tough graphs havek-factors if they satisfy trivial necessary conditions. In this paper, we prove the following stronger result: Suppose|V(G)| k + 1,k |V(G)| even, and|S| k w(G – S) – 7/8k ifw(G – S) 2, wherew(G – S) is the number of connected components ofG – S. ThenG has ak-factor.     

Ganzgeschlossene und Prädikatengeschlossene Logiken. II

Ohne ZusammenfassungEs handelt sich um eine Fortsetzung von Teil I in Heft 14/1–2, pp. 24–37 dieser Zeitschrift.