On N. N. Bogolyubov's works in classical and quantum statistical mechanics

A review of N. N. Bogolyubov's works in classical and quantum statistical mechanics is presented.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 155–201, February, 1993.     

On a certain boundary value problem of N. P. Vekua with a piecewise-smooth shift on a piecewise Lyapunov contour

Odessa. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 2, pp. 108–115, March–April, 1992.     

On a problem of A. N. Kolmogorov

The paper is devoted to the derivation of the "possibly sharpest" estimates at the investigation of the deviation of the distribution function of the normalized sum of independent random variables from the distribution function of the normal law.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 289–319, 1990.     

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 a theorem of S. N. Bernshtein

This note contains the proof of a theorem on the characterization of a normal law on locally compact Abelian groups, this theorem being a generalization of S. N. Bernshtein's wellknown characterization of the normal law by the property that sums and differences of two independent random variables are also random variables.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 301–307, September, 1969.The author wishes to thank A. M. Vershik and the referee for a number of valuable comments.     

AC-Removability,Hausdorff dimension,and property (N)

Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 6, pp. 1325–1334, November–December, 1994.     

Quasi-Frobenius Algebras and Their Integrable N-Parameter Deformations Generated by Compatible N×N Metrics of Constant Riemannian Curvature

We prove that the equations describing compatible N×N metrics of constant Riemannian curvature define a special class of integrable N-parameter deformations of quasi-Frobenius (in general, noncommutative) algebras. We discuss connections with open–closed two-dimensional topological field theories, associativity equations, and Frobenius and quasi-Frobenius manifolds. We conjecture that open–closed two-dimensional topological field theories correspond to a special class of integrable deformations of associative quasi-Frobenius algebras.     

Peculiarities of Decomposition of the SU(N) Yang–Mills Field

Several aspects of extension of the Faddeev–Niemi decomposition for the SU(2) Yang–Mills field to the case of the SU(N) gauge group are discussed. Bibliography: 4 titles.     

Types of N=2 superconformal transformations

Khar'kov State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 86, No. 2, pp. 202–209, February, 1991.     

N. N. Bogolyubov's research in mathematics and theoretical physics

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1156–1164, September, 1989.     

A result of L. N. Bol'shev from the theory of the statistical testing of hypotheses

One gives a proof of L. N. Bol'shev's result on the existence of an optimal statistical test for the problem of the testing of two simple hypotheses under the condition that there exists the possibility to decline the taking of a decision.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 129–137, 1986.     

Constructive models of N1-categorical theories

An example of an N₁-categorical, but not N₀-categorical, theory, for which only a prime model is constructible, is constructed.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 885–888, June, 1978.     

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.     

Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

A. N. Whitehead (1861–1947) contributed notably to the foundations of pure and applied mathematics, especially from the late 1890s to the mid 1920s. An algebraist by mathematical tendency, he surveyed several algebras in his book Universal Algebra (1898). Then in the 1900s he joined Bertrand Russell in an attempt to ground many parts of mathematics in the newly developing mathematical logic. In this connection he published in 1906 a long paper on geometry, space and time, and matter. The main outcome of the collaboration was a three-volume work, Principia Mathematica (1910–1913): he was supposed to write a fourth volume on parts of geometries, but he abandoned it after much of it was done. By then his interests had switched to educational issues, and especially to space and time and relativity theory, where his earlier dependence upon logic was extended to an ontology of events and to a general notion of “process,” especially in human experience. These innovations led to somewhat revised conceptions of logic and of the philosophy of mathematics. © 2002 Elsevier Science (USA).A. N. Whitehead (1861–1947) contribuiu de forma marcante para os Fundamentos da Matemática Pura e Aplicada, especialmente entre o fim da década de 1890 e meados da década de 1920. Sendo um algebrista na sua vertente matemática, fez um levantamento de diversas álgebras no seu livro Universal Algebra (1898). Pouco depois de 1900 juntou-se a Bertrand Russell numa tentativa para basear várias partes da matemática sobre a lógica matemática, que se começava então a desenvolver. Nesse âmbito publicou em 1906 um longo artigo sobre geometria, espaço e tempo, e matéria. O principal resultado da colaboração foi um trabalho em três volumes, Principia Mathematica (1910–1913): estava previsto que Whitehead escrevesse um quarto volume sobre aspectos das geometrias, mas abandonou-o depois de uma boa parte já estar escrita. Por essa altura os seus interesses tinham-se voltado para questões educacionais; especialmente para o espaço e o tempo e para a teoria da relatividade, onde a sua anterior dependência da lógica se estendeu a uma ontologia de acontecimentos e a uma noção geral de “processo” especialmente na experiência humana. Estas inovações levaram a concepções um pouco revistas da lógica e da filosofia da matemática. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 00A30; 01A60; 03-03; 03A05.     

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.     

A question of L. A. Shemetkov

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

Monotonicity conditions for a sequence of generalized polynomials of S. N. Bernshtein and A. O. Gel'fond

It is shown that the sequence of generalized polynomials of S. N. Bernshtein and A. O. Gel'fond, which corresponds to a functionf(x), is an increasing or decreasing sequence depending on the concavity or convexity of the function. Analogous results are given in the case of functions of two variables.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 747–752, June, 1969.In conclusion, the author wishes to express his thanks to I. I. Ibragimov and A. D. Gadzhiev for their interest in his work.     

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.     

New examples of imbedded spherical soap bubbles inS N (1)

A. D. Alexandrov proved that the only imbedded soap bubbles in euclidean or hyperbolicn-space are round spheres. This also holds for ann-dimensional hemisphere. In this paper we give many new examples of differentiable (n–1)-spheres imbedded as soap bubbles in sphericaln-space. Thus we show Alexandrov's theorem is false for sphericaln-space even under the added assumption that the soap bubbles are differentiable (n–1)-spheres.The author would like to thank the referee for many helpful comments and corrections