Bicompact manifolds

The paper is an introduction to the theory of bitopological manifolds. It contains the basic definitions and assertions of the theory and also some examples of bitopological manifolds.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 26–68, 1985.     

Problems of the theory of bitopological spaces. 2

This paper is the second part of a survey of recent results and new problems in the theory of bitopological spaces. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 5–67. Translated by A. A. Ivanov.     

The notion of exactness of bitopological extensions

The notion of exactness of topological extensions is generalized to bitopological spaces, and some bitopological notions of similar type are considered. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 242, 1997, pp. 230–236. Translated by N. Yu. Netsvetaev.     

Bitopologies on products and ratios

The purpose of the paper is to extend the notions of splitting and jointly continuous topologies on function spaces to products of spaces, to present a new theory of duality between topologies on function spaces and topologies on products of spaces, and to generalize these theories to bitopological spaces. The author starts from the definitions well known in the literature. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 242, 1997, pp. 217–229. Translated by N. Yu. Netsvetaev.     

Geometry of Poisson brackets and methods of Liouville integration of systems on symmetric spaces

A survey is given of various methods of integrating Hamiltonian systems on symmetric spaces and Lie algebras, and an analysis is also presented of applications of the technique developed to some mechanical problems. Some questions of nonintegrability of Hamiltonian systems on symplectic manifolds are considered, and, in particular, the Morse theory of completely integrable Hamiltonian systems is expounded.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 29, pp. 3–108, 1986.     

Space Structures: Theory and Applications

This is the first part of the book Space Structures: Theory and Applications, which provides an introduction to general topology. It consists of six chapters: structures and categories, structures of metric type, continuity structures, extension stuctures, classical bitopological structures, and the general theory of bitopological spaces.     

Certain properties of bitopological spaces

Some connections and interrelations between T_i-separation axioms (i=1,2,3) for bitopological spaces are considered. In particular, four different versions of the definition of the pairwise Hausdorff separation axion are discussed. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 135–140. Translated by O. A. Ivanov.     

Totally Bounded Quasi-Uniformities and Pseudocompactness of Bispaces

 We characterize pairwise Tychonoff bispaces that admit only totally bounded quasi-uniformities in terms of a suitable notion of bitopological pseudocompactness. We also show that a pairwise Tychonoff bispace has a unique (up to equivalence) bicompactification if and only if it admits a unique totally bounded quasi-unifomity. These results extend classical theorems of R. Doss for uniform spaces to the quasi-uniform (bitopological) setting, and are applied to the study of T ₀ topological spaces that admit a unique quasi-uniformity and a unique bicompactification, respectively. Finally, we discuss the problem of extending the classical Glicksberg theorem on product of pseudocompact spaces to bispaces and a partial solution is obtained. Supported by the Spanish Ministry of Science and Technology, grant BFM2000-1111. Supported by a grant from Generalitat Valenciana. Received November 7, 2001; in revised form August 14, 2002     

On some properties of bitopological QHC spaces

Some important questions connected with bitopological QHC spaces are investigated. New conditions are found, under which such spaces are compact with respect to one component of the topology. It is shown that a pairwise extremal disconnected bitopological QHC space is S-closed in the sense of [4]. Theorems on the second category of a base set and on the almost Baire property of bitopological QHC spaces are proved. Also, several properties of QHC bitopological spaces are found under some known bitopological mappings. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 186–192, April–June, 2006.     

Singularities of Solution Surfaces for Quasilinear First-Order Partial Differential Equations

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.     

Bitopological spaces

In this paper we recount some initial concepts and results of the general theory of bitopological spaces, understood as pairs (X, ), where is a topological structure on X×X.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 30–55, 1982.     

Bibliography on bitopological spaces

A bibliography on bitopological spaces.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 63–78, 1988.     

Category of topological jet manifolds and certain applications in the theory of nonlinear infinite-dimensional dynamical systems

A new category of topological jet manifolds is proposed for the purpose of investigating exact finite-dimensional approximations of nonlinear dynamical systems on infinite-dimensional functional manifolds. Differential geometry structures on these manifolds and their applications to the theory of integrability in quadratures of nonlinear dynamical Lax-type systems are studied.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1242–1256, September, 1992.     

The RAGE theorem in Banach spaces

A generalization of the so called RAGE theorems from scattering theory in Hilbėrt spaces to a more general class of uniformly boundedC ₀-semigroups on Banach spaces is given. The extension is based on the notion of weak almost periodicity (in the sense of Eberlein). This research is based on parts of Chapter 3 of the author’s dissertation at Essen University. The work of the author was supported by the Deutsche Forschungsgemeinschaft (DFG).     

Problems of the theory of bitopological spaces

This paper gives a survey of and problems in the study of bitopological spaces.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 5–62, 1988.     

Bitopological and topological ordered k-spaces

Domain theory, in theoretical computer science, needs to be able to handle function spaces easily. It also requires asymmetric spaces, and these are necessarily not T₁. At the same time, techniques used with the higher separation axioms are useful there (see [Topology Appl. 199 (2002) 241]). In order to handle all these requirements, we develop a theory of k-bispaces using bitopological spaces, which results in a Cartesian closed category. The other well-known way to combine asymmetry and separation is ordered topological spaces [Nachbin, Topology and Order, Van Nostrand, 1965]; we define the category of ordered k-spaces, which is isomorphic to that found among bitopological spaces.     

Pairwise weakly and pairwise strongly irresolute functions

In this paper we consider a new weak and strong forms of irresolute functions in bitopological spaces, namely, ij-quasi-irresolute functions and strongly irresolute functions. Several characterizations and basic properties of these functions are given. We investigate the relationships among some weak forms of continuity and other generalizations of continuous mappings in bitopological spaces.     

The main problems of the multiple coverage method of a common depth point and data processing for this method

A flaw of fundamental nature in foundations and pratical applications of efficient seismic prospecting is discussed. The authors invite the geophysical community to organize a broad discussion in the seismic literature devoted to basic problems of seismic prospecting and its applications. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1997, pp. 240–262. Translated by N. S. Zabavnikova and T. N. Surkova     

Boundary control and quasiphotons in the problem of reconstruction of a Riemannian manifold via dynamic data

The problem of reconstruction of a Riemannian manifold via a nonstationary response operator is considered. An approach to solving this problem which is based on methods of boundary control theory is suggested. The procedure of reconstruction makes use of nonstationary Gaussian beams (quasiphotons). Locality is a significant feature of the procedure. A class of manifolds which can be reconstructed within the framework of this approach is described in terms of boundary control theory. For example, this class includes all analytic manifolds. Bibliography: 21 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 21–50. Translated by A. P. Katchalov.     

Solutions of the Yang-Baxter equation

We give the basic definitions connected with the Yang-Baxter equation (factorization condition for a multiparticle S-matrix) and formulate the problem of classifying its solutions. We list the known methods of solution of the Y-B equation, and also various applications of this equation to the theory of completely integrable quantum and classical systems. A generalization of the Y-B equation to the case ofZ ₂-graduation is obtained, a possible connection with the theory of representations is noted. The supplement contains about 20 explicit solutions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 95, pp. 129–160, 1980.