On the Lipschitz stability of complex systems of differential equations

Mathematical models of real physical processes and phenomena can be described by the so-called complex systems of differential equations. Among them are systems that decompose into interacting subsystems and intrasystemic connections. They have significant nonlinearities. In this connection the application of the existing methods of study of the dynamics of the behavior of such systems is quite difficult. We propose some approaches to the study of Lipschitz stability connected with the technique of applying Lyapunov functions. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 12–16.     

Constancy of upper-continuous two-valued mappings into the Sorgenfrey line

By using the Sierpiński continuum theorem, we prove that every upper-continuous two-valued mapping of a linearly connected space (or even a c-connected space, i.e., a space in which any two points can be connected by a continuum) into the Sorgenfrey line is necessarily constant. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1034–1039, August, 2007.     

Functions of the first Baire class with values in metrizable spaces

We show that every mapping of the first functional Lebesgue class that acts from a topological space into a separable metrizable space that is linearly connected and locally linearly connected belongs to the first Baire class. We prove that the uniform limit of functions of the first Baire class ƒ _n: X → Y belongs to the first Baire class if X is a topological space and Y is a metric space that is linearly connected and locally linearly connected. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 568–572, April, 2006.     

An example of a nonarcwise connected rim-countable locally connected continuum

An example of a rim-countable, locally connected, but nonarcwise connected continuum is constructed, which is important for the classification of locally connected continua. The point of interest in this result is that there are no metrizable locally connected continua with the above properties. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 659–666, May, 1999.     

Hyperspaces of nowhere topologically complete spaces

It is proved that ifX is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace 2 ^X of all nonempty compact subsets ofX is strongly universal in the class of all coanalytic spaces. Moreover, 2 ^X is homeomorphic to Π₂ ifX is a Baire space, and toQ∖Π₁ ifX contains a dense absoluteG _δ-setG ⊂X such that the intersectionG ∩U is connected for any open connectedU ⊂X. (Here Π₁, Π₁⊂X are the standard subsets of the Hilbert cubeQ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 35–51, July, 1997. Translated by O. V. Sipacheva     

Records in the Fα-scheme. I. Martingale properties

Connections between the records in various models is discussed. Martingale properties for some sequences of random variables connected with the record moments in the F^α-scheme are established. Bibliography: 24 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 19–36, 1993. Translated by A. Sudakov.     

Polynomiality criterion for entire functions of several complex variables

The “radial” polynomiality criterion for entire functions of several complex variables is proved. Translated fromMatermaticheskie Zametki, Vol. 66, No. 4, pp. 500–502, October, 1999.     

Connected extensions of metric spaces and fixed points of Banach contractions

Two theorems announced by the author are proved. The first theorem concerns the existence of connected metrizable extensions of metric spaces. The second theorem is a slightly paradoxical assertion about the validity of the fixed-point principle. Bibliography: 4 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 227–237.     

On orientable real algebraicM-surfaces

Here we study relations between homology classes determined by real points of a real algebraicM-surface. We prove new congruences involving the Euler characteristics of the connected components of the set of these real points. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–526, October, 1997. Translated by S. S. Anisov     

What information about a wave function is given by “measuring” it with the help of the tomographic method of recovering the Wigner distribution?

Mathematical problems connected with “measuring” a wave function by the tomographic method of recovering the Wigner distribution are considered. Inequalities showing what information is contained in a finite number of projections of the Wigner distribution are obtained. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 189–200.     

Real hypersurfaces of a complex space form

In this paper, we show that an n-dimensional connected non-compact Ricci soliton isometrically immersed in the flat complex space form ){(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}, with potential vector field of the Ricci soliton is the characteristic vector field of the real hypersurface is an Einstein manifold. We classify connected Hopf hypersurfaces in the flat complex space form (C ^{á ñ\fracn+12},J, á , ñ ){(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )} and also obtain a characterization for the Hopf hypersurfaces in (C^\fracn+12,J, á , ñ ) {(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle ) }.     

An approach to the zeta function of a finite factorgraph via the Markov topological chains

An approach to the zeta and the L-function of a finite connected graph via the theory of Markov topological chains (symbolic dynamics) is presented. The case of the L-function of a finite connected graph with labeling is considered. Bibliography: 12 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Published inZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 226–245. Translated by A. M. Nikitin.     

Abstract isomorphisms of quasiminimal groups

A connected solvable algebraic group is called minimal if its center is trivial and it has no proper connected subgroups with trivial center. We show that abstract isomorphisms of minimal groups defined over fields of characteristic zero are standard. Supported by RFFR grants Nos. 96-01-01678 and 96-01-01675, and by the State Committee for Higher Education of Russia, grant No. 2. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 567–586, September–October, 1996.     

Applied aspects of studying the roots of dispersion equations on nonprincipal sheets of a complex plane

In quantitatively estimating interference wave fields by methods of function theory, the initial integration path is subjected to deformation in the complex (ζ) plane. As a result of this deformation, sometimes the contour λ may go into other sheets of the Riemann (ζ) surface. This forces us to study the integrands and, in particular, the dispersion equation of the problem on the other sheets of the (ζ) plane. Similar questions are discussed for the case where a layer SH is in contact with two elastic half-spaces. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 62–90. Translated by T. N. Surkova.     

Minimality and unique ergodicity of homogeneous actions

Consider a connected Lie groupG, a lattice Γ inG, a connected subgroupH ofG, and the adjoint representation Ad ofG on its Lie algebra g. Suppose that Ad(H) splits into a semidirect product of a reductive subgroup and the unipotent radical. We prove that the minimality of the leftH-action onG/Γ then implies its unique ergodicity. Simultaneously, we suggest a reduction of the study of finite ergodic measures for an arbitrary action (G/Γ,H), where the subgroupH∈G is connected and Γ∈G is discrete, to the case of an Abelian subgroupH. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 293–301, August, 1999.     

On convergence to the poisson law for distributions of the sum of random variables connected into a chain

This paper deals with the triangular array of random variables where every sequence is connected into a stationary Markov chain. For the uniformly strongly mixing chain we find conditions under which the distributions of the sum converge to the Poisson law. A more general case of theL ^p-regular Markov chain is also considered. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 3, pp. 297–307, July–September, 1995. Translated by R. Lapinskas     

The role of solvable groups in quantization of Lie algebras

A wide class of quantum universal enveloping algebras uniquely corresponding to Hopf algebras H with spectrum Q(H) in the category of groups is defined. Such quantum algebras are the quantum groups of the simply connected solvable Lie groups P(H). Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 168–178 Translated by V. D. Lyakhovskii     

Primitive connected and additive theories of polygons

We study into monoids S the class of all S-polygons over which is primitive normal, primitive connected, or additive, that is, the monoids S the theory of any S-polygon over which is primitive normal, primitive connected, or additive. It is proved that the class of all S-polygons is primitive normal iff S is a linearly ordered monoid, and that it is primitive connected iff S is a group. It is pointed out that there exists no monoid S with an additive class of all S-polygons. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 300–313, May–June, 2006.     

Coherent states for a generalized oscillator with a finite-dimensional Hilbert space

A construction of oscillator-like systems connected with a given set of orthogonal polynomials and coherent states for such systems developed by the authors is extended to the case of systems with a finite-dimensional state space. As an example, we consider a generalized oscillator connected with Krawtchouk polynomials. Bibliography: 24 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 75–99.