Showing papers in "Mathematical Notes in 1995"

Homogeneous Riemannian manifolds of positive Ricci curvature

Abstract: We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π1(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber.

On complex structures on two-dimensional tori admitting metrics with nontrivial quadratic integral

Abstract: In this paper the set of complex structures on a torus admitting Riemannian metrics (consistent with these complex structures) with nontrivial quadratic integrals is described.

TheN−1-property of maps and Luzin's condition (N)

Abstract: A functionf∶G→ℝ n , whereG is an open set in ℝ n , has theN −1-property if for allE⊂ℝ n we have {¦E¦=0⇒¦f −1(E)¦=0} (¦·¦ is the Lebesgue measure). The article is concerned with the relations between theN −1-property of functions, the maximal rank of derivatives, and the differentiability almost everywhere of composite functions.

Singular integral operators on complicated contours and pseudodifferential operators

Abstract: We apply the pseudodifferential operator technique to the study of algebras of singular operators on complicated contours. This technique is used to construct a symbolic calculus for theC *-algebra generated by singular integral operators whose coefficients may have singularities of the second kind on complicated contours; the curves forming a node are not required to have a tangent at the node.

Finite groups with a given set of Schmidt subgroups

Abstract: We obtain a description of finite groups all of whose Schmidt subgroups are either supersolvable or not supersolvable. In particular, we show that these groups are always solvable.

Distribution of values of linear functions and asymptotic behavior of trajectories of some dynamical systems

Abstract: The asymptotic behavior of mean values for integrals of quasiperiodic functions, which characterizes the uniformity of the distribution of irrational windings on a torus, is shown to be essentially dependent on the dimension of the torus. We prove the nonrecurrence of mean values for arbitrarily smooth three-frequency quasiperiodic functions. We also present a series of results concerning the distribution of fractional parts for systems of linear functions.

Uniform estimates and stabilization of solutions to equations of one-dimensional motion of a multicomponent barotropic mixture

Abstract: Numerous papers deal with equations of motion of a barotropic gas (see [1]-[3] and the references therein). The asymptotic behavior as t --* +oo of the solutions as well as their existence and uniqueness, is of great interest, especially if (all or some of) the data are not assumed to be small. These topics were studied for the one-dimensional motion in [4-7] and in some other papers. Equations of motion of gas mixtures are more complicated and also play an important role. The unique solvability "in the large" was proved in [8] for the initial-boundary value problem for model equations describing a multicomponent mixture of viscous barotropic gases. The present paper contains some results concerning the asymptotic behavior of the solutions to a similar (but more general) system. These results have been obtained by combining the methods of [7] with the use of Lagrangian mass coordinates [8] for each component of the mixture. Note that attempts to solve this problem by using the methods in [4] have failed (even in the absence of mass forces). w In the half-strip H = (0, V) x R + , we study the system of quasilinear differential equations

Distributive semiprime rings

Abstract: It is proved that a right distributive semiprime PI ringA is a left distributive ring and for each elementx ∈A there is a positive integern such thatx n A=Ax n . We describe both right distributive right Noetherian rings algebraic over the center of the ring and right distributive left Noetherian PI rings. We also characterize rings all of whose Pierce stalks are right chain right Artin rings.

On trigonometric integrals

Abstract: Given a function composed of some trig functions, one generally must perform adhoc techniques. In the next two section we deal with some very specific cases that tend to cover a lot of integrals one encounters due to trigonometric substitution (a technique we have not yet learned). The next techniques will also inspire what things may be necessary. In general, converting all trigonometric function to sin’s and cos’s and breaking apart sums is not a terrible idea when confronted with a random integral. It may be easier, however, to view the problem in a different light (as is the case with integrals involving products of sec’s and tan’s).

Two-gap elliptic solutions to integrable nonlinear equations

Abstract: We study spectral surfaces associated with elliptic two-gap solutions to the nonlinear Schrodinger equation (NLS), the Korteweg-de Vries equation (KdV), and the sine-Gordon equation (SG). It is shown that elliptic solutions to the NLS and SG equations, as well as solutions to the KdV equation elliptic with respect tot, can be assigned to any hyperelliptic surface of genus 2 that forms a covering over an elliptic surface.

Factor-powers of finite symmetric groups

Abstract: To a transformation semigroup (S, M) we assign a new semigroupFP(S) called the factor-power of the semigroup (S, M). Then we apply this construction to the symmetric groupSn. Some combinatorial properties of the semigroupFP(Sn) are studied; in particular, we investigate its relationship with the semigroup of 2-stochastic matrices of ordern and the structure of its idempotents. The idempotents are used in characterizingFP(Sn) as an extremal subsemigroup of the semigroupBn of all binary relations of ann-element set and also in the proof of the fact thatFP(Sn) contains almost all elements ofBn.

Structure of subsemigroups of factor-powers of finite symmetric groups

Abstract: For the factor-powerFP(Sn) of the symmetric groupSn, we describe regular elements, maximal subgroups, isolated and fully isolated subsemigroups, and also maximal nilpotent subsemigroups whose zero elements coincide with the zero element of the semigroupFP(Sn).

Exact Solutions of G-Invariant Chiral Equations

Abstract: A method is suggested for solving the chiral equations (αg,zg −1),¯z+(αg,¯zg −1),z=0, whereg belongs to some Lie groupG. The solution is written out in terms of harmonic maps. The method can be used even for some infinite-dimensional Lie groups.