The curvature of the Sasaki metric of tangent sphere bundles

We study the sectional curvaturesK of the Sasaki metric of tangent sphere bundles over spaces of constant curvatureK(T ₁(M ⁿ, K)). We give precise bounds on the variation of the Ricci curvature and a bound on the scalar curvature ofT ₁ (M ⁿ, K) that is uniform onK. In an appendix we calculate and give lower bounds for the lengths of closed geodesics onT ₁ S ⁿ. titles.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 132–145.     

A system of basis invariants of the group Bn

In 1968 Flatto announced the conjecture that the polynomials _±(±x ₁±x ₂±±x _n )^2s ,s=1,n, are algebraically independent. This conjecture was confirmed by Haeuslein. A new proof of this result is given.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 54–55.     

A geometric interpretation of stationary-wave type solutions of the sine-Gordon equation

Using the connection of solutions of the sine-Gordon equationz _xy =sinz with the construction of a grid of asymptotic lines on surfaces of constant negative curvature inE ³ we show that those surfaces and only those that have curvature –1 and any two asymptotic lines of a single family of which are congruent correspond to solutions of stationary-wave typez=z(x+by).Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 81–87.     

Maximally mobile Davies metric spaces

The spaces whose linear elements are given by a homogeneous function of degree two with respect to the direction arguments are studied. The maximal order of the group of motionsG _r (r=n(n–1)/2+2) in Davies spaces is determined. A tensor characterization of maximally mobile spaces is given.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 47–54.     

On the geodesic curvature of an integral curve of a two-dimensional pfaffian manifold inE 4

The concept of generalized geodesic curvature is introduced along with the invariant mean geodesic curvature vector for a two-dimensional Pfaffian manifold inE ₄, and the indicatrix of generalized geodesic curvature of integral curves of this manifold is constructed.It is proved that the ratio of the fourth quadratic form of the manifold to its first fundamental form is equal to the square of the modulus of the generalized geodesic curvature vector, and an analog of a theorem of Euler is deduced.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 22–26.     

The limit distribution for the emptying times of a single-channel queueing system with Markov arrival stream admitting consolidation of states

We study the limit behavior of the emptying times of anM _r /G/1/n single-channel queueing system under heavy load conditions. It is assumed that the arrival stream is governed by a Markov chain admitting state consolidation.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 55–61, 1986.     

On rigidity and analytic inflexibility of surfaces of negative Gaussian curvature under a given direction of displacement of boundary points

We study the infinitesimal bendings of first and second order of regular (classC ²) surfaces of negative Gaussian curvature, which are bounded either a) by four asymptotic linesg ₁,g ₂,g ₁ , andg ₂ , or b) by two asymptotic linesg ₁ andg ₂ and a lineg which nowhere has asymptotic directions, under the assumption that connections are imposed on the surface permitting displacement of the points of the linesg ₁ andg ₂ (in the first case) and the lineg (in the second) in a constant direction.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 82–89.     

An estimate for the region of regularity of solutions of certain nonlinear differential inequalities

In the diskx ²+y ²R ² of thex, y-plane we consider the differential inequalityz _xxz_yy–z _xy ² –(1+z _x ^/2 +z _y ^/2 )^k, where the constants >0 andk>1. In the case =1 andk=2 this inequality means that the surfacez(x, y) has Gaussian curvatureK1. Efimov has shown that in this case the radius of the disk has an upper bound. In the present article we establish an analogous upper bound for the radiusR of the disk in which the functionz(x, y) satisfies the differential inequality above.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 19–21.     

On immersions ofn-dimensional Lobachevskií space in 2n-dimensional Euclidean space withn principal direction fields

In this article we prove that the principal direction fields are holonomic and use them to introduce curvilinear coordinates on an immersed region in terms of which the linear element of Lobachevskií space is written in the form where The fundamental system of equations is established for an immersion ofn-dimensional Lobachevskií space into 2n-dimensional Euclidean space withn principal direction fields for the functions _i and, and a way of constructing an arbitrary local analytic immersion is shown.Translated from Ukrainskií Geometricheskií Sbornik, No. 28, pp. 3–8, 1985.     

On the volume of certain manifolds with closed geodesics

It is proved that the ratio of the volume of a Riemannian manifold of cohomology typeCP ² orCP ³ for which all geodesics are closed and of length 2 to the volume of the standard sphere assumes a fixed value.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 102–106     

On the reductibility of the general existence problem for closed convex surfaces to Miranda's equation

It is proved that the solution of the general existence problem for closed convex surfaces with prescribed local propertiesf(R ₁ R ₂,R ₁+R ₂,n)=(n) can be obtained as the solution of Miranda's equationR ₁ R ₂+(f)+cn=((n),(n)) with right-hand side depending on the unknown surface under the hypothesis that the latter satisfies the closure condition , where is the unit sphere andd is its element of area.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 103–112.     

Yuan's alternative theorem and the maximization of the minimum eigenvalue function

LetA ₁ andA ₂ be two symmetric matrices of ordern×n. According to Yuan, there exists a convex combination of these matrices which is positive semidefinite, if and only if the functionxR ⁿ max {x ^T A ₁ x,x ^T A ₂ x} is nonnegative. We study the case in which more than two matrices are involved. We study also a related question concerning the maximization of the minimum eigenvalue of a convex combination of symmetric matrices.This research was partially supported by Dirección General de Investigación Científica y Técnica (DGICYT) under Project PB92-0615.     

A generating process for the Trotter product of stochastic semigroups

We find the form of the generating processW _t for the Trotter productX _s ^t Z _s ^t of stochastic semigroups:W _t =Y _t +V _t +[Y, V] _t , whereY _t andV _t are generating processes of the semigroupsX _s ^t andZ _s ^t respectively and [Y, V] _t is their mutual quadratic variation.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 94–96, 1986.     

The Gauss-Bonnet theorem for the torsion-free hyperbolic connection space τ 3 0

We consider the torsion-free hyperbolic connection space ₃ ⁰ . In this space we introduce the concept of Lobachevskií parallelism. Using Lobachevskií parallelism we define the geodesic curvature of a curve, find a formula for computing it, and prove a Gauss-Bonnet theorem for the space in question.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 96–102.     

The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

Special conformal mappings in the general theory of relativity

Some special conformal mappings of relativistic spaces are studied. The amount of arbitrariness with which one can describe a nontrivial special conformai mapping is determined. Necessary and sufficient conditions are found under which a space-timeV admits a special conformal mapping to space-time ¯V in which the metric tensor of bothV and ¯V satisfy the Einstein equations with the energy-momentum tensor of an ideal fluid having a geodesic velocity field.It is proved that such spaces are spaces of standard cosmological models.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 43–50.     

Hausdorff separation in categories

Considering subobjects, points and a closure operator in an abstract category, we introduce a generalization of the Hausdorff separation axiom for topological spaces: the notion ofT ₂-object. We discuss the properties ofT ₂-objects, which depend essentially on the behaviour of points, and finally we relate them to the well-known separated objects.The results of this paper are essentially taken from the author's Ph. D. Thesis written under the supervision of Professors M. Sobral and W. Tholen and partially supported by a scholarship of I.N.I.C.-Instituto Nacional de Investigação Científica.     

Connections on tangent bundles of higher order associated to regular Lagrangians

Given a regular Lagrangian L of order k on M we show that there exists a canonical connection _L on T ^2k–1 M whose paths are the solutions of the Euler-Lagrange equations for L. If L is the kinetic energy defined by a Riemannian metric then _L is the Riemannian connection.Partially supported by Subdirección General de Cooperación Internacional, Programa de Cooperación Científica con Iberoamérica, SPAIN and CNPq-BRAZIL, Proc. 30.1115/79.     

On a problem in the statistics of strong martingales in the plane

We study the problem of finding an explicit form of the conditional mathematical expectation M{|f _x }, where is a function of an unknown strong martingale on the plane andf _x is the-algebra generated by the observed strong martingale.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 71–75, 1986.