On integrability of an almost complex structure on a strictly nearly Kähler 6-manifold

We prove that an almost complex structure positively associated with a Kähler 2-form in a strictly nearly Kähler 6-manifold is not integrable.     

Simons-type inequalities for minimal surfaces with constant Kähler angle in a complex hyperquadric

In this paper, we establish Simons-type inequalities and obtain some pinching theorems for compact minimal surfaces with constant Kähler angle immersed into a complex hyperquadric, and we characterize all these minimal surfaces when equality holds in the pinching theorems.     

Some new domains with complete Kähler metrics of negative curvature

We add to the known examples of complete Kähler manifolds with negative sectional curvature by showing that the following three classes of domains in euclidean spaces also belong: perturbations of ellipsoidal domains in ?ⁿ, intersections of complex-ellipsoidal domains in ?², and intersections of fractional linear transforms of the unit ball in ?². In the process, we prove the following theorem in differential geometry: in the intersection of two complex-ellipsoidal domains in ?², the sum of the Bergman metrics is a Kähler metric with negative curvature operator.     

Existence of a complete holomorphic vector field via the Kähler–Einstein metric

In this paper, we study the existence of a complete holomorphic vector field on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kähler–Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kähler–Einstein metric whose differential has a constant length. Then, we will construct a complete holomorphic vector field from the gradient vector field of the potential function.

     

Invariant submanifolds and properCR foliations on a para-coKählerian manifold with concircular structure vector field

After recalling the basic properties of para-coKählerian manifolds \(\tilde M\) with concircular structure vector field ξ, the infinitesimal auto morphismsX of the structure 1-form \(\tilde \eta \) are considered. One of the results is that the Lie derivative of all powers of the structure 2-form \(\tilde \Omega ,\) i.e. \(\mathcal{L}x\tilde \Omega ^p ;p = 1,...,m,\) is exterior recurrent. Further two types of horizontal distributionsD _n which are normal to ξ. IfD _t (resp.D _n ) is involutive, the corresponding leafM _t (resp.M _n ) is a minimal submanifold of \(\tilde M\) . FurtherM _n is a symplectic submanifold and ξ is an umbilical normal section ofM _n . Finally proper immersion \(M \to \tilde M\) are discussed, whereM is aCR-sub-manifold whose horizontal distribution isD _t . It is shown that the vertical distribution is involutive, and the restriction of ξ toM is an symptotic direction. Some interesting special cases are treated.     

Local structure of graphs with λ=μ=2,a 2=4

Several properties of graphs with ==2,a ₂=4 are studied. It is proved that such graphs are locally unions of triangles, hexagons or heptagons. As a consequence, a distance regular graph with intersection array (13,10,7;1,2,7) does not exist.     

Some properties of polynomials orthogonal in a space with “intermediate” topology. Kernel function and extremal properties

A space of holomorphic functions is considered. A topology in this space is intermediate between the topology of uniform convergence and the topology of uniform convergence on compact sets. The properties of systems of orthonormal polynomials are studied in Hilbert spaces with this topology.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 915–923, July, 1993.     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

Erratum to “Affine surfaces in R4 with planar geodesics with respect to the affine metric”

In this work a mistake in the paper is corrected. There is also a new proof of the main theorem which classifies the non-degenerate affine surfaces in R ⁴ having planar geodesics with respect to the affine metric.     

Mixing “in the sense of ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. II

In the first part of the present paper, we established estimates for the rate of approach of the integrals of a family of “physical” white noises to a family of Wiener processes. We use this result to establish the estimate for the rate of approach of a family of solutions of ordinary differential equations perturbed by some “physical” white noises to a family of solutions of the corresponding It? equations. We consider both the case where the coefficient of random perturbation is separated from zero and the case where it is not separated from zero.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

On a “Separating” set for higher-order equations of elliptic type with discontinuous nonlinearities

We study the existence of semiregular solutions of the Dirichlet problem for a higher-order equation of elliptic type with a spectral parameter and a discontinuous nonlinearity. By a variational method, we obtain a theorem on the existence of solutions and on the properties of a “separating” set for this problem.     

Minimizing the expected discounted time to ruin for a company managing N distinct funds with a “superclaims”process

1.IntroductionOne of the central problems in classical ruin theory of a( non- life) insurance businesshas been the determination of the initial reserve size to ensure minimal probability of ruinover some time interval.If the comulative claims process X ( t) is compouns Possion,and ifthe premium rate,p.is a constant,then for an initial reserve u,the( Levy) processR( t) =u + pt- X( t) ( 1 )defines the‘risk- reserve’ or state of the company attime t.the company is ruined at epoch T=T( u) wher…     

Integrating the Korteweg–de Vries Equation with a Self-Consistent Source and “Steplike” Initial Data

It is shown that steplike solutions of the Korteweg–de Vries equation with a self-consistent source can be found by the inverse scattering method for the Sturm–Liouville operator on the entire real line.