Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1443–1453
We obtain an estimate for the ruin probability of an insurance company that invests a part of its capital in stocks and puts the rest of the capital in a bank account. An insurance premium is established depending on the capital of the insurance company.
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1454–1472
We find sufficient conditions on coefficients of diffusion equation on noncompact manifold, that guarantee non-explosion of solutions in a finite time.
This property leads to the existence and uniqueness of solutions for corresponding stochastic differential equation with globally non-Lipschitz coefficients.
Proposed approach is based on the estimates on diffusion generator, that weakly acts on the metric function of manifold. Such estimates enable us to single out a manifold analogue of monotonicity condition on the joint behaviour of the curvature of manifold and coefficients of equation.
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1473–1484
We establish relations for the distribution of functionals associated with the behavior of a risk process with random premiums after ruin and for a multivariate ruin function.
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1485–1509
We solve main two-boundary problems for a random walk. The generating function of the joint distribution of the first exit time of a random walk from an interval and the value of the overshoot of the random walk over the boundary at exit time is determined. We also determine the generating function of the joint distribution of the first entrance time of a random walk to an interval and the value of the random walk at this time. The distributions of the supremum, infimum, and value of a random walk and the number of upward and downward crossings of an interval by a random walk are determined on a geometrically distributed time interval. We give examples of application of obtained results to a random walk with one-sided exponentially distributed jumps.
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1510–1522
We find an explicit formula for finding pairs of cocycles for the construction of examples of locally compact quantum groups by using the bicrossed product of Lie groups.
Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1523–1531
We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to space variables. We consider the case of an arbitrary number of space variables. We obtain conditions for the existence and uniqueness of the solution of this problem independent of the behavior of solution as $|x| \rightarrow +\infty$. The indicated classes of the existence and uniqueness are defined as spaces of local integrable functions. The dimension of the domain in no way limits the order of nonlinearity.
Group classification of quasilinear elliptic-type equations. I. Invariance with respect to Lie algebras with nontrivial Levi decomposition
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1532–1545
The problem of group classification of quasilinear elliptic-type equations in the two-dimensional space is considered. The list of all equations of this sort is obtained, which admit the semisimple Lie algebras of symmetry operators and the Lie algebras of symmetry operators with nontrivial Levi decomposition.
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1546–1556
A plane stationary convective Stefan problem is analyzed in the case where the convection is caused by the presence of a prescribed rotation of intensity μ. A method of studying this problem is proposed which consists in a series expansion of the solution in terms of powers of a small parameter μ. The null expansion term is defined by the Rietz method. The formula describing the dependence of free boundary equation on μ is obtained.
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1557–1573
We propose and investigate a mathematical model of an open bilinear control system for the conversion of heat energy in a coherent form. We show that the use of a combinational parametric resonance formed by the control system in a one-temperature ensemble of weakly dissipative elastic-gyroscopic subsystems enables one to obtain a positive energy output without using any cooling device apart from the control system.
Properties of parabolic Kählerian spaces admitting an almost geodesic mapping of the type π2 with degenerate affinor structure
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1574–1579
We study an almost geodesic mapping of Riemann spaces with parabolic affinor structure. Some properties of parabolic Kählerian spaces admitting an almost geodesic mapping are established.
Ukr. Mat. Zh. - 2007νmber=6. - 59, № 11. - pp. 1580–1584
We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator.