Ukr. Mat. Zh. - 2003νmber=4. - 55, № 12. - pp. 1587-1657
We present main recent results on the generalization of white-noise analysis related to a family of generalized translation operators.
Asymptotic Representations for Solutions of One Class of Systems of Quasilinear Differential Equations
Ukr. Mat. Zh. - 2003νmber=4. - 55, № 12. - pp. 1658-1668
We establish asymptotic representations for solutions of one class of systems of differential equations appearing in the investigation of the asymptotic behavior of nth-order quasilinear differential equations.
Ukr. Mat. Zh. - 2003νmber=4. - 55, № 12. - pp. 1669-1677
We prove that a conditional expectation on a Kac algebra, under certain conditions, decomposes into a composition of two conditional expectations of a special type and gives rise to a compact quantum hypergroup connected to a quantum Gelfand pair.
Ukr. Mat. Zh. - 2003νmber=4. - 55, № 12. - pp. 1678-1796
We consider initial-value problems for infinite systems of first-order partial functional differential equations. The unknown function is the functional argument in equations and the partial derivations appear in the classical sense. A theorem on the existence of a solution and its continuous dependence upon initial data is proved. The Cauchy problem is transformed into a system of functional integral equations. The existence of a solution of this system is proved by using integral inequalities and the iterative method. Infinite differential systems with deviated argument and differential integral systems can be derived from the general model by specializing given operators.
Ukr. Mat. Zh. - 2003νmber=4. - 55, № 12. - pp. 1697-1703
We present a variational generalization of the problem of infinitesimal geodesic deformations of surfaces in the Euclidean space E 3. By virtue of rotary deformation, the image of every geodesic curve is an isoperimetric extremal of rotation (in the principal approximation). The results are associated in detail with rotary-conformal deformations. The application of these results to the mechanics of elastic shells is given.
Ukr. Mat. Zh. - 2003νmber=4. - 55, № 12. - pp. 1704-1719
On the basis of generalized Lagrange identity for pairs of formally adjoint multidimensional differential operators and a special differential geometric structure associated with this identity, we propose a general scheme of the construction of corresponding transformation operators that are described by nontrivial topological characteristics. We construct explicitly the corresponding integro-differential symbols of transformation operators, which are used in the construction of Lax-integrable nonlinear two-dimensional evolutionary equations and their Darboux–Bäcklund-type transformations.
Ukr. Mat. Zh. - 2003νmber=4. - 55, № 12. - pp. 1720-1723
We prove the existence and uniqueness of a continuously differentiable solution with required asymptotic properties.
Ukr. Mat. Zh. - 2003νmber=4. - 55, № 12. - pp. 1724-1728