# Volume 55, № 12, 2003

### Spaces of Test and Generalized Functions Related to Generalized Translation Operators

Berezansky Yu. M., Tesko V. A.

Ukr. Mat. Zh. - 2003. - 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. - 2003. - 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 *n*th-order quasilinear differential equations.

### Factorization of Conditional Expectations on Kac Algebras and Quantum Double Coset Hypergroups

Chapovsky Yu., Kalyuzhnyi A. A.

Ukr. Mat. Zh. - 2003. - 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.

### Infinite Systems of Hyperbolic Functional Differential Equations

Ukr. Mat. Zh. - 2003. - 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.

### Infinitesimal Rotary Deformations of Surfaces and Their Application to the Theory of Elastic Shells

Ukr. Mat. Zh. - 2003. - 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.

### Structure of Binary Transformations of Darboux Type and Their Application to Soliton Theory

Prykarpatsky Ya. A., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 2003. - 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.

### Qualitative Investigation of the Singular Cauchy Problem *F*(*t*, *x*, *x*′) = 0, *x*(0) = 0

Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1720-1723

We prove the existence and uniqueness of a continuously differentiable solution with required asymptotic properties.

### Index of volume 55 of „Ukrainian Mathematical Journal"

Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1724-1728