# Volume 55, № 2, 2003

### A Nonparametric Test for the Equivalence of Populations Based on a Measure of Proximity of Samples

Klyushin D. A., Petunin Yu. I.

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 147-163

We propose a new measure of proximity of samples based on confidence limits for the bulk of a population constructed using order statistics. For this measure of proximity, we compute approximate confidence limits corresponding to a given significance level in the cases where the null hypothesis on the equality of hypothetical distribution functions may or may not be true. We compare this measure of proximity with the Kolmogorov–Smirnov and Wilcoxon statistics for samples from various populations. On the basis of the proposed measure of proximity, we construct a statistical test for testing the hypothesis on the equality of hypothetical distribution functions.

### Positive and Monotone Systems in a Partially Ordered Space

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 164-173

We investigate properties of positive and monotone differential systems with respect to a given cone in the phase space. We formulate criteria for the stability of linear positive systems in terms of monotonically invertible operators and develop methods for the comparison of systems in a partially ordered space.

### Model BCS Hamiltonian and Approximating Hamiltonian in the Case of Infinite Volume. IV. Two Branches of Their Common Spectra and States

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 174-196

We consider the model and approximating Hamiltonians directly in the case of infinite volume. We show that each of these Hamiltonians has two branches of the spectrum and two systems of eigenvectors, which represent excitations of the ground states of the model and approximating Hamiltonians as well as the ground states themselves. On both systems of eigenvectors, the model and approximating Hamiltonians coincide with one another. In both branches of the spectrum, there is a gap between the eigenvalues of the ground and excited states.

### Dirichlet Problem for the Stokes Flow Function in a Simply-Connected Domain of the Meridian Plane

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 197-231

We develop a method for the reduction of the Dirichlet problem for the Stokes flow function in a simply-connected domain of the meridian plane to the Cauchy singular integral equation. For the case where the boundary of the domain is smooth and satisfies certain additional conditions, the regularization of the indicated singular integral equation is carried out.

### The Reduction Method in the Theory of Lie-Algebraically Integrable Oscillatory Hamiltonian Systems

Prykarpatsky A. K., Samoilenko V. G., Taneri U.

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 232-240

We study the problem of the complete integrability of nonlinear oscillatory dynamical systems connected, in particular, both with the Cartan decomposition of a Lie algebra \(G = K \oplus P{\text{, where }}K\) is the Lie algebra of a fixed subgroup \(K \subset {\text{G}}\) with respect to an involution σ : *G* → *G* on the Lie group *G*, and with a Poisson action of special type on a symplectic matrix manifold.

### Multipoint Problem for Nonisotropic Partial Differential Equations with Constant Coefficients

Ptashnik B. I., Symotyuk M. M.

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 241-254

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable *t* and periodic conditions with respect to coordinates *x* _{1},..., *x* _{p} for a nonisotropic (concerning differentiation with respect to *t* and *x* _{1},..., *x* _{p}) partial differential equation with constant complex coefficients. We establish conditions for the existence and uniqueness of a solution of this problem and prove metric theorems on lower bounds for small denominators appearing in the course of the construction of its solution.

### On the Stability of an Equilibrium State of Gyroscopic Coupled Systems

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 255-263

We investigate the stability of an equilibrium state of gyroscopic coupled conservative systems in the case where the force function does not attain a local maximum in this state. We consider the situation where the gyroscopic coupling is weak with respect to a part of coordinates and strong with respect to the other part.

### Spaces $S^p$ with Nonsymmetric Metric

Rukasov V. I., Stepanets O. I.

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 264-277

We determine exact values of the best approximations and Kolmogorov widths of $q$-ellipsoids in spaces $S_\phi ^{p,{\mu}}$ defined by anisotropic metric.

### Approximate Solution of the de la Vallée Poussin Problem for a Differential Equation of Neutral Type by the Projection-Iterative Method

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 278-283

We solve the de la Vallée Poussin problem for a functional-differential equation by the projection-iterative method. We construct an algorithm, establish conditions sufficient for the convergence of the method, and present a computational scheme.

### Geometric Form of the Hahn–Banach Theorem for Generalized Convexity

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 284=287

We investigate a class of compact sets convex with respect to a certain family of planes. For compact sets that satisfy the condition of acyclicity of sections by a certain collection of two-dimensional planes, we prove their generalized convexity.

### A Note on *FC*-Groups

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 288-289

Let *G* be an arbitrary *FC*-group, let *R* be its locally soluble radical, and let *L*/*R* = *L*(*G*/*R*). We prove that, for *N* ⊲ *G*, *G*/*N* is residually finite if *R* \(\subseteq\) *N* \(\subseteq\) *L*.