# Volume 53, № 8, 2001

### Mykhailo Vasyl'ovych Ostrograds'kyi and his role in the development of mathematics

Gorbachuk M. L., Samoilenko A. M.

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1011-1023

### Ostrohrads'kyi Formalism for Singular Lagrangians with Higher Derivatives

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1034-1037

We generalize the Ostrohrads'kyi method for the construction of the Hamiltonian description of a nondegenerate (regular) variational problem of arbitrary order to the case of degenerate (singular) Lagrangians. These Lagrangians are of major interest in the contemporary theory of elementary particles. For simplicity, we consider the Hamiltonization of a variational problem defined by a singular second-order Lagrangian. Generalizing the Ostrohrads'kyi method, we derive equations of motion in the phase space. We determine a complete collection of constraints of the theory.

### M. V. Ostrogradsky As Probabilist

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1038-1047

We review the writings on probabilistic topics of M. V. Ostrogradsky (1801–1862) in the bicentenary year of his birth from a standpoint different from the sesquicentenary article of Gnedenko. Ostrogradsky's statistical technology follows closely that of Laplace's *Théorie Analytique des Probabilités* in its use of the Bayes theorem together with the principle of insufficient reason. He makes more precise or modifies certain of Laplace's application-oriented conclusions. The more striking results relate to sampling for attributes without replacement in a finite population and to the probability of error by a panel of judges, anticipating Poisson.

### Transport Theory of Homogeneous Reacting Solutes

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1048-1052

We consider the one-dimensional convection (advection)-dispersion equation of the transport theory of reacting solutes in porous media. A method is given for the best approximation of the numerical solution both in absence of interaction with the solid phase and in presence of discontinuous initial conditions. The class of solutions is determined by the multiresolution analysis of the partial differential operator, using Haar wavelets and splines, and it is compared with the Fourier solution.

### Group Classification of Nonlinear Schrödinger Equations

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1053-1060

We propose an approach to problems of group classification. By using this approach, we perform a complete group classification of nonlinear Schrödinger equations of the form *i*ψ_{ t } + Δψ + *F*(ψ, ψ*) = 0.

### On the Existence of Periodic Solutions for Certain Classes of Systems of Differential Equations with Random Pulse Influence

Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1061-1079

We establish conditions for the existence of periodic solutions for systems of differential equations with random right-hand side and random pulse influence at fixed times. We consider the case of small pulse perturbation and weakly nonlinear systems.

### Spectrum and States of the BCS Hamiltonian in a Finite Domain. II. Spectra of Excitations

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1080-1100

We establish that the averages per volume of the BCS and approximating Hamiltonians over all excited states coincide in the thermodynamic limit. Earlier, this was established only for the ground state.

### Exponential Estimate for the Fundamental Matrix of a Linear Impulsive System

Petryshyn R. I., Sopronyuk Т. M.

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1101-1108

We investigate the properties of oscillating sums and integrals dependent on parameters. These properties are used for the estimation of the normal fundamental matrix of a linear system with rapidly oscillating coefficients and pulse influence at fixed moments of time.

### Generalized Lagrangians and Spinning Particles

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1109-1120

We show that the spin structure of elementary particles can be naturally described by the generalized Ostrogradskii Lagrangians depending on higher-order derivatives. One component of a spin is related to the rotation of a particle and the other one, caused by the dependence of a Lagrangian on the acceleration, is known as a zitterbewegung component of spin.

### Approximation Characteristics of the Spaces $S_p^{ϕ}$ in Different Metrics

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1121-1146

We continue the investigation of the approximation characteristics of the spaces $S_p^{ϕ}$ introduced earlier. In particular, we establish direct and inverse theorems on the approximation of elements of these spaces. We also determine the exact values of upper bounds of $m$-term approximations of $q$-ellipsoids in the spaces $S_p^{ϕ}$ in the metrics of the spaces $S_p^{ϕ}$.

### On the Unique Solvability of a Nonautonomous Functional Differential Equation of Neutral Type

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1147-1152

We present conditions for the existence and uniqueness of solutions of functional differential equations of the neutral type and their continuous dependence on the initial data.