# Volume 48, № 9, 1996

### On one problem with free boundary for a nonlinear system

Bazalii B. V., Krasnoshchok M. V.

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1155–1165

We formulate the filtration problem with free boundary as a problem with discontinuous nonlinearity for a degenerate elliptic or parabolic system. We prove that a solution of the Dirichlet problem exists in both cases. We study some qualitative properties of these solutions, e.g., the existence of “dead cores”.

### On the existence of entire functions of bounded *l*-index and *l*-regular growth

Bordulyak M. T., Sheremeta M. M.

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1166–1182

We prove that, under certain conditions on a positive function*l* continuous on [0, +∞], there exists an entire transcendental function*f* of bounded*l*-index such that lnln*M* _{f(r)}ln*L(r)*,*r*→∞, where*M* _{f} *(r)*=max {|*f(z)*|: |*z*|=*r*} and*L(r)*=∫ _{0} ^{ r } *l(t)dt*. If*l(r)=r* ^{ p-1 } for*r≥1*, 0<ρ<∞, then there exists an entire function*f* of bounded*l*-index such that*M* _{ f }(*r*)≈*r* ^{ p }.

### Basic properties of the problem of the best simultaneous approximation of several elements

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1183–1193

For the problem of the best approximation of several elements with respect to the maximum of convex-concave fractional functions with additional restrictions, we establish duality relations and criteria for the element of the best approximation.

### Unbounded branches of solutions of some boundary-value problems

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1194–1199

For nonlinear equations of a special type, in the case of double degeneration of a linearized problem, we prove the existence of unbounded branches of solutions originating at a bifurcation point.

### Optimal control over the heat field in thin bodies local constraints on control

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1200–1208

Asymptotic solutions of problems of optimal locally constrained control over the heat field in thin bodies are constructed and justified. These problems relate to the critical case of singularly perturbed systems (the degenerate problem has a family of solutions).

### Power moments of negative order for the principal spectral function of a string

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1209–1222

We established necessary and sufficient conditions for the existence of finite power moments of all integer negative orders for the principal spectral function of a string. The necesity of this problem is explained by its relation to the so-called strong Stieltjes moment problem.

### Multiserial rings

Kirichenko V. V., Yaremenko Yu. V.

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1223–1235

We introduce the notion of multiserial (*n*-serial) rings and study their properties. The second-order minors of such rings are investigated. We also find all possible forms of quivers for Noetherian and hereditary*n*-serial rings and describe the structure of semiprime and hereditary*n*-serial rings.

### γ-Convergence of integral functionals and the variational dirichlet problem in variable domains

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1236–1254

By using special local characteristics of domains Ω_{ s }⊂Ω,*s*=12,..., we establish necessary and sufficient conditions for the γ-convergence of sequences of integral functionals*I* _{λs }:*W* ^{ k,m }(Ω_{ s })→ℝ, λ⊂Ω to interal functionals defined on W^{ k,m }(Ω).

### On linear widths of classes $H^ω$

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1255-1264

We obtain new results related to the estimation of the linear widths $λ_N$ and $λ^N$ in the spaces $C$ and $L_p$ for the classes $H^ω$ (in particular, for $H^α,\; 0 < α < 1$).

### Symmetry and exact solution of heat-mass transfer equations in thermonuclear plasma

Cherniga R. M., Wilhetmsson H.

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1265–1277

For the nonlinear system of partial differential equations, which describes the evolution of temperature and density in TOKAMAK plasmas, multiparameter families of exact solutions are constructed. The solutions are constructed by the Lie-method reduction of initial systems of equations to a system of ordinary differential equations. Examples of non-Lie ansätze and exact solutions are also presented.

### On asymptotic formulas for solutions of systems of linear differential equations with a degerate matrix with derivatives

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1278–1285

In this paper, we suggest a method for the construction of asymptotic formulas for solutions of systems of differential equations in the case where the roots of the characteristic equation are simple

### On the concept of generalized solution of operator equations in banach spaces

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1286–1290

We introduce a new concept of generalized solution of operator equations with closed linear operator in a Banach space as an element of the completion of the space in certain locally convex topology. We prove a theorem on the existence and uniqueness of a generalized solution and give examples of finding the generalized solution for infinite systems of the linear algebraic equations.

### Parasupersymmetric quantum mechanics of arbitrary order with*N* parasupercharges

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1291–1294

Parasupersymmetric quantum mechanics is generalized to the case of an arbitrary number of parasuperchanrges *N* and the order of paraquantization *p*. We show that parasuperpotentials can be explicitly expressed via a single arbitrary function.

### Hidden symmetries of the two-particle dirac equation with linear interaction

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1295–1296

We investigate Lie and non-Lie symmetries of the two-particle Dirac equation with linear interaction in the one-dimensional case. The integrals of motion and hidden parasupersymmetries are found. By using algebraic method and non-Lie symmetries, we obtain the energy spectra of the considered system