# Volume 56, № 3, 2004

### Dmytro Yakovych Petryna (on his 70 th birthday)

Gorbachuk M. L., Khruslov E. Ya., Lukovsky I. O., Marchenko V. O., Mitropolskiy Yu. A., Pastur L. A., Samoilenko A. M., Skrypnik I. V.

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 291-292

### Creative Contribution of D. Ya. Petrina to the Development of Contemporary Mathematical Physics

Gerasimenko V. I., Malyshev P. V.

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 293-308

This is a brief survey of the results obtained by Prof. D. Ya. Petrina in various branches of contemporary mathematical physics.

### BCS Model Hamiltonian of the Theory of Superconductivity as a Quadratic Form

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 309-338

Bogolyubov proved that the average energies (per unit volume) of the ground states for the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide in the thermodynamic limit. In the present paper, we show that this result is also true for all excited states. We also establish that, in the thermodynamic limit, the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide as quadratic forms.

### On the Optimal Coefficient of Efficiency of a Semi-Markov System in the Scheme of Phase Lumping

Shlepakov L. N., Vovkodav N. G.

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 339-345

By using methods of the theory of semi-Markov processes, we analyze the problem of detecting signals in a multichannel system. We construct an optimal strategy for the motion of a search device in a multichannel system and obtain the corresponding estimate for the search efficiency.

### On Permutable Congruences on Antigroups of Finite Rank

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 346-351

We find necessary and sufficient conditions for any two congruences on an antigroup of finite rank to be permutable.

### Coconvex Approximation of Functions with More than One Inflection Point

Dzyubenko H. A., Zalizko V. D.

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 352-365

Assume that *f* ∈ *C*[−1, 1] belongs to *C*[−1, 1] and changes its convexity at *s* > 1 different points *y* _{i}, \(\overline {1,s} \) , from (−1, 1). For *n* ∈ *N*, *n* ≥ 2, we construct an algebraic polynomial *P* _{n} of order ≤ *n* that changes its convexity at the same points *y* _{i} as *f* and is such that $$|f(x) - P_n (x)|\;\; \leqslant \;\;C(Y)\omega _3 \left( {f;\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right),\;\;\;\;\;x\;\; \in \;\;[ - 1,\;1],$$ where ω_{3}(*f*; *t*) is the third modulus of continuity of the function *f* and *C*(*Y*) is a constant that depends only on \(\mathop {\min }\limits_{i = 0,...,s} \left| {y_i - y_{i + 1} } \right|,\;\;y_0 = 1,\;\;y_{s + 1} = - 1\) , *y* _{0} = 1, *y* _{ s + 1} = −1.

### Nevanlinna–Pick Problem for Stieltjes Matrix Functions

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 366-380

We consider the Nevanlinna–Pick interpolation problem for Stieltjes matrix functions. We obtain two criteria for the indeterminacy of the Nevanlinna–Pick problem with infinitely many interpolation nodes. In the indeterminate case, we describe the general solution of the Nevanlinna–Pick problem in terms of fractional-linear transformations.

### Boundary Functionals of a Semicontinuous Process with Independent Increments on an Interval

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 381-398

We investigate boundary functionals of a semicontinuous process with independent increments on an interval with two reflecting boundaries. We determine the transition and ergodic distributions of the process, as well as the distributions of boundary functionals of the process, namely, the time of first hitting the upper (lower) boundary, the number of hittings of the boundaries, the number of intersections of the interval, and the total sojourn time of the process on the boundaries and inside the interval. We also present a limit theorem for the ergodic distribution of the process and asymptotic formulas for the mean values of the distributions considered.

### Euler Approximations of Solutions of Abstract Equations and Their Applications in the Theory of Semigroups

Mishura Yu. S., Shevchenko H. M.

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 399-410

Using the Euler approximations of solutions of abstract differential equations, we obtain new approximation formulas for *C* _{0}-semigroups and evolution operators.

### On the Discreteness of the Structural Space of Weakly Completely Continuous Banach Algebras

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 411-418

We consider a class of Banach algebras with irreducible finite-dimensional representations and prove that, for amenable Banach algebras from this class, the weak complete continuity implies the discreteness of their structural space.

### On the Decomposition of an Operator into a Sum of Four Idempotents

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 419-424

We prove that operators of the form (2 ± 2/*n*)*I* + *K* are decomposable into a sum of four idempotents for integer *n* > 1 if there exists the decomposition *K* = *K* _{1} ⊕ *K* _{2} ⊕ ... ⊕ *K* _{n}, \(\sum\nolimits_1^n {K_i = 0} \) , of a compact operator *K*. We show that the decomposition of the compact operator 4*I* + *K* or the operator *K* into a sum of four idempotents can exist if *K* is finite-dimensional. If *n* tr *K* is a sufficiently large (or sufficiently small) integer and *K* is finite-dimensional, then the operator (2 − 2/*n*)*I* + *K* [or (2 + 2/*n*)*I* + *K*] is a sum of four idempotents.

### Interpolation Sequences for the Class of Functions of Finite η-Type Analytic in the Unit Disk

Sheparovych I. B., Vynnyts’kyi B. V.

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 425-430

We establish conditions for the existence of a solution of the interpolation problem *f*(λ_{ n }) = *b* _{n} in the class of functions *f* analytic in the unit disk and such that $$\left( {\exists \;c_1 > 0} \right)\;\left( {\forall z,\;|\;z\;| < 1} \right):\;\;\left| {f\left( z \right)} \right|\;\; \leqslant \;\;\;\exp \left( {c_1 \eta \left( {\frac{{c_1 }}{{1 - \left| z \right|}}} \right)} \right).$$ Here, η : [1; +∞) → (0; +∞) is an increasing function convex with respect to ln *t* on the interval [1; +∞) and such that ln *t* = *o*(η(*t*)), *t* → ∞.