# Volume 60, № 9, 2008

### Yurii Oleksiiovych Mytropol's'kyi (Obituary)

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1155-1156

### $(\min, \max)$-equivalence of posets and nonnegative Tits forms

Bondarenko V. M., Stepochkina M. V.

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1157–1167

We study the relationship between the (min, max)-equivalence of posets and properties of their quadratic Tits form related to nonnegative definiteness. In particular, we prove that the Tits form of a poset *S* is nonnegative definite if and only if the Tits form of any poset $(\min, \max)$-equivalent to *S* is weakly nonnegative.

### Inverse Sturm-Liouville problem on a figure-eight graph

Gomilko A. M., Pivovarchik V. N.

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1168–1188

We study the inverse problem for the Strum-Liouville equation on a graph that consists of two quasione-dimensional loops of the same length having a common vertex. As spectral data, we consider the set of eigenvalues of the entire system together with the sets of eigenvalues of two Dirichlet problems for the Sturm-Liouville equations with the condition of total reflection at the vertex of the graph. We obtain conditions for three sequences of real numbers that enable one to reconstruct a pair of real potentials from L 2 corresponding to each loop. We give an algorithm for the construction of the entire set of potentials corresponding to this triple of spectra.

### Weak local homeomorphisms and B-favorable spaces

Karlova O. O., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1189–1195

Let *X* and *Y* be topological spaces such that every mapping *f* : *X* → *Y* for which the set *f *^{ - 1}(*G*)
is an *f *_{σ} -set in *X* for any set *G* open in *Y* can be represented as a pointwise limit of continuous mappings *f _{n}* :

*X*→

*Y*. The question of subspaces

*Z*of the space

*Y*for which mappings

*f*:

*X*→

*Z*have the same property is investigated.

### Best *M*-term trigonometric approximations of the classes of periodic functions of many variables in the space *L*_{q}

_{q}

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1196 – 1214

We obtain order estimates for the best *M*-term trigonometric approximations of classes *B*^{ Ω}_{p,θ}
of periodic multivariable functions in the space *L _{q}* for some values of the parameters

*p*and

*q*.

### On scalar-type spectral operators and Carleman ultradifferentiable *C*_{0}-semigroups

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1215 – 1233

Necessary and sufficient conditions for a scalar-type spectral operator in a Banach space to be a generator of a Carleman ultradifferentiable *C*_{0}-semigroup are found. The conditions are formulated exclusively in terms of the spectrum of the operator.

### On the Skitovich-Darmois theorem and Heyde theorem in a Banach space

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1234–1242

According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space.

### Spectrum and states of the BCS Hamiltonian with sources

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1243–1269

We consider the BCS Hamiltonian with sources, as proposed by Bogolyubov and Bogolyubov, Jr. We prove that the eigenvectors and eigenvalues of the BCS Hamiltonian with sources can be exactly determined in the thermodynamic limit. Earlier, Bogolyubov proved that the energies per volume of the BCS Hamiltonian with sources and the approximating Hamiltonian coincide in the thermodynamic limit.

### Equivalent definition of some weighted Hardy spaces

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1270–1274

We present the equivalent definition for spaces of functions analytic in the half-plane ${\mathbb C}_+ = \{z: Re z > 0 \}$ for which $$\sup_{|\varphi| < \frac{\pi}2} \left\{\int\limits_0^{+\infty}\left| f(r e^{i\varphi})\right|^p e^{-p\sigma r|\sin \varphi|} dr \right\} < +\infty.$$

### Regularization inertial proximal point algorithm for unconstrained vector convex optimization problems

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1275–1281

The purpose of this paper is to investigate an iterative regularization method of proximal point type for solving ill posed vector convex optimization problems in Hilbert spaces. Applications to the convex feasibility problems and the problem of common fixed points for nonexpansive potential mappings are also given.

### Convergence of an impulsive storage process with jump switchings

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1282–1286

We investigate an impulsive storage process switched by a jump process. The switching process is, in turn, averaged. We prove the weak convergence of the storage process in the scheme of series where a small parameter ε tends to zero.

### Periodic solutions of linear impulsive differential inclusions

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1287–1296

We establish sufficient conditions for the existence of periodic *R*-solutions of linear differential inclusions with impulses at fixed times.