# Volume 57, № 11, 2005

### Theorems on the existence and nonexistence of solutions of the Cauchy problem for degenerate parabolic equations with nonlocal source

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1443–1464

We consider the Cauchy problem for a doubly nonlinear degenerate parabolic equation with nonlocal source under the assumption that the initial function is integrable. We establish the existence and nonexistence of time-global solutions of the problem.

### Ruin problem for a generalized Poisson process with reflection

Bratiichuk N. S., Lukovych O. V.

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1465–1475

We consider a generalized Poisson process with reflection at the level *T* > 0. Under certain conditions on the distribution of the values of positive jumps of the process, we obtain representations for the characteristic functions of functionals associated with the exit of the indicated process to the negative semiaxis.

### Linear groups with minimality condition for some infinite-dimensional subgroups

Dixon M. R., Evans M. J., Kurdachenko L. A.

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1476–1489

Let $F$ be a field, let $A$ be a vector space over $F$, and let $GL(F, A)$ be the group of all automorphisms of the space $A$. If $H$ is a subgroup of $GL(F, A)$, then we set aug $\dim_F (H) = \dim_F (A(ωFH))$, where $ωFH$ is the augmentation ideal of the group ring $FH$. The number ${\rm{aug} \dim}_F (H)$ is called the augmentation dimension of the subgroup $H$. In the present paper, we study locally solvable linear groups with minimality condition for subgroups of infinite augmentation dimension.

### Interpolation of operators of weak type $(ϕ_0, ψ_0, ϕ_1, ψ_1)$ in Lorentz spaces

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1490–1507

We prove theorems of interpolation of quasilinear operators of weak type (φ_{0 }, ψ_{0 }, φ_{1 }, ψ_{1 }) in the Lorentz spaces.
The operators considered are the analogs of the Calderon and Benett operators for concave and convex functions φ_{0 }(*t*), ψ_{0 }(*t*), φ_{1 }(*t*), ψ_{1 }(*t*).

### New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap”

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1508–1533

The BCS Hamiltonian of superconductivity has the second branch of eigenvalues and eigenvectors. It consists of wave functions of pairs of electrons in ground and excited states. The continuous spectrum of excited pairs is separated by a nonzero gap from the point of the discrete spectrum that corresponds to the pair in the ground state. The corresponding grand partition function and free energy are exactly calculated. This implies that, for low temperatures, the system is in the condensate of pairs in the ground state. The sequence of correlation functions is exactly calculated in the thermodynamic limit, and it coincides with the corresponding sequence of the system with approximating Hamiltonian. The gap in the spectrum of excitations depends continuously on temperature and is different from zero above the critical temperature corresponding to the first branch of the spectrum. In our opinion, this fact explains the phenomenon of “pseudogap.”

### Dynamics of neighborhoods of points under a continuous mapping of an interval

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1534–1547

Let $\{ I, f Z^{+} \}$ be a dynamical system induced by the continuous map $f$ of a closed bounded interval $I$ into itself. In order to describe the dynamics of neighborhoods of points unstable under $f$, we suggest a notion of $\varepsilon \omega - {\rm set} \omega_{f, \varepsilon}(x)$ of a point $x$ as the $\omega$-limit set of $\varepsilon$-neighborhood of $x$. We investigate the association between the $\varepsilon \omega - {\rm set}$ and the domain of influence of a point. We also show that the domain of influence of an unstable point is always a cycle of intervals. The results obtained can be directly applied in the theory of continuous time difference equations and similar equations.

### On Sylow subgroups of Shunkov periodic groups

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1548–1556

We study the structure of Sylow 2-subgroups in Shunkov periodic groups with almost layer-finite normalizers of finite nontrivial subgroups.

### On the embedding of Waterman class in the class *H*_{p}^{ω}

_{p}

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1557–1562

In this paper the necessary and sufficient condition for the inclusion of class ЛBV in the class *H _{p}*

^{ω}is found.

### Inverse problem for the heat equation with degeneration

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1563–1570

We consider the inverse problem of determining the time-dependent thermal diffusivity that is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration.

### Stabilization of the Cauchy problem for integro-differential equations

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1571–1576

In the present paper, we obtain a criterion for the stabilization of the Cauchy problem for an integro-differential equation in the class of functions of polynomial growth γ ≥ 0.

### On modified Picard and Gauss—Weierstrass singular integrals

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1577–1584

We introduce a certain modification of the Picard and Gauss—Weierstrass singular integrals and prove approximation theorems for them.