B. Zabavsky, A. Gatalevych. New characterizations of commutative clean rings. P.115-118.
A ring is called clean if every its element is the sum of a unit and an idempotent. We introduce the
notion of an avoidable element and describe class of the commutative clean rings as the rings in
which zero is an avoidable element.
SubjClass 06F20, 13F99

V.V. Lyubashenko. Homotopy cooperads. P.119-160.
The theory of 2-monads is used as a ground to study non-symmetric cooperads. We give a new
definition of homotopy cooperads. Ordinary and homotopy cooperads are placed in lax $Cat$-
operads which are lax algebras over the free-operad strict 2
-monad. We give an~example of a
homotopy cooperad cofree with respect to ordinary cooperads.
SubjClass 18D50

O.M. Mulyava, M.M. Sheremeta. On a new condition of finite Lipschitz of Orlicz-Sobolev class. P.161
-170.
For a~function $P$ continuos on $(-|, +|)$ increaing to $+|$ the lower and upper estimates for
$frac{a(P(qs))}{a^p(P(s))}$ are found, where $p>1, q>1$ and $a$ is positive function continuous
on $[x_0,+|)$, increasing to $+|$. The above results applied to Dirichlet series with positive
exponents.
SubjClass 30B50

E.A. Sevost`yanov, D.S. Dolya. On removable singularities of a class of mappings. P.171-184.
A paper is devoted to study of local behavior of so-called $Q$-mappings including quasiconformal
mappings and mappings with bounded distortion. It is showed that, such mappings have removable
isolated singularities whenever the growth of the mappings is not more than some function of a~radius
of a~ball.
SubjClass 30С65, 31A15, 32U20

S.V. Litynskyy, A.O. Muzychuk. Solving of initial-boundary value problems for the wave equation using
retarded surface potential and Laguerre transform. P.185-203.
Approach for solving of initial-boundary value problems for the homogeneous wave equation is
described and proved. It is based on the Laguerre transform in the time domain and the boundary
integral equations. Retarded potentials are used for representation of generalized solutions of such
problems. The densities of retarded potentials are expanded in Fourier-Laguerre series which
coefficients have special convolution form. As a~result, initial-boundary value problems are reduced to
a~sequence of boundary integral equations.
SubjClass 31B10, 35D30, 35J50, 35J57, 45B05

M.A. Belozerova, G.A. Gerzhanovskaya. Asymptotic representations of solutions of second order
differential equations with nonlinearities close to regularly varying. P.204-214.
The asymptotic representation, necessary and sufficient conditions of the existence of sufficient broad
classes of the solutions are found for differential equations of the second order that are in some sense
similar to equations with nonlinearities, that are regularly varying at the singular points.
SubjClass 34C41, 34Е10

H.P. Lopushanska, A.O. Lopushansky, O.M. Myaus. Classical solution of the inverse problem for
fractional diffusion equation under time-integrated over-determination condition. P.215-220.
We prove the correctness of the inverse problem on determination of a~pare of functions: a~classical
solution $u$ of the first boundary value problem for linear diffusion equation $ D^alpha_t u-u_{xx}
=F_0(x), (x,t)in (0,l) imes (0,T]$ with regularized fractional derivative of order $alphain (0,1)$ with
respect to time and function $F_0(x)$ under integral by time over-determination condition.
SubjClass 35S15

INDEX, 2015. P.221-224.

Additional information:

Matematychni Studii
Proceedings of the Lviv Mathematical Society

Journal is devoted to research in all fields of mathematics. Original papers of moderate length are accepted; exception is possible for survey articles. Languages accepted are: English, French, German, Russian, and Ukrainian. Published quarterly.

Published for the Lviv Mathematical Society by VNTL Publishers with finansial and organizing support of the Lviv National University.

Address:
Matematychni Studii
Dept. of Mechanics and Mathematics,
Lviv National University,
1 Universytetska St.,
79000, Lviv, Ukraine
e-mail: matstud@franko.lviv.ua