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Title: Matematychni Studii, Vol.20, No.2
ISSN: 1027-4634
Price: 20.00 USD
Year: 2003
  
Contents:

A. M. Stepin. 50-th anniversary of R. I. Grigorchuk. P.115-118.
M. D. Pochynaiko, Yu. M. Sydorenko. Integrating of some (2+1)-dimentional integrable systems by methods of inverse scattering problem and binary Darboux transformation. P.119-132. (in Ukrainian)>
For space-twodimentional symetric generalizations of the $KdV(2dKdV)$ and modified $KdV$ equation $2dmKdV$ for alternative by known $L - P - Q$ Manacov's triad the Lax pairs $L - A$ are constructed. The binary Darboux transformations for the associated linear systems with $2dKdV, 2dmKdV$ and formulas for solution the studied equations in explicit forms are found. It is shown that explicit solutions of $2dKdV, 2dmKdV$ equations and solutions of space-twodimentional nonlinear Schroedinger equations obtained by the inverse scattering method for Dirac's operator are contained among solutions obtained by binary Darboux transformation as a special case.
Subjclass: 12G99, 14H05.

O. B. Skaskiv, Ya. Z. Stasyuk. On the Wiman theorem for absolutely convergent Dirichlet series. P.133-142.
For functions $f$ analytic in the unit disc and such that $(1-r)K_f(r) to +infty $ $(r to 1-0),$ where $K_f(r)=(ln M_f(r))',$ $M_f(r)=max{|f(z)|: |z|=r}$ it is proved that $$ M_f(r)=(1+o(1))max{re f(z):|z|=r}=-(1+o(1))min{re f(z):|z|=r} $$ as $rto 1-0$ $(r notin E)$, $me (Ecap [r,1))=o(1-r)$ $(r to 1-0).$
Subjclass: 30B10, 30B50, 30D40.

M. M. Sheremeta, O. M. Sumyk, M. M. Zelisko. On the boundedness of $l$-$M$- and $l$-$mu$-index of the Dirichlet series. P.143-150.
We find conditions on the exponents of the Dirichlet series with an arbitrary abscissa of absolute convergence in order that the relations $ln M(sigma,F)=O(P(sigma))$, $sigmauparrow A$ and $lnmu(sigma,F)=O(P(sigma))$, $sigmauparrow A$ be equivalent, where $P$ is a certain convex function defined on $(-infty,A)$, $Ain(-infty,+infty]$. The obtained result was applied to proving equivalence of the boundedness of $l$-$M$ and $l$-$mu$ index of the Dirichlet series.
Subjclass: 30B50.

N. M. Pyrch. Orthogonal retractions and $M$-equivalence. P.151-161.
Two Tychonoff spaces are $M$-equivalent if their free topological groups are topologically isomorphic. We introduce the notion of orthogonal retracts in a Tychonoff space and as a development of Okunev's construction show that every pair of orthogonal retracts leads to a pair of $M$-equivalent spaces.
Subjclass: 22A05.

L. Halbeisen. A set-theoretic approach to complete minimal systems in Banach spaces of bounded functions. P.162-166.
Using independent families from combinatorial set theory, it is shown that for every infinite cardinal $kappa$, $ellinf(kappa)^*$ contains a subspace which is isomorphic to a Hilbert space of dimension $tokappa$. This provides a new proof for the first step in the construction of complete minimal systems in Banach spaces of bounded functions.
Subjclass: 46B15, 03E05, 46B26.

K. Grasela. Generalized derivations and Fourier transform of polynomial ultradistributions. P.167-178.
Algebras of symmetric ultradistributions on the spaces of functions of infinitely many variables are investigated. Linear representation of such algebras as convolution algebras of symmetric tensor products are constructed. Generalized derivations and Fourier transform of such algebras are described.
Subjclass: Primary 46F25, Secondary 58C35.

O. H. Bihun. Modification of the Lie-algebraic scheme and approximation error estimations. P.179-184.
A new modification of the Lie-algebraic scheme for solving partial differential equations with initial and boundary conditions based on constructing quasirepresentations of the Heisenberg-Weyl algebra operators involving boundary conditions is proposed. Approximation errors for the modified scheme are evaluated.
Subjclass: 65M99, 35G10.

Ya. V. Mykytyuk. Factorization of Fredholm operators. P.185-199. (in Ukrainian)
We consider a factorization problem for Fredholm operators along the chains of orthogonal projectors in Banach operator algebras and describe a class of such algebras in which the factorization problem has a solution.
Subjclass: 47A68, 47A46.

T. I. Zvozdetskyi. Description of solutions of an operator equation which contains Gelfond-Leont'ev integration. P.200-204. (in Ukrainian)
We describe all linear continuous solutions $Tcolon {cal H}(G_1) to {cal H}(G_2)$ of the operator equation $T {cal I /}_{varrho_1,mu_1}= {cal I /}_{varrho_2,mu_2}T$, where ${cal I}_{varrho_i,mu_i}$ is the Gelfond-Leont'ev integration in the space of analytic functions ${cal H}(G_i)$ ($iin{1,2}$).
Subjclass: 47B38.

B. V. Oliynyk. Group of isometries of infinite biotop space. P.205-209. (in Ukrainian)
A group of isometries of the space of biotops $B(X)$ for an arbitrary set $X$ is characterized.
Subjclass: 20F29, 20F38.

D. I. Bodnar, N. P. Hoyenko. Approximation of ratio of Lauricella functions by branched continued fraction. P.210-214. (in Ukrainian)
The convergence of ratio of Lauricella hypergeometric functions $F_D^{(N)}(a,b_1,b_2,dots,b_N;,c;,z_1,z_2,dots,z_N)/ F_D^{(N)}(a+1,b_1+1,b_2,dots,b_N;,c+1;,z_1,z_2,dots,z_N)$ noi expansion into branched continued fraction is investigated for certain restrictions of function parameters. It is shown that the fraction converges to a holomorphic function being analytic continuation of the ratio into the region ${(z_1,ldots,z_N) in {Bbb C}^N: Re z_k<1/2,: 1le k le N }$ label{end}.
Subjclass: 30E10, 40A15.

P. V. Filevych. The Baire categories and Wiman's inequality for entire functions. P.215-221.
By the classical Wiman-Valiron theorem, for every entire function $f$ there exists an increasing to $+i$ sequence $(r_n)$ such that $ln M(r_n,f)-lnmu(r_n,f)leleft(frac12+o(1)right)lnlnmu(r_n,f)$, $ntoi$, where $M$ is the maximum modulus and $m$ is the maximal term of $f$. It is known that in this theorem the constant $frac12$ cannot be replaced by a smaller number. Using the concept of the Baire category, it is shown in the presented paper that for the majority of entire functions the constant $frac12$ may be replaced by $frac14$.
Subjclass: 30B10, 30B20, 30D20.

INDEX, 2003. P.222-224.
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.


Editor-in-Chief:
M.M.Sheremeta (Lviv)

Associate Editors:
M.M.Zarichnyi (Lviv), O.B.Skaskiv (Lviv)


Editorial Board:
T.O.Banakh (Lviv), M.M.Bokalo (Lviv), A.A.Goldberg (Netanya, Israel), R.I.Grigorchuk (Moscow,Russia), I.Yo.Guran (Lviv),Yu.A.Drozd (Kyiv), S.D.Ivasyshen (Chernivtsi), V.V.Kirichenko (Kyiv), M.Ya.Komarnytskyi (Lviv), B.I.Kopytko (Lviv), R.Cauty (Paris, France), S.P.Lavrenyuk (Lviv), O.V.Lopushanskyi (Lviv), Ya.V.Mykytyuk (Lviv), I.V.Ostrovskii (Kharkiv; Ankara, Turkey), O.A.Pankov (Vinnytsia), I.V.Protasov (Kyiv), B.Yo.Ptashnyk (Lviv), V.G.Samoilenko (Kyiv), O.G.Storozh (Lviv), V.I.Sushchansky (Kyiv), S.Yu.Favorov (Kharkiv)


Technical Board:
I.E.Chyzhykov, Ya.Z.Stasyuk

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


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