2017
Том 69
№ 4

All Issues

Volume 62, № 11, 2010

Article (English)

Characterization of $A_{16}$ by a noncommuting graph

Darafsheh M. R., Davoudi Monfared M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1443–1450

Let $G$ be a finite non-Abelian group. We define a graph $Γ_G$ ; called the noncommuting graph of $G$; with a vertex set $G − Z(G)$ such that two vertices $x$ and $y$ are adjacent if and only if $xy ≠ yx$. Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If $S$ is a finite non-Abelian simple group and $G$ is a group such that $Γ_S ≅ Γ_G$; then $S ≅ G$. It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except $A_{10}, L_4(8), L_4(4)$, and $U_4(4)$. In this paper, we prove that if $A_{16}$ denotes the alternating group of degree 16; then, for any finite group $G$; the graph isomorphism $Γ_{A_{16}} ≅ Γ_G$ implies that $A_{16} ≅ G$.

Article (Russian)

On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations

Ignat'ev A. O.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1451–1458

A system of linear differential equations with pulse action at fixed times is considered. We obtain sufficient conditions for the existence of a positive-definite quadratic form whose derivative along the solutions of differential equations and whose variation at the points of pulse action are negative-definite quadratic forms regardless of the times of pulse action.

Article (Ukrainian)

Regular orthoscalar representations of extended dynkin graphs $\widetilde{E}_6$ and $\widetilde{E}_7$ and *-algebras associatedwith them

Livins'kyi I. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1459–1472

We obtain a classification of indecomposable orthoscalar representations of the extended Dynkin graphs $\widetilde{E}_6$ and $\widetilde{E}_7$ with a special character and of the *-algebras associated with them, up to the unitary equivalence.

Article (Ukrainian)

Principle of localization of solutions of the Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type

Litovchenko V. A., Strybko O. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1473–1489

In the case where initial data are generalized functions of the Gevrey-distribution type for which the classical notion of equality of two functions on a set is well defined, we establish the principle of local strengthening of the convergence of a solution of the Cauchy problem to its limit value as $t → +0$ for one class of degenerate parabolic equations of the Kolmogorov type with $\overrightarrow{2b}-$parabolic part whose coefficients are continuous functions that depend only on $t$.

Article (Russian)

Integral inequalities and stability of an equilibrium state on a time scale

Luk’yanova T. A., Martynyuk A. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1490–1499

We present some integral inequalities on a time scale and establish sufficient conditions for the uniform stability of an equilibrium state of a nonlinear system on a time scale.

Article (Russian)

Metric properties of functions defined by partial automata

Nekrashevich V. V., Oliinyk A. S., Sushchanskii V. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1500–1510

We characterize natural categories in which morphisms are defined by partial automata of the following three types: asynchronous automata, window automata, and automata synchronous over finite alphabets. We distinguish subcategories whose morphisms are defined by finite automata.

Article (Russian)

Riemann boundary-value problem on an open rectifiable jordan curve. I

Kud'yavina Yu. V., Plaksa S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1511–1522

The Riemann boundary-value problem is solved for the classes of open rectifiable Jordan curves extended as compared with previous results and functions defined on these curves.

Article (English)

On the asymptotic extension dimension

Repovš D., Zarichnyi M. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1523–1530

We introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes the relationship between the asymptotic extensional dimension of a proper metric space and the extension dimension of its Higson corona.

Article (Russian)

Estimation of dilatations for mappings more general than quasiregular mappings

Salimov R. R., Sevost'yanov E. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1531–1537

We consider the so-called ring $Q$-mappings, which are natural generalizations of quasiregular mappings in a sense of Väisälä’s geometric definition of moduli. It is shown that, under the condition of nondegeneracy of these mappings, their inner dilatation is majorized by a function $Q(x)$ to within a constant depending solely on the dimension of the space.

Article (English)

Well-posed reduction formulas for the $q$-Kampé-de-Fériet function

Chu W., Zhang W.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1538–1554

By using the limiting case of Watson’s $q$-Whipple transformation as $n → ∞$, we investigate the transformations of the nonterminating $q$-Kampé-de-Fériet series. Further, new formulas for the transformations and well-posed reduction formulas are established for the basic Clausen hypergeometric series. Several remarkable formulas are also found for new function classes beyond the $q$-Kampé-de-Fériet function.

Brief Communications (English)

A note on invariant submanifolds of $(k, μ)$-contact manifolds

Avik De

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1555–1560

The object of the present paper is to study invariant submanifolds of a $(k, μ)$-contact manifold and to find the necessary and sufficient conditions for an invariant submanifold of a $(k, μ)$-contact manifold to be totally geodesic.

Brief Communications (Ukrainian)

Analog of the Cayley–Sylvester formula and the Poincaré series for an algebra of invariants of ternary form

Bedratyuk L. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1561–1570

An explicit formula is obtained for the number $ν_d(n)$ of linearly independent homogeneous invariants of degree $n$ of a ternary form of order $d$. A formula for the Poincaré series of the algebra of invariants of the ternary form is also deduced.

Brief Communications (Ukrainian)

Nonlinear equations with essentially infinite-dimensional differential operators

Bogdanskii Yu. V., Statkevych V. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1571–1576

We consider nonlinear differential equations and boundary-value problems with essentially infinite-dimensional operators (of the Laplace–Lévy type). An analog of the Picard theorem is proved.

Brief Communications (Ukrainian)

Fading evolutions in multidimensional spaces

Pogorui A. О.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1577–1582

We study fading random evolutions in multidimensional spaces. By reducing multidimensional cases to the one-dimensional case, we calculate the limit distributions of fading evolutions for some semi-Markov media.

Brief Communications (English)

Characterization of $M_{11}$ and $L_3(3)$ by their commuting graphs

Salarian M. R.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1583–1584

For the groups $M_{11}$ and $L_3(3)$, we show that their commuting graphs are unique.