# Volume 62, № 11, 2010

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

Darafsheh M. R., Davoudi Monfared M.

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 $A_{16} ≅ G$.

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

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.

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

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.

### 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.

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 $t$.

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

Luk’yanova T. A., Martynyuk A. A.

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.

### Metric properties of functions defined by partial automata

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

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.

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

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

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.

### On the asymptotic extension dimension

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.

### Estimation of dilatations for mappings more general than quasiregular mappings

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

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.

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

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.

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

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.

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

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.

### Nonlinear equations with essentially infinite-dimensional differential operators

Bogdanskii Yu. V., Statkevych V. M.

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.

### Fading evolutions in multidimensional spaces

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.

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

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.