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Vol. 24 (63), No. 3. 2011

Physics and Mathematics Sciences.

Astakhova T.N., Zuev A.L.
Stabilization of Nonlinear Systems in the Class of Control Functions with Discrete Switching
A problem on the construction of a control function that ensures the existence of a limit point x = 0 for all solutions x(t) of a nonlinear system is considered. A local approach to the solvability of this problem is proposed in the class of sampled controls with switching at discrete time instants. The result obtained is applied for a nonholonomic model with constraints on the control. Simulation results are presented.

Artamonov S.Yu.
On Some Properties of Modulus of Continuity Related to Riesz Derivative
A modulus of continuity related to the Riesz derivative is constructed. Its properties are studied in the spaces Lp of periodic functions with 1 ≤ p ≤ ∞. The direct Jackson type estimate, the inverse Bernstein type estimate and equivalence to the K-functional related to the Riesz derivative are proved.

Karpenko I.I., Goncharenko A.M.
In this paper it’s considered problems of simultaneous diagonalization for quaternionic matrixes which are a self-adjoint concerning the non-Hermitian involution in the real algebra of quaternions. Some criterions of unitary and nonsingular diagonalizations for such matrixes are received.

Kudryashov Yu.L.
Isomorphism of Two Presentations of Self-Conjugate Dilatation of Dissipative Operator
Spectral presentation of self-conjugate dilatation of dissipative operator and presentation is in-process examined only for the limited dissipative operator. In the case of the limited operator the isomorphism of the indicated presentations of dilatation is directly built.

Kuzmenko E.M.
Conditions of Compact Differentiability and Repeated Compact Differentiability of Variational Functionals in Sobolev Spaces W1,p of Functions of Several Variables
Weierstrass classes W1Kp(z) and W2Kp(z) which have been previously researched for the occasion of Sobolev space W1,2 over segment are introduced for the integrands f (x, y, z) of variational functionals ∫D f (x, y, y')dx, acting within of Sobolev space W1,p (D); p ≥ 1 over the compact area D ⊂ Rn . Belonging the integrant to the correspondent Weierstrass class is shown to guarantee compact differentiability of correspondent order for the variational functional.

Pogrebitskaya A.M., Smirnova S.I.
An Analytical Estimate of the Double Hybrid VKB-Galerkin Solution for the Nonlinear Homogeneous Differential Equation with the Variable Coefficients
In this paper the analytical estimate of an double hybrid WKB-Galerkin solution for the nonlinear second order differential equation, arising in some mathematical physics problems, is presented. Asymptotic character of  corresponding hybrid solution is proved.

Romanenko I.A.
Fundamental Systems of Compacta in Integral Spaces
Description of appropriate fundamental systems of compacta in general integral spaces Lp and Sobolev spaces Wn;p of functions of one variable is given. The properties of scales of subspaces generated by the fundamental systems of compacta were researched.

Statkevich V.M.
Cramer’s Rule Analog for Simultaneous Linear Differential Equations with Nonregular Elliptic Operator
Simultaneous linear differential equations for functions on an infinite-dimensional Hilbert space with nonregular elliptic operator (Lu)(x) = j(u'' (x)) polynomial coefficients are proposed. Cramer’s rule for such simultaneous equations is proved.

Stonyakin Ph.S.
About Differentiability of Indefinite Pettis Integral by Upper Limit
In this paper two new properties for Pettis integrable mappings acting from a real segment into Frechet spaces are investigated: almost everywhere weak integral boundedness and σ-compact measurability. The sufficient condition for differentiability of indefinite Pettis integral in terms of almost everywhere weak integral boundedness is obtained. The necessary condition for differentiability of indefinite Pettis integral in terms of σ-compact measurability is proved.

Khalilova Z.I.
K - Sublinear Multivalued Operators and their Properties
The sublinear multivalued operators with compact convex values are studied in the work . It’s shown that in the case of Banach spaces such operators form a ordered Banaсh cone.