Following ideas of A.N. Kolmogorov we consider stochastic sequences as elements of an abstract complex Hilbert space. We study classes of sequences connected with systems of orthogonal polynomials on the real line and on the unit circle (polynomial and P-stationary sequences). A notion of a canonical integral representation for an arbitrary sequence in a Hilbert space is introduced. Using methods of the theory of operators and the theory of orthogonal polynomials we obtain some results which are analogous to the results of the theory of stationary sequences (difference equations for the correlation function, a series expansion of the spectral function of the sequence, laws of large numbers and other). 2000 Mathematics Subject Classification: 42C05, 33C45, 60G. [ Full-text available (PDF) ] Top of the page.
In this paper the controllability problem for the wave equation controlled by the boundary conditions is investigated on a segment. Bang-bang controls solving approximate null-controllability problem of this equation, are built with the aid of the solutions of the Markov trigonometric moment problem. Moreover, the error of computation and the accuracy of approximation for the end state of the control system are estimated. 2000 Mathematics Subject Classification: 93B05, 35B37, 35L05. [ Full-text available (PDF) ] Top of the page.
We introduce a class of $S$-sets in $\mathbb{R}^k$. This is a special class of discrete sets that generalizes the class of discrete periodic sets. We investigate the properties of $S$-sets. In particular, we prove that the class of $S$-sets coincides with the class of uniformly spread discrete sets in the sense of Laczkovich. 2000 Mathematics Subject Classification: 28E99, 03E99. [ Full-text available (PDF) ] Top of the page.
The existence of the asymptotics of the basic function $\varphi_{n,0}$ and $\psi_{n,\alpha_n}$ of a generalized Taylor series for the $H_{\rho,2}$ function class is proved. The first term of the asymptotic expansions of these functions is obtained. 2000 Mathematics Subject Classification: 42A70. [ Full-text available (PDF) ] Top of the page.
In this article possibility of a substantiation of the theorem of continuous dependence of decisions on parameter and themethod of averaging for the differential equations and inclusions from derivative Hukuhara in a case when the right part does not satisfy the Lipshitz condition on a phase variable is considered. 2000 Mathematics Subject Classification: 34A60, 34C29. [ Full-text available (PDF) ] Top of the page.
For the functional model of nondissipative operator, the Lax -- Fillips scattering scheme is constructed. The wave operators and the scattering operator are calculated. 2000 Mathematics Subject Classification:47A40. [ Full-text available (PDF) ] Top of the page.
In this paper the degenerate generalized interpolation problem has been considered for Stieltjes functions. A new method is developed for the reduction of the degenerate problems to the complete indeterminate of interpolation problems. The all solution of the degenerate interpolation problem is described in terms of a linear fractional transformations. 2000 Mathematics Subject Classification: 41A05, 30E05. [ Full-text available (PDF) ] Top of the page.
Let $a$ be a quasinilpotent element of a Banach algebra and $f(\zeta)= \sum\limits_{n=0}^\infty c_n \zeta^n$ a formal power series over $\mathbb{C}$. Some analogs of the Taylor expansion $f(a+h)=\sum_{n=0}^\infty \frac{1}{n!}f^{(n)}(a)h^n$ are obtained. 2000 Mathematics Subject Classification: 4630, 47A60. [ Full-text available (PDF) ] Top of the page.
We study filters $\F$ for which the equality $LPR=SR_\F$ holds. We characterize filters $\F$ for which every $\F$-convergent series has a null subsequence. This property with the one which we call the unbounded gap property of $\F$ imply that $LPR=SR_\F$. The unbounded gap property of $\F$ (existence of $A\in\F$, $A=(a_n)$ such that the sequence of $g_n=a_{n+1}-a_n$ is unbounded) is a necessary condition for this equality. The equality does not hold for any ultrafilter. 2000 Mathematics Subject Classification: 40A05, 54A20. [ Full-text available (PDF) ] Top of the page.
The set of the nonisotropic planes of the pseudo-Euclidean space of index 1 is studied. The smooth structure is introduced. The metric is defined, the kind of metric in the local metric is found. The expressions for the Christoffel symbols of the first and the second kind and the equation of the geodesic lines are obtained. 2000 Mathematics Subject Classification: 58A05. [ Full-text available (PDF) ] Top of the page.
The system of boundary integral equations for materials with interface cracks under dynamic loading is presented. The expressions for singular integral kernels are given for the case of a plane interface and harmonic loading. The fracture mechanics parameters for the interface penny-shaped crack under longitudinal and shear waves are computed by the method of boundary elements. 2000 Mathematics Subject Classification: 74R10, 74M99, 45E99. [ Full-text available (PDF) ] Top of the page.
Wave processes in two-layer-system of immiscible current-carrying liquids which fill the cylindrical tank with rectangular cross-section are studied. It is shown that the equilibrium state corresponded to the flat interface of liquids and uniform current distribution can become unstable. This instability reveals itself in appearance of interfacial waves under certain conditions. On a base of Galerkin's approximations for governing equations of magnetohydrodynamics, calculations of stability region boundary in the space of dimensionless parameters of the system are carried out. The influence of basic parameters on wave-generation thresholds is clarified. 2000 Mathematics Subject Classification: 76W05. [ Full-text available (PDF) ] Top of the page.
For a metric space $K$ we define by $D(K)$ the space of all functions on $K$, whose modulus of continuity is satisfied the Dini condition at 0. We prove that $D(K)$ is dual if $K$ is compact and $D(K)$ is separable if $K$ is a convex compact subset of a Banach space. Other properties are studied for the special case when $K=[0,1]$. 2000 Mathematics Subject Classification: 42A70. [ Full-text available (PDF) ] Top of the page.
We study the effects which appear when the selection of $\pm 1$ coefficient in the definition of unconditional convergence is replaced by selection of operators from a fixed collection of operators. The analogues of the theory of cotype and the famous Kadets's theorem about unconditionally convergent series in uniform convex space are obtained. 2000 Mathematics Subject Classification: 46B20. [ Full-text available (PDF) ] Top of the page.