Showing papers in "Siberian Mathematical Journal in 1987"

Averaging of symmetric diffusion in random medium

Abstract: Let a~j(y, ~)), y ~ B d, ~ , be random fields, homogeneous with respect to integral shifts, let a~j = aj~, i, ]= i~ d and let, with probability I, almost everywhere in R d, the condition hold: A_J~l~ <~ a,j(y, ~o ) ~,~ <~ A+ 8 ~ I ~ ( 0 . 1 ) for any ~{~}~R ~ where A+ are positive nonrandom constanLs; here and throughout, repetition of indices is understood as summation from I to d; i~E 2 = ~i~iWe know [I-3] that operators L~u ~t /2 . D,(a,j(g, o ) Dju)', (0 .2) D~ = O/Ox~, g -= x/e, approximate as s + 0 to an "averaged" operator with constant coefficients

A linear-quadratic optimization problem and the frequency theorem for nonperiodic systems. i

Abstract: Here x ~ R ~, ~ R ~, the matrices A(t), b(t), G(t) = G(t)*, g(t), F(t) = F(t)* ~ ~01 m > 0 have appropriate orders and their elements are real, measurable, bounded, periodic functions with period T. (Equalities and inequalities hold almost everywhere. An asterisk denotes transposition, while in the case of complex vectors and matrices, it denotes Hermitian conjugation, i.e. transposition and complex conjugation.) Below we consider the following two problems. I. The optimization problem x (0) = a t $ = S $ (t, x (t), u (t)) dt--)- rain (0.3) 0 (a is a given vector), while those x(.), u(.) are admissible for which (0.1) and the stability condition Ix(.)l~L~(O,~), fu(.)l~L~(O, ~) hold. 2. The problem of the determination of the form V(t, x) = x*R0(t)x (the Lyapunov function) such that R0(t) = R0(t + T) = R0(t)*, R0(t) is an absolutely continuous matrix function and for the derivative we have, by virtue of Eq. (0.1), the equality

The Grunsky coefficient conditions

Abstract: Starting from [3], several authors have repeatedly posed the following problem (see [6, p. 126; 7] about this): Does the inequality (4) (and those equivalent to it) characterize the class E(k), i.e., is it also sufficient for the k-quasiconformal extendability of f? Pommerenke [2] has established that in each case the inequality (4) ensures the k'-quasiconformal extendability of f with k'~k. Kiihnau [6], using the connection between the Grunsky matrix and the least (nontrivial) Fredholm eigenvalue ~ of the curve f(Izl = I) (found in [8]), has shown that the posed problem Qhas a negative solution, namely: The function

On potentials of measures in Banach spaces

Abstract: in M with respect to the Kobayashi metric, such that the closure of V in M is compact (the existence of such a ball follows from the condition that M is hyperbolic), U be a ball of the same radius in A with respect to the hyperbolic metric. Since a holomorphic map from A to M does not increase the distance with respect to these metrics [7, p. 311], the inequality []/oNrJ~llfl] v is valid (the norms are defined as in Corollary i). It follows from the explicit expressions for the Bloch seminorm (i) and the element of length ((i [zIg)Idz[) that the Lipschitz coefficient coincides with the Bloch seminorm b. Using Banach's open mapping theorem, it is easy to prove that the norms I['II U and b on the space of Bloch functions on A, equal to zero at zero, are equivalent, from which we get the assertion formulated above.

Weakly compact embeddings of symmetric spaces

Abstract: Let E and F be Banach spaces, j:E § F be a continuous embedding of E into F, and (E, F) 0 q be the space constructed by the method of real interpolation. It has been shown in [l]'that (E, F)0, q is a reflexive space for 0 < 8 < 1 and 1 < q < ~ if and only if j is a weakly compact operator, i.e., j transforms each norm-bounded set in E into a set that is relatively compact in F in the o(F, F*)-topology. The following problem is interesting in connection with this statement. For what symmetric spaces E and F is the embedding E c F weakly compact? In the present article, ~ we prove that the embeddings of certain well-known symmetric spaces are weakly compact.

The inverse problem of value-distribution theory for meromorphic functions of finite order

Abstract: In this article we give an almost complete solutiom of the inverse problem of the value distribution theory in the class of meromorphie fcnr of finite order. The problem that corresonds, roughly speaking, to the s~imss of entire functions remains open. i. We use the standard notation of the Nevan1~a theory (see, e.g., [i]). A mero-morphic function, if not stated anything to the comtrary, means afunetionthat ismeromorphic in the finite plane. The following deficiency relation is one of the maim results of the theory of meromorphic functions: For each meromorphic function f a(a,O