The Navier-Stokes system is considered in a plane domain that has several exits to infinity having the form of channels of bounded width. It is assumed that the external force decays sufficiently fast at infinity. Solutions are considered that are defined and bounded for all t is-an-element-of R. Such solutions lie on an attractor of the system. An asymptotic expansion as Absolute value of x --> infinity is obtained for these solutions. The presence of this expansion indicates, in particular, that turbulence in this situation does not propagate to infinity.
Semicontinuous real functions are considered. The following property is established for the Dini directional semiderivative and the Dini semidifferential (the subdifferential). If at some point the semiderivative is positive in a convex cone of directions, then arbitrarily close to the point under consideration there exists a point at which the function is subdifferentiable and has a subgradient belonging to the positively dual cone. This result is used in the theory of the Hamilton-Jacobi equations to prove the equivalence of various types of definitions of generalized solutions.
A special commutative Moufang loop H of order 3(4) is described. With the help of this loop, a trilinear Dickson form is constructed whose automorphism group is a Chevalley group of type E6. Next, with the help of H, a 27-dimensional representation is constructed for O7(3) over Z[zeta] , zeta3 = 1 . This makes it possible to prove anew the embedding O7 (3) subset-or-equal-to 2 E6(2). A similar construction concerning the embedding L4(3) subset-or-equal-to F4(2) is described.
A description is given of situations in which a subspace invariant with respect to a differential operator with constant coefficients admits spectral synthesis. Bibliography: 22 titles.
The author establishes that, for every function f(z) that is analytic inside the unit disk D and belongs to the space L(p)(D) with p > 1, the equation[GRAPHICS]is satisfied, where L(p)E(n)(f, D) and L(p)R(n)(f, D) are the minimal deviations of f from polynomials of degree at most n and from rational functions of order at most n . In particular, rho < 1 if and only if f can be continued analytically over the disk Absolute value of z < 1/rho There is also a similar proposition for the approximation of functions in the spaces H(p), p > 1.
It is shown that if a quasiconformal automorphism f : B(n) --> B(n) of the unit ball in R(n) (n greater-than-or-equal-to 2) has coefficient of quasiconformality K(f)(r) = sup(Absolute value of x less-than-or-equal-to r) k(f, x) in the ball of radius r < 1 with asymptotic growth such that integral1 K(f)(r) dr < infinity, then it has a radial limit at almost every point of the boundary. This asymptotic growth of K(f)(r) is sharp in a certain sense.
Random walks in a random environment are considered on the set Z of integers when the moving particle can go at most R steps to the right and at most L steps to the left in a unit of time. The transition probabilities for such a random walk from a point x is-an-element-of Z are determined by the vector P(x) is-an-element-of R(R+L+1). It is assumed that the sequence {p(x), x is-an-element-of Z) is a sequence of independent identically distributed random vectors. Asymptotic properties with probability 1 are investigated for such a random process. An invariance principle and the law of the iterated logarithm for a product of independent random matrices are proved as auxiliary results.
The concept of oriented degree is extended to the class of mappings of the form f - g, where f is a proper Fredholm mapping of nonnegative index and g a continuous f-compactly restrictable mapping. In the case when f is a Fredholm mapping of zero index and f and g are equivariant with respect to the action of the circle and the toms, formulas are obtained which express the degree of these mappings in terms of invariants of representations of the corresponding groups. An application to the investigation of the global behavior of a bifurcation branch of a certain nonlinear boundary value problem is given.
Steady-state and time-dependent problems are studied for the equation partial derivative(t)u + PI(del(u)u) = - sigmau + f, where u is-an-element-of TM, M is a two-dimensional closed manifold, and PI is the projection onto the subspace of solenoidal vector fields that admit a single-valued flow function. Existence of steady-state solutions is proved. For the evolution problem Lyapunov stability of the zero solution in Sobolev-Liouville spaces is proved by the method of vanishing viscosity. The existence of generalized weak (PIW2k1, PIW2kw1) attractors, k > 1 an integer, is proved. A *-weak (L(infinity), L(infinity*-w) attractor is constructed in the phase space L(infinity) for the velocity vortex equation.
The set of Lebesgue points of a locally integrable function on N-dimensional Euclidean space R(N), N greater-than-or-equal-to 1 , is an F(sigmadelta)-set of full measure. In this article it is shown that every F(sigmadelta)-set of full measure is the set of Lebesgue points of some measurable bounded function, and, further, that a set with these properties is the set of points of convergence and nontangential (stable) convergence of a singular integral of convolution type: phi(epsilon)*f(x), phi(epsilon)(t) = epsilon(-N) phi(t/epsilon) is-an-element-of L(R(N)), epsilon --> +0, for some measurable bounded function f . On the basis of this result the set of points of summability of a multiple Fourier series by methods of Abel, Riesz, and Picard types is described.
An investigation is made of the geometry of the multiplication mappings muX for monads T = (T, eta, mu) whose functorial parts are (weakly) normal (in the sense of Shchepin) functors acting in the category of compacta. A characterization is obtained for a power monad as the only normal monad such that the multiplication mapping muI(tau) is soft for some tau > omega1. It is proved that the multiplication mappings mu(G)X and mu(N)X of the inclusion hyperspace monad and the monad of complete chained systems are homeomorphic to trivial Tychonoff fibrations for openly generated continua that are homogeneous with respect to character.
Let phi(i) be continuous mappings of a compactum X onto compacta Y(i), i = 1, ... , n. The following theorem is known for n = 2: if any bounded function f on X can be represented in the form f = g1 . phi1 + g2 . phi2, where g1 and g2 are bounded functions on Y1 and Y2, then any continuous f can be represented in the same form with continuous g1 and g2 . An example is constructed showing that the analogous theorem is false for n > 2.
A semigroup is called filtering if each of its subsemigroups has the smallest (with respect to inclusion) generating set. It is proved in this article that every maximal chain of nonempty subsemigroups of a finite filtering semigroup has length equal to the order of the semigroup, and that filtering semigroups are characterized by this property in the class of finite semigroups. The main result is a characterization of the class of finite filtering semigroups by means of forbidden divisors, to which end the author finds all finite nonfiltering semigroups all of whose proper divisors are filtering semigroups.
This is a detailed study of the problem of the existence and characterization of finite-dimensional Chebyshev subspaces of the spaces phi(L) and L(p(t)) on the interval I = [-1, 1], where phi(t) is an even nonnegative continuous nondecreasing function on the half-fine [0, +infinity) , and the function p(t) is measurable, finite, and positive almost everywhere on I. If phi is an N-function, it is characterized as a Chebyshev subspace of the Orlicz spaces with the Luxemburg norm.
It is determined under what conditions the standard problem of extension of a mapping to the whole space is solvable for any closed subset . For finite-dimensional metric compacta and -complexes this is equivalent to the system of inequalities -. The result is applied to finding conditions for general position of a compactum in a Euclidean space.
The famous Steiner problem in the Euclidean plane, which is that of investigating minimal nets spanning fixed finite subsets M of points in the plane, is solved when M is extremal, i.e. when M lies on the boundary of its convex hull, and the nets are nondegenerate, i.e. have no vertices of degree 2.
After giving a brief introduction to our new ''theory of dual systems of integer vectors'', we give the first applications to the theory of positive quadratic forms. We consider the question of enumerating the L-polytopes of lattices, paying particular attention to the case of five-dimensional lattices. The results reported in this paper were announced earlier by the authors in Doklady [1]; here we give the details.
With denoting the error of best uniform rational approximation from to on , we determine the numbers , where each of these numbers was calculated with a precision of at least 200 significant digits. With these numbers, the Richardson extrapolation method was applied to the products , and it appears, to at least 10 significant digits, that which gives rise to an interesting new conjecture in the theory of rational approximation.
The concept of mean dimension is introduced for a broad class of subspaces of L(p)(R) , and analogues of the Kolmogorov widths, Bernstein widths, Gel'fand widths, and linear widths are defined. The precise values of these quantities are computed for Sobolev classes of functions on R in compatible metrics, and the corresponding extremal spaces and operators are described. A closely related problem of optimal recovery of functions in Sobolev classes is also studied.
An infinitesimal description is obtained for fractional iterates of analytic functions on the unit disk under the condition that the functions and their iterates do not move fixed points on the unit circle at which they have finite angular derivatives.
