Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

New Proof of the Semmes Inequality for the Derivative of the Rational Function

In the open disk of the complex plane, we consider the following spaces of functions: the Bloch space ; the Hardy--Sobolev space ; and the Hardy--Besov space . It is shown that if all the poles of the rational function R of degree n, , lie in the domain , then , where and depends only on . The second of these inequalities for the case of the half-plane was obtained by Semmes in 1984. The proof given by Semmes was based on the use of Hankel operators, while our proof uses the special integral representation of rational functions.     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .     

Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

Dehornoy constructed a right invariant order on the braid group B _n uniquely defined by the condition 1{\text{ if }}\beta _0 ,\beta _1$$ " align="middle" border="0"> are words in . A braid is called strongly positive if 1$$ " align="middle" border="0"> for any . In the present paper it is proved that the braid is strongly positive if the word does not contain . We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.     

The Multidimensional Weyl Theorem and Covering Families

The well-known theorem of Weyl about the essential self-adjointness of the Sturm--Liouville operator in with , and is generalized to second-order elliptic operators in . The multidimensional Weyl theorem is derived from a more general theorem; to state and prove the latter, a special covering family is constructed. The results obtained imply the known multidimensional analogs of the Weyl theorem and, unlike these analogs, apply to open proper subsets in .     

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

Representability of Analytic Functions in Terms of Their Boundary Values

Suppose that is an arbitrary finite complex Borel measure on the interval is its Poisson integral, and are the conjugate harmonics of , and is the nontangential limiting value of the analytic function as . In this paper, we consider the problem of representing the analytic function in terms of its boundary values .     

On the Growth of Entire Functions of Exponential Type near a Straight Line

Let be a sequence of complex numbers such that as . For close to the imaginary axis, upper bounds of the indicator of a nonzero entire function of exponential type with minimal growth vanishing on is obtained. These bounds give sufficient conditions for the system of exponents to be incomplete in an unbounded domain in .     

Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series and whose coefficients satisfy the condition for , where the sequence can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If as , then the series is uniformly convergent. If for all , then the sequence of partial sums of this series is uniformly bounded. If the series is convergent for and as , then this series is uniformly convergent. If the sequence of partial sums of the series for is bounded and for all , then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence is lacunary. In the general case, they are not necessary.     

Augmentation and Modification Problems for Hermitian Matrices

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian -matrix with a given -block in block -representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian -matrix of rank not greater than so that the obtained matrix, being added to a given Hermitian -matrix , will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems.     

Norm Estimates for Multiplication Operators in Hilbert Algebras

In this paper, it is proved that for the bilinear operator defined by the operation of multiplication in an arbitrary associative algebra with unit over the fields or , the infimum of its norms with respect to all scalar products in this algebra (with ) is either infinite or at most . Sufficient conditions for this bound to be not less than are obtained. The finiteness of this bound for infinite-dimensional Grassmann algebras was first proved by Kupsh and Smolyanov (this was used for constructing a functional representation for Fock superalgebras).     

The Ingham Divisor Problem on the Set of Numbers without kth Powers

Suppose that k and l are integers such that and , M _k is a set of numbers without kth powers, and . In this paper, we obtain asymptotic estimates of the sums over      

On the Similarity of Some Differential Operators to Self-Adjoint Ones

The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form , in the space with weight . As is well known, the answer to this problem in the case is positive; it was obtained by using delicate methods of the theory of Hilbert spaces with indefinite metric. The use of a general similarity criterion in combination with methods of perturbation theory for differential operators allows us to generalize this result to a much wider class of weight functions .     

Implementation of the Sparr Interpolation Method in Classes of Spaces of Smooth Functions

The multidimensional Sparr interpolation method is implemented in the Besov spaces and the Lizorkin--Triebel spaces . It is shown that this results in Besov spaces of type . An interpolation theorem for Besov spaces using weak conditions of the form is formulated.     

On a Characterization of Spaces of Differentiable Functions

In this paper, we generalize Bernstein's theorem characterizing the space by means of local approximations. The closed interval is partitioned into disjoint half-intervals on which best approximation polynomials of degree divided by the lengths of these half-intervals taken to the power are considered. The existence of the limits of these ratios as the lengths of the half-intervals tend to zero is a criterion for the existence of the th derivative of a function. We prove the theorem in a stronger form and extend it to the spaces .     

Morrey Regularity of Nonlinear Elliptic Systems of High Order under Degeneration of Ellipticity

We study nonlinear elliptic systems of the form , even, , with the natural energy space . We establish that for solutions from belong to the Morrey space and the Morrey exponent does not tend to zero under the degeneration of ellipticity. In the case , a similar result is obtained under an additional structure condition on the system.     

Bases in Sobolev Spaces on Bounded Domains with Lipschitzian Boundary

In the Sobolev space , where is a bounded domain in ⁿ with a Lipschitzian boundary, for an arbitrarily given , we construct a basis such that the error of approximation of a function the Nth partial sum of its expansion with respect to this basis can be estimated in terms of the modulus of smoothness of order .     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

The Order of Best Approximation in Some Classes of Functions

Starting from the equivalence between the Ditzian–Totik modulus and , where , in this article large classes of functions are introduced for which the modulus can be easily calculated. As a consequence, very good estimates for the bestapproximation are obtained. The attempts to estimate or calculate themodulus can be a very intricateproblem.     

Intersections of Convex Bodies

Let be nonempty convex bodies in . Let be vectors in , let , and let . Then is a convex set, and the family of sets is concave. Let . Then for the mean cross-sectional measures W_v (\Phi (\rho )), , the functions are concave on D. (Note that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca% WGxbWaaSbaaSqaaiaaicdaaeqaaOGaaiikaiabfA6agjaacIcacqaH% bpGCcaGGPaGaaiykaiabg2da9iaabAfacaqGVbGaaeiBamaaBaaale% aatCvAUfKttLearyqr1ngBPrgaiuGacqWFRbWAaeqaaOGaeuOPdyKa% aiikaiabeg8aYjaacMcaaaa!4EE7!\[W_0 (\Phi (\rho )) = {\text{Vol}}_k\Phi (\rho )\] is the k-volume.) Bibliography: 2 titles.