On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane of the spectrum of the operator , generated by the closure in of the operation originally defined on smooth functions with values in a Hilbert space satisfying the Dirichlet conditions . Here and A is a model operator acting in . Criterial conditions on the parameter for the eigenfunctions of the operator to form a complete and minimal system as well as a Riesz basis in the Hilbert space H are given.     

On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions

We obtain sufficient conditions for the continuity of the general nonlinear superposition operator (generalized Nemytskii operator) acting from the space of differentiable functions on a bounded domain to the Lebesgue space . The values of operators on a function are locally determined by the values of both the function itself and all of its partial derivatives up to order inclusive. In certain particular cases, the sufficient conditions obtained are proved to be necessary as well. The results are illustrated by several examples, and an application to the theory of Sobolev spaces is also given.     

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 .     

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.     

Joint Approximations of Distributions in Banach Spaces

For a given homogeneous elliptic partial differential operator with constant complex coefficients, two Banach spaces and of distributions in , and compact sets and in , we study joint approximations in the norms of the spaces and (the spaces of Whitney jet-distributions) by the solutions of the equation in neighborhoods of the set . We obtain a localization theorem, which, under certain conditions, allows one to reduce the above-cited approximation problem to the corresponding separate problems in each of the spaces.     

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      

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.     

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 .     

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 .     

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).     

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.     

Some Generalizations of the Notion of Length

Two numerical characteristics of a nonrectifiable arc generalizing the notion of length are introduced. Geometrically, this notion can naturally be generalized as the least upper bound of the sums , where are the lengths of segments of a polygonal line inscribed in the curve and is a given function. On the other hand, the length of is the norm of the functional in the space ; its norms in other spaces can be considered as analytical generalizations of length. In this paper, we establish conditions under which the generalized geometric rectifiability of a curve implies its generalized analytic rectifiability.     

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.     

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 .     

Cotangent Bundle over Projective Space and the Manifold of Nondegenerate Null-Pairs

A nondegenerate null-pair of the real projective space consists of a point and of a hyperplane nonincident to this point. The manifold of all nondegenerate null-pairs carries a natural Kählerian structure of hyperbolic type and of constant nonzero holomorphic sectional curvature. In particular, is a symplectic manifold. We prove that is endowed with the structure of a fiber bundle over the projective space , whose typical fiber is an affine space. The vector space associated to a fiber of the bundle is naturally isomorphic to the cotangent space to . We also construct a global section of this bundle; this allows us to construct a diffeomorphism between the manifold of nondegenerate null-pairs and the cotangent bundle over the projective space. The main statement of the paper asserts that the explicit diffeomorphism is a symplectomorphism of the natural symplectic structure on to the canonical symplectic structure on .     

A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in S 3

We study a version of the Gauss map for a surface immersed in and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff has a constant mean curvature. As a corollary, we conclude that an embedded flat torus with constant mean curvature is a spherical Delonay surface.     

A Remark on Subgroup Separability in the Class of Finite π-Groups

We prove that if a group G is residually , then for every -subgroup of the group G, the set of -roots from this subgroup is a -separable -subgroup.     

On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum

We study trigonometric sums in finite fields . The Weil estimate of such sums is well known: , where f is a polynomial with coefficients from F(Q). We construct two classes of polynomials f, , for which attains the largest possible value and, in particular, .     

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.