Table of contents

Decompositions of spectra corresponding to cobordism theories with singularities are examined, and also relations between the Adams and Adams-Novikov spectral sequences and spectral sequences of singularities. A multiplicative variant of the indecomposable spectrum BoP is constructed, this being a direct summand in the splitting of the spectrum MSU of special unitary cobordism theory. Bibliography: 14 titles.

Let f be a bounded holomorphic function in the region of a capacitor. An estimate is given for the oscillation of the components of f on continua belonging to the plates of the capacitor. Bibliography: 4 titles.

A potential is constructed for the Weil-Petersson metric on the Teichmüller space of marked Riemann surfaces of genus in terms of the density of the Poincaré metric on the region of discontinuity of the corresponding normalized marked Schottky group. It is proved that the difference between the projective connections corresponding to the Fuchsian uniformization and the Schottky uniformization for a marked Riemann surface of genus is the -derivative of this potential, and the Weil-Petersson symplectic form on Teichmüller space is the -derivative of the Fuchsian projective connection. The results establish how the accessory parameters of the Fuchsian uniformization and the Schottky uniformization of a Riemann surface are connected with the geometries of Teichmüller space and Schottky space. Bibliography: 31 titles.

The author analyzes the behavior as of the fundamental solution of the Cauchy problem for the equation with infinitely differentiable coefficients and decreasing as . For the case when the functions and can be expanded as in asymptotic series of the form

where , as , , , she constructs and justifies asymptotic expansion of the fundamental solution to within any power of uniformly with respect to all and in . Bibliography: 12 titles.

It is proved that there exist integers such that the system of congruences

where denotes the exponent of the highest power of 2 dividing , is solvable in integers only if the necessary condition holds, where

From this the estimate is derived for the number of terms in the Hilbert-Kamke problem. Combined with a result from the previous paper, this gives the formula for . Bibliography: 5 titles.

Let be a convex domain and a convex compact set in ; let be the space of analytic functions in , provided with the topology of uniform convergence on compact sets, and the space of germs of analytic functions on with the natural inductive limit topology; and let be the space dual to . Each functional generates a convolution operator , , , which acts continuously from into . Further let be the Fourier-Borel transform of the functional . In this paper the following theorem is proved: Theorem.Let be a bounded convex domain in with boundary of class or , where the are bounded planar convex domains with boundaries of class , and let . In order that it is necessary and sufficient that 1) for all , and 2) be a function of completely regular growth in in the sense of weak convergence in . Here

is the regularized radial indicator of the entire function , and is the support function of the compact set . Bibliography: 29 titles.

The article consists of three sections. In §1, relations among the stationary subgroups are proved, and a method of computing from the structure of the algebra of covariants is presented. §2 contains a proof of a reduction theorem for covariants. In §3, some examples are collected and some consideration given to the connection between the algebra of covariants and the algebra of invariants . Bibliography: 16 titles.

The main result is Theorem 1.If is a holomorphic function on the polydisk in , and for each fixed in some nonpluripolar set the function can be continued holomorphically to the whole plane with the exception of some polar set of singularities, then can be continued holomorphically to , where is a closed pluripolar subset of . Some generalizations are also given, along with corollaries on extension of functions with analytic sets of singularities. Bibliography: 13 titles.

This paper investigates rational approximations of a Markov function that have the highest order of contact with it at infinity, and whose denominators are invariant under multiplication of their argument by a root of unity of some fixed degree (such approximations are used in many number-theoretical problems). The approximations converge under mild restrictions on the measure. Moreover, the denominators of the approximants and the corresponding functions of the second kind have logarithmic asymptotics expressible in terms of a certain extremal measure which, in the simplest case, is the Tchebycheff measure. An explicit form is found for the extremal measure in the general case; in fact, the inverse of the distribution function is expressed in terms of elementary functions, the power moments are calculated, and the Markov function of the extremal measure is connected with algebraic equations and generalized hypergeometric functions. Bibliography: 10 titles.

The author establishes the regularity of maps from Riemannian manifolds of small dimensions into certain homogeneous spaces with metrics near to the invariant metrics that locally minimize the multidimensional Dirichlet functional (energy). Bibliography: 21 titles.

The article provides a construction of an infinite-dimensional compact space of dimension 2 modulo p for any p. A characterization of compact spaces n-dimensional modulo p in terms of inverse spectrum of polyhedra is given. It is proved that compact spaces n-dimensional modulo p, and only these spaces, are images of n-dimensional compact spaces under maps acyclic in the sense of cohomology with coefficients in Z_p. Bibliography: 18 titles.

Let , , be a function of bounded variation on the line. This paper investigates whether convolutions of the form , , are uniquely determined from their values on the semiaxis . As a corollary to one of the results a conjecture of Kruglov is proved: if is a distribution function, is the standard normal distribution function, and , , then the equality

implies that . Bibliography: 11 titles.

The authors study equations of the form

where the and are elements in some Hilbert space , and the are bounded linear operators on . It is assumed that the corresponding operator symbol

is a holomorphic Fredholm operator-valued function which is normal in some neighborhood of zero. Bibliography: 9 titles.

The system of equations

is considered with initial data in the form of a wave packet of small amplitude

The asymptotics of the solution as which is uniform in the strip , is constructed by the method of multiscale expansions. The coefficients of the asymptotics are a system of wave packets traveling with group velocities; the leading term is determined from a system of nonlinear equations of Schrödinger type. Bibliography: 32 titles.

A criterion for the validity of the -Liouville theorem is proved. In §1 it is shown that the question of - and -Liouville theorems reduces to the study of the so-called massive sets (in other words, the level sets of harmonic functions in the classes and ). In §2 some properties of capacity are presented. In §3 the criterion of -massiveness is formulated — the central result of this article — and examples are presented. In §4 a criterion for the -Liouville theorem is formulated, and corollaries are derived. In §§5-9 the main theorems are proved. Figures: 5. Bibliography: 17 titles.

A mixed boundary value problem in a perforated domain is considered for the system of linear elasticity theory with nonuniformly oscillating coefficients. The coefficients of the system depend on fast and slow independent variables and are periodic functions of fast variables with period ε. For small ε, estimates are obtained for the difference between the eigenvalues of this problem and the eigenvalues of the corresponding averaged problem. Estimates of the same kind are obtained for eigenfunctions. The methods worked out in the paper are general, and they are applicable to a large class of problems on averaging of eigenvalues for equations and systems of elliptic type. Bibliography: 5 titles.

The structure of separable closed braids is described. It is proved that for every separable braid with

n + 1

threads and length of expression d one can indicate an equivalent separated closed braid whose length of expression does not exceed

(d + 2)(n + 1)n

. Bibliography: 3 titles.

Let be a representation space of a Lie group , a differential form of the first degree on and a field of closed convex cones on . Problem 1 is the minimization of the integral of the differential form along curves which satisfy certain boundary conditions and are solutions of the differential inclusion . This problem is assumed to be equivariant in the sense that the field of cones and the differential form are invariant under the action of . For Problem 1 the author introduces the concept of a totally extremal manifold, which is an analogue of the concept of a totally geodesic manifold in a Riemannian or Finsler space. Some theorems are proved on totally extremal manifolds for fundamental representations of Lie groups. These theorems are used along with techniques developed in previous papers by the author to construct a synthesis of optimal trajectories for some multidimensional equivariant problems. Figures: 4. Bibliography: 13 titles.

Criteria are established for linear independence of Keldysh derived chains constructed from the root vectors of functions analytic in the left half-plane with values in the set of operators acting in a Hilbert space . In particular, an operator-valued function is considered. Let for and suppose that zero does not belong to the numerical range of the operator for some . Denote by an eigenvector corresponding to an eigenvalue , and by the subset of eigenvalues for which and for . Then it is proved that the vectors that belong to the direct sum of copies of the space are linearly independent when while . If, moreover, the operator , then this assertion holds also for . A connection is exhibited between the results obtained here and the question of uniqueness of the solution of a problem on the half-line for systems of ordinary differential equations with constant coefficients. Bibliography: 7 titles.

It is proved that almost stable group representations over a field have a finite basis of identities. Moreover, a variety generated by an arbitrary almost stable representation is Specht and all of its subvarieties have a finite uniformly bounded basis rank. In particular, the identities of an arbitrary representation of a finite group are finitely based. Bibliography: 17 titles.