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It is shown that an arbitrary locally flat imbedding of one piecewise linear manifold into another of codimension greater than two can be changed to a piecewise linear imbedding by means of an arbitrarily small isotopy of the ambient manifold. Bibligraphy: 5 entries.

In this paper a local unsolvability theorem is proved for differential equations with weighted derivatives that generalize the class of equations of principal type. In contrast to the latter, here the lower terms have an essential influence on local solvability. In this paper a one-parameter family of hamiltonians is constructed that plays the same role as the principal symbol for equations of principal type, and conditions on the behavior of this family are indicated under which there is no local solvability. Bibliography: 6 titles.

This paper considers a boundary value problem for the equation in some conical domains , where , , is a homogeneous polynomial of degree with real coefficients, and . An essential restriction on the domain is the following condition: the boundary contains no rays parallel to the -axis. The first part of the paper studies, for a wide class of domains , the asymptotics of a fundamental solution and the solution of a boundary value problem subject to the condition that the right-hand side and the boundary data tend rapidly to zero at infinity. In §3, for a specific domain and , a more involved case is examined, in which the right-hand side and the boundary data are unbounded.Bibliography: 13 titles.

In this paper, infinitesimal deformations which preserve the area element of a surface in (A-deformations) which also preserve the lengths of lines of curvature are studied. Here A-deformations are considered up to infinitesimal bendings (which constitute the trivial case for the problem posed). Such A-deformations are also called canonical.For regular surfaces of nonzero total curvature (without umbilic points) the problem indicated reduces to a homogeneous second order partial differential equation of elliptic type. In this paper a series of results about the existence and arbitrariness of canonical A-deformations is obtained. The basic results are valid for surfaces in the large.Bibliography: 20 titles.

In this paper, we present a method of introducing a one-dimensional analytic structure into the maximal ideal space of a uniform algebra, based on the use of subharmonic functions. Let be a uniform algebra on a compact Hausdorff space , the maximal ideal space of , the Gel'fand transform of a function , , and a continuous function on which locally belongs to the algebra . In ? 1, we introduce certain functions which estimate the dimensions of the images of the fibers , , under mappings , and prove that these functions are subharmonic. The results obtained on subharmonicity are used to prove the principal theorems of the paper, on finiteness of the fibers and analytic structure (Theorems 1-4). In ? 2, these theorems are applied to the study of the maximality properties of algebras of analytic functions on compact subsets of the Riemann sphere. Bibliography: 20 titles.

In this paper we consider the system (1)where , and the are homogeneous polynomials of degree () with complex coefficients. Let be the space of coefficients of the right-hand sides of the system (1). Any point defines a system of the form (1).Our aim in this paper is to show that the property of the solutions of the system (1) being dense in is locally characteristic, i.e. we prove that in there exists an open set such that the solutions of the system (1) with right-hand side are everywhere dense in .This result can be extended without difficulty to the case in which the degree of the homogeneous polynomials appearing in the right-hand side of the system (1) is odd.Bibliography: 4 titles.

This paper continues an earlier one (Math. USSR Sb. 44 (1983), 471-481). A function measuring the extent to which a Riemannian space is nonisotropic at the point is studied. Using , definitions of the notion of correctness of Schur's theorem are given in the multidimensional case. The relations between these definitions are clarified, and sufficient conditions for the correctness of Schur's theorem are given. It is shown that by a small deformation of the given metric it is possible to obtain one in which Schur's theorem is not correct. The methods developed in the paper are applied to study some geometric properties of geodesically parallel surfaces.Bibliography: 11 titles. Figures: 1.

In this paper the author constructs a holomorphic analytic continuation onto the whole complex plane of special Euler products-symmetric squares-corresponding to Siegel modular forms for congruence-subgroups of .The proof of this theorem is based on the analytic properties of "mixed" Eisenstein series for "arithmetic" congruence-subgroups of with character . The paper contains a proof that holomorphic analytic continuation onto the whole complex plane is possible for these series, and a derivation of their functional equation in the case of primitive .Bibliography: 13 titles.

In this paper the solution to some problems concerning rational approximation in the -metric () is given. The following is a typical problem: to describe the closure in the space of the linear hull of the Cauchy family In the paper it is shown that this closure consists of all functions for which there exists a function , analytic in , decreasing to zero at infinity, and such that for almost all .Bibliography: 6 titles.

In a recent paper by the author a new geometric definition of deficient values for a function meromorphic in was introduced, and with its aid a connection between the geometric structure of and the distribution of values of was established. In the present paper definitions characterizing the structure of , more delicately are introduced, and a more detailed study of these connections is carried out. As a by-product a theorem of Miles is obtained as a corollary. This theorem complements, in a sense, Ahlfors' second fundamental theorem of the theory of covering surfaces.Bibliography: 3 titles.

This paper indicates a method of calculating the defect numbers of the boundary value problem in terms of the -numbers of the Hankel operator constructed in a specified way with respect to the coefficients and . On the basis of this result the authors establish that the estimates, obtained in 1975 by A. M. Nikolaĭchuk and one of the authors (Ukrainian Math. J. 27 (1975), 629-639), of the defect numbers in terms of the number of coincidences in a disk of the solutions of certain approximating problems are sharp. This paper also establishes, in passing, criteria for the solvability of the problem of approximating a function , specified on a circle, by a function , meromorphic in a disk, for which a portion of the poles (along with the principal parts of the Laurent series at these poles) is assumed to be given.As auxiliary results expressions for partial indices are obtained, and properties of factorizing multipliers of Hermitian matrices of the second order with a negative determinant and a sign-preserving diagonal element are established.Bibliography: 27 titles.

In this paper the methods of M. Karoubi (MR 41 #6205) are generalized to the case of equivariant K-theory. The sets of Fredholm operators in certain (Hilbert) spaces of representations of finite groups G are described which are classifying spaces for equivariant K-functors. The results were announced in the paper MR 46 #2702. Bibliography: 16 items.

The paper is a continuation of work (MR 49 #801) in which in the case of a noncontracting unbounded domain there is distinguished a geometric characteristic of the domain that determines (under the fulfillment of a certain condition of regularity of the domain) the rate of stabilization for of the solution in 0)\times\Omega$ SRC=http://ej.iop.org/images/0025-5734/30/4/A01/tex_sm_2281_img4.gif/> of the following second boundary value problem for a parabolic equation: in which the initial function decreases sufficiently rapidly as . It is proved in the present paper that the same characteristic also determines the rate of stabilization of the solution in a class of contracting domains . In this case, as in the case of a noncontracting domain, tends to zero as like : there exist estimates of the function from above and from below having such an order of decrease.Bibliography: 11 titles.

This paper is devoted to the coincidence theory of two continuous mappings.A definition is given, in cohomological terms, of the coincidence index of two continuous mappings , where and are connected (not necessarily compact), orientable, -dimensional topological manifolds without boundary, is a compact mapping and is a proper mapping.Invariance of the index under compact homotopies of and proper homotopies of is proved. It is shown that is a sufficient condition for the existence of coincidence points of and . The Lefschetz number for and is also defined. The main result of the paper is a theorem on the coincidence of the numbers and .Bibliography: 7 titles.

In the paper we study purities in categories of Abelian groups having the property that the union of an increasing chain of -pure subgroups of an Abelian group is itself an -pure subgroup of . Such purities are called inductive. For every prime number we set if for , , there is an and an such that . Head purities are defined as purities of the form , where is a set of prime numbers. Head purities and -purities, evidently, are inductive. In the paper we show that every inductive purity in the category of all torsion-free Abelian groups is a certain -servancy, every inductive purity in the category of all periodic Abelian groups is a certain -purity, and every inductive purity in the category of all Abelian groups is the intersection of a certain -purity and a certain Head purity.Bibliography: 8 items.

This paper is devoted to the substantiation of a criterion for the quasisymmetric conjugacy of an arbitrary group of homeomorphisms of the real line to a group of affine transformations (the Ahlfors problem). In a criterion suggested by Hinkkanen the constants in the definition of a quasisymmetric homeomorphism were assumed to be uniformly bounded for all elements of the group. Subsequently, for orientation-preserving groups this author put forward a more relaxed criterion, in which one assumes only the uniform boundedness of constants for each cyclic subgroup. In the present paper this relaxed criterion is proved for an arbitrary group of line homeomorphisms, which do not necessarily preserve the orientation. Bibliography: 4 titles. Introduction A homeomorphism g : R → R is said to be quasisymmetric [1] if it satisfies the condition M−1 g g(x+ t)− g(x) g(x) − g(x− t) Mg . (1) If g is a quasisymmetric homeomorphism, then the homeomorphism g−1 is also quasisymmetric. For arbitrary quasisymmetric homomorphisms g1 and g2 their composite is also a quasisymmetric homeomorphism and Mg1g2 Mg1Mg2 [1]. Since the constant Mg in condition (1) for a homeomorphism g is not unique, this means that there exists a constant Mg1g2 for the homeomorphism g1g2 such that the inequality holds. We say that a group G consisting of quasisymmetric homeomorphisms is quasisymmetric. The following basic result for a quasisymmetric group of line homeomorphisms was obtained in [2]. Theorem 1. Let G be a group of line homeomorphisms. Then a quasisymmetric homeomorphism η such that η ◦G ◦ η−1 is a group of affine transformations exists if and only if G is a quasisymmetric group such that Mg = M for all g ∈ G, where M is a fixed constant. This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 53-01-00174) and the Programme of Support of Leading Scientific Schools of RF (grant no. NSh-457.2003.1). AMS 2000 Mathematics Subject Classification. Primary 54H15; Secondary 20F38, 28D99.

We have earlier proved (Dokl. Akad. Nauk SSSR 164 (1965), 40-42; Mat. Sb. 70 (112) (1966), 132-144) a theorem on the representation of an arbitrary function analytic in a closed convex region by a Dirichlet series in the open region . In this paper we prove that any function analytic in an open convex finite region and continuous in can be represented by a Dirichlet series with coefficients which can be computed by means of specific already-known formulas.We also prove that if the convex region is bounded by a regular analytic curve, then any function analytic in can be expanded in a Dirichlet series in . These two theorems are based on the following theorem from the theory of entire functions:Let be a finite open region, the support function of , , and a function satisfying the conditions Then there exists an entire function of exponential type with growth indicator and completely regular growth, which satisfies the following conditions:1) All the zeros of are simple, and 0$ SRC=http://ej.iop.org/images/0025-5734/9/1/A05/tex_sm_2048_img10.gif/>.2) We have the estimate r_0.$ SRC=http://ej.iop.org/images/0025-5734/9/1/A05/tex_sm_2048_img11.gif/>3) The sequence is part of a sequence , , which depends on the region but not on the function .In this paper we prove an analogous theorem for entire functions of arbitrary finite order .

In this paper the group of homeomorphisms of an arbitrary topological manifold is considered, with either the compact-open, uniform (relative to a fixed metric), or majorant topology. In the latter topology, a basis of neighborhoods of the identity is given by the strictly positive functions on the manifold, a homeomorphism being in the neighborhood determined by such a function if it moves each point less than the value of this function at the point. The main result of the paper is the proof of the local contractibility of the group of homeomorphisms in the majorant topology. Examples are easily constructed to show that this assertion is false for the other two topologies for open manifolds. In the case of a compact manifold the three topologies coincide. In conclusion a number of corollaries are given; for example, if a homeomorphism of a manifold can be approximated by stable homeomorphisms then it is itself stable.

This paper considers the Banach algebra with the usual norm and convolution as multiplication. A characterization is given for closed ideals of which are rotation invariant and have as spectrum, in terms of annihilators of certain collections of pseudomeasures. The main result of the paper is connected with a construction which yields an uncountable chain of closed ideals intermediate between neighboring invariant closed ideals with spectrum . This construction associates an ideal with a closed subset . It is shown that if then . Another result is the lack of a continuous projection from the largest to the smallest ideal when , and when , from an invariant ideal onto the neighboring smaller invariant ideal. A certain algebra of functions on the sphere which arises naturally in the construction of the intermediate ideals is also studied.Bibliography: 18 items.

In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of investigation go back originally to the work of Dyson [i] and I. M. Lifsic [2], [3] on the energy spectra of disordered systems, although in their probability character our sets are more similar to sets studied by Wigner [4]. Since the approaches to the sets we consider are the same, we present in detail only the most typical case. The corresponding results for the other two cases are presented without proof in the last section of the paper. §1. Statement of the problem and survey of results We shall consider as acting in iV-dimensiona l unitary space ///v, a selfadjoint operator BN (re) of the form

First part of paper. The paper is concerned with the study of asymptotic properties of the Green function for the Neumann problem in the exterior of a planar convex domain for the Helmholtz equation. Figures: 1. Bibliography: 10 items.

Recently the paper of Homma (Yokohama Math. J. 14 (1966), 47-54; MR 36 #892) which implies the possibility of piecewise linear approximation of piecewise linear manifolds in codimensions higher than two was found to contain an error, so that it is at present unclear whether the proof of this result can be completed using Homma's method. The present paper gives a proof of this result for the case of the elementary manifolds (cells and spheres), thus preserving the validity of two recently proved results whose proof were based on Homma's theorem. The method of proof used in this paper differs from Homma's method and is close to Connell's proof for approximation of stable homeomorphisms (Ann. of Math. (2) 78 (1963), 326-338; MR 27 #4238). Bibliography: 19 items

The paper is a continuation of the author's paper [5] (Math. USSR Sbornik 6 (1968), 97-114).§1 concerns the iterations of a polynomial of degree on a singular set . It is assumed that the critical points of lie either in the domains of attraction of finite attracting cycles or at infinity. The theorems of [5] (Theorem 1 concerning the topological isomorphism of the transformation and of a shift on the space of one-sided -ary sequences with a finite number of identifications; Theorem 2: ) are generalized for the case of a disconnected .In §2 the author investigates the iterations of on the entire plane . He shows (Theorem 3) that the dynamical systems and are topologically isomorphic for sufficiently small in the case of polynomials satisfying one of the hypotheses of §1 and a certain "coarse" condition of "nonconjugacy" of the iterations of distinct critical points.Hypothesis: the set of structurally stable mappings investigated in the paper is everywhere dense in the space of coefficients.Nine figures; bibliography of eight titles.

In this paper it is established that for a certain class of homogeneous spaces generators of the ring can be obtained with the help of two simple constructions, which are given in terms of the theory of representations. Some applications of the results and methods of the paper are indicated. Bibliography: 27 items.

In this paper some questions concerning the removal of singular sets for quasiconformal mappings are considered. Unlike previously existing results, in which it was required that the mapping be a homeomorphism or that the capacity of the singular points be zero, in this paper the restriction is weaker: the Hausdorff measure of the singular points is less than n – 1. A series of examples is given which show how to construct a set of singular points. In addition, theorems on removable singular sets are proved in which the quasiconformal mapping always has a continuous extension. In particular, the principle of symmetry for quasiconformal mappings is proved.Bibliography: 16 items.

In this paper we examine a problem connected with Cauchy's problem for linear elliptic equations.Let be a bounded region of , and let be its boundary. In we consider the elliptic equation (1)where is a regular elliptic expression with complex coefficients. Let , be a piece of the surface . The coefficients of the expression , the surface , and the boundary are assumed to be infinitely smooth. We are concerned with Cauchy's problem on with the initial conditions , , where designates the direction normal to . In this paper we prove that under our assumptions the set of Cauchy initial data for solutions of (1) in is dense in for any integer if Cauchy's problem is unique for the formal conjugate operator , as is the case, for example, when has no multiple complex characteristics.In addition, in this paper we give conditions under which the analogous assertion holds for certain elliptic systems.Bibliography: 4 items.

Let f : X→S be a characteristic, holomorphic mapping of complex spaces (with nilpotent elements). The paper proves that, if f is a flat mapping and all its fibers are equivalent to one and the same compact complex space X0, then, with respect to this mapping, X is equivalent to a holomorphic fibering over S with fiber X0 and structure group Aut(X0). It is further proved that, if the base S is reduced, the assertion remains true for any holomorphic mapping f, at least in the case when the fiber X0 is an irreducible space. This is a strong generalization of the corresponding result of Fischer and Grauert, in which a similar assertion is proved for the case when X and S are complex manifolds and f is a locally trivial mapping. This paper also proves that, if the compact complex space X0 satisfies the condition H1(Ω, X0) = 0, where Ω is the sheaf of germs of holomorphic vector fields on X0, then any locally trivial deformation of the space X0, with arbitrary parameter space, is trivial. This generalizes Kerner's result, in which the parameter space is assumed to be a manifold. Bibliography: 7 titles.

This paper gives a construction leading to Deheuvels homology of compact metric spaces with coefficients in copresheaves of abelian groups. We construct an epimorphic map of Deheuvels homology onto Aleksandrov–Cech homology, whose kernel is expressed in terms of the derived functors of the inverse limit functor. We consider projective objects in the category of copresheaves of abelian groups and homology with coefficients in projective copresheaves. The fundamental result of the paper is the theorem that the Deheuvels homology of compact metric spaces with coefficients in copresheaves of abelian groups is a universal extension of Aleksandrov–Cech homology among homologies which satisfy the exactness condition and other natural requirements. Bibliography: 5 items.

The paper considers the properties of vector analogs of the Sobolev spaces, , which appear in the study of various models of continuous media. Korn's inequality, proved in the paper, makes it possible to reduce the problem of compactness of the imbedding operators in these spaces to the scalar case and, consequently, to apply the scalar imbedding theorem of S. L. Sobolev. Heretofore, Korn's inequality in the general form was known only for .Bibliography: 18 items.

The criterion for the admissibility of spectral synthesis which was established in the first part of this paper is employed in the solution of a series of problems; in particular, it is employed in the investigation of the homogeneous convolution equation and in the investigation of systems of such equations.Let be the space of functions holomorphic in a convex region . Let be a continuous linear functional on . Then the subspace of solutions of the equation is invariant and always permits spectral synthesis. However, the system of equations , ..., does not always admit spectral synthesis. In this paper we determine in terms of characteristic functions the precise conditions for the possibility of spectral synthesis for this situation. If is an unbounded convex region, then spectral synthesis is always possible.Bibliography: 24 items.

In this paper a classification is given of actions of finite cyclic groups on quasicomplex manifolds in terms of the invariants of cobordism theory. Moreover, the methods of the paper allow one to understand the geometric nature of known results of a series of authors on actions of cyclic groups of prime order. Bibliography: 11 items.

In a domain we consider the first boundary value problem for a quasilinear parabolic fourth-order equation with a small parameter in the highest derivatives, which degenerates for into a second order equation. It is well known that the semigroup corresponding to this problem has an attractor, that is, an invariant attracting set in the phase space. In this paper we investigate the structure of this attractor by means of an asymptotic expansion in .The dominant term of the asymptotics is the solution of a second-order equation. The asymptotic expansion also contains boundary layer functions, which are responsible for the deterioration of the differential properties of the elements of the attractor near the boundary. The asymptotics constructed in this way (with an estimate of the remainder) enable us to study the differential properties of attractors and their behavior as in any interior subdomain , .For simplicity, the investigation is carried out in the case when is a bounded cylindrical domain. The generalization to does not present any difficulties.

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.

Sufficient conditions for an extremum of second-order type are considered on the basis of the concepts (introduced in the paper) of -Taylor mappings and lower -semi-Taylor mappings, where is a norm in a linear space , and is a topology in a subset . Examples are given illustrating the novelty of the results obtained. Among the references devoted to sufficient conditions for an extremum, [1] (§3.4) and [2] (§6) should be mentioned.

Let be a natural number, . Then we shall say that a function of period in each variable is monotonic if there exist an open rectangular parallelepiped and numbers , each of which is either 0 or 1 , such that for , and if , and for , then .The main result of this paper is that the multiple trigonometric Fourier series of an integrable monotonic function is Pringsheim convergent almost everywhere, in particular at each point of continuity of in the interior of .

Let and be subharmonic functions of finite order on . The main theorem of this paper shows that, if , the relation "" is preserved, in a certain sense, for mass distributions and . This result yields new uniqueness theorems for both subharmonic and entire functions on the complex plane.Corollaries include a broad class of sufficient conditions for the completeness of systems of exponential functions in a complex domain . The conditions for completeness are stated entirely in terms of the distribution of the points of the sequence in the neighborhood of infinity and in terms of the geometric properties (mixed areas) of .

In this paper all primitive representations of braid groups on strings of surfaces of finite type in groups of permutations of symbols are found. As an application it is proved that the groups of pure braids of surfaces are characteristic subgroups of the braid groups. These results generalize Artin's classical results on Artin's braid groups.

This paper is devoted to the development of methods of investigating the stability and global minimality of specific surfaces in Euclidean space and more generally in the Riemannian manifold. The author has obtained an effective sufficient condition for the stability of symmetric cones of any codimension in Euclidean space. By means of this sufficient condition he has proved the stability of several new series of cones of codimension two and higher. The author has constructed a new class of globally minimal surfaces in locally trivial vector bundles. The proof of the basic theorems is carried out by means of the construction of suitable calibration forms.

Weakly nonlinear semi-Hamiltonian systems of n differential equations of hydrodynamic type in Riemann invariants are considered, and the geometry of the (n + 2)-web formed by the characteristics and the level lines of the independent variables are studied. It is shown that the rank of this web on the general solution of the system is equal to n. This result is used to obtain formulas for the general integral of the systems under consideration, with the necessary arbitrariness in n functions of a single argument. Separate consideration is given to the cases n = 3 and n = 4, for which it is possible not only to integrate the corresponding systems, but also to give a complete classification of them to within so-called transformations via a solution (reciprocal transformations). It turns out that for n = 3 they can all be linearized (and are thus equivalent), while for n = 4 there exist exactly five mutually nonequivalent systems, and any other system can be reduced to one of them by a transformation via a solution. There is a discussion of the connection between weakly nonlinear semi-Hamiltonian systems and Dupin cyclides-hypersurfaces of Euclidean space whose principal curvatures are constant along the corresponding principal directions. Some unsolved problems are formulated at the end of the paper.

This paper is devoted to spectral synthesis of the operator adjoint to multiplication by a power of the independent variable in weighted spaces of entire functions of a single complex variable, and is closely connected with equations of convolution type, and with the general theory of subspaces invariant under a multiple differentiation operator. The problem of approximating solutions of a homogeneous equation of -sided convolution type by elementary solutions is solved. Certain systems of such equations are investigated.

By the KAM theory most of the nonresonant invariant tori of a nondegenerate integrable Hamiltonian system are preserved under a small perturbation of the Hamiltonian. On the other hand, Poincare's theorem states that the invariant tori of the unperturbed system, foliated by periodic solutions, are not completely scattered under a perturbation: as a rule, several periodic solutions are preserved and become nondegenerate. This paper fills out the gap between these two results. Namely, it is shown here that the resonant tori of an integrable Hamiltonian system that are fibered in more than one-dimensional ergodic components are not completely destroyed under a small perturbation of the Hamiltonian: as a rule, several of their nonresonant subtori are preserved and become hyperbolic. In concrete systems it is shown that there exist a large number of hyperbolic systems, and that this effect is an obstruction for the integrability of these systems. Bibliography: 16 titles.

The purpose of this paper is to construct harmonic analysis in a multiply connected plane domain Ω+ bounded by n piecewise-analytic curves. Part I (§§1-3) appeared in the preceding issue of this journal.

Let denote the supremum of the numbers for which there is a function on the closed unit disk such that is analytic inside and the set possesses an interior point. In 1945, Salem and Zygmund proved that , and asked for the value of . It is proved in this paper that .

This paper studies the class of superintuitionistic logics and the class of normal extensions of the modal system S4, and the syntactic and semantic connections between the two classes, given by the mapping (which assigns to every modal logic its superintuitionistic fragment) and by the mappings and (which assign to every superintuitionistic logic its smallest and its greatest companion, respectively). It is shown that from classes of relational models with respect to which a logic is complete, one can construct a class of models with respect to which the logics and are complete. The relationship of inference (of canonical formulas) in logics , , and is also described. As a consequence, preservation theorems are obtained for finite approximability, for Kripke completeness and for the disjunction property at the transition from to , and also for decidability at the transition to and .Bibliography: 21 titles.

This paper gives an explicit classification of the maximal arithmetic subgroups of the group of K-points of an indefinite orthogonal K-group of type (Dl), where K is a global field, charK ≠ 2.

The paper is devoted to the construction of affine Kac-Moody algebras via defining relations arising from an integral affine Cartan matrix, in the case where the ground field has positive characteristic p > 7. Explicit constructions are given for algebras obtained in this way; in distinction with the zero characteristic case they have infinite-dimensional center. A theorem on the universality of this central extension is proved, and a series of finite-dimensional factors of affine modular algebras over the ideals having trivial intersection with Cartan's subalgebra is constructed. Moreover, a construction for the PI-envelopes of affine Kac-Moody algebras is given.

The paper is devoted to a study of singular optimal trajectories by the chronological calculus methods developed in the school of R. V. Gamkrelidze.

Given a positive Borel measure on the real line and a function on , the main purpose of the paper is to prove (under certain assumptions on ) relative asymptotic formulas of the type where , is Szego's function corresponding to and the function , and are polynomials orthonormal relative to the measures and respectively.Bibliography: 15 titles.

Let n be a natural number, q a prime power, and θ a primitive element of the field GF(qn). This paper shows that there exist absolute constants c1, c2 > 0 such that for N ≥ max(exp exp(c1ln2n), c2n ln q) the set of elements θ1, ..., θN includes at least one which generates a primitive normal basis of GF(qn) over GF(q). For fixed n, this gives a polynomial time algorithm in ln q which, given an arbitrary primitive element θGF(qn), finds an element which generates a primitive normal basis for GF(qn) over GF(q). Bibliography: 17 titles.

In this paper there is an investigation, for the case of a compact group G, of the orbit space X/G of a given G-space X, from the point of view of the theory of retracts. A particular case of the main result asserts that if one of the spaces X and G has countable weight and X is a G-A(N)R for metrizable spaces, then X/G is an A(N)R for metrizable spaces. New results about the equivariant embedding of metrizable G-spaces are also obtained. Bibliography: 28 titles.