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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 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.

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.

This paper is devoted to problems of the representation and best approximation of the Cauchy kernel on the whole real axis - ∞ < x < + ∞. The method proposed here for the solution of such problems is based on the attraction of systems of rational functions with fixed poles which are orthogonal or biorthogonal on the whole axis. Bibliography: 9 items.

In this paper the existence of discontinuous solutions xn+1 = u(x), x ?, of a positive definite quasiregular n-dimensional variational problem is established when the order of growth of the integrand of the functional degenerates up to unity on non-self-intersecting (n - 1)-dimensional surfaces lying in the region ? or on its boundary S.

This paper treats the questions of convergence and summability on a convex polygon of the Dirichlet series of a function which is analytic in and continuous on . Necessary and sufficient conditions for convergence are given for the case of a square; in the general case, if the necessary conditions for convergence are satisfied, it is sufficient that the integral converge.Bibliography: 7 items.

Let be the Fourier coefficients of the solution of the Cauchy problem for a nonlinear parabolic equation of the form where is a linear elliptic operator of order and is the nonlinear part of the equation. Then are the moment functions of the equation, i.e. the average of the function with respect to a probability measure , where characterizes the degree of concentration of the measure. In this paper we give an asymptotic expansion for the functions as .Bibliography: 8 items.

This paper is devoted to a study of the conditions under which one algebraic operation can be expressed in terms of others by some arrangement of parentheses. The terminology is mainly that of Frenkin (MR 44 #6888). It is shown that the class of ?-reducible n-groupoids is axiomatizable, but not elementary, and the class of ?-uniformly reducible n-groupoids is not axiomatizable; a criterion for ?-uniform reducibility in terms of pseudo-isotopies (a generalization of the concept of isotopy) between ?-reducing operations is obtained. It is shown that a free n-groupoid of finite rank is not ?-uniformly reducible, but one of infinite rank is ?-uniformly reducible; as a consequence, any n-groupoid is a homomorphic image of one which is ?-uniformly reducible. Some results on algebras with unary operations are also obtained. Bibliography: 7 items.

Let G(t, x) be the Green's function of a parabolic differential operator ∂/∂t + P(- i∂/∂x). In a previous article of the authors (Math. USSR Sb. 20 (1973), 519-542) estimates for G are obtained by means of a convex function νp invariantly defined by P, and the saddle points are distinguished under the assumption that νp is smooth. In the present paper the question of the existence of a finite number of saddle points is studied without assuming the smoothness of νp; an example of a polynomial P is constructed for which the function νp is not smooth. It is shown that for almost all polynomials P the function νp is strictly convex almost everywhere. Bibliography: 13 items.

In this paper an estimate is given for the ρ-invariant of the Mazur module for abelian varieties over Γ-extensions. For an elliptic Fermat curve the group of points at each level of a noncyclotomic Γ-extension is computed. Bibliography: 6 items.

In this paper the group of birational automorphisms of a cubic surface of Markov type is computed, and it is proved that its action on the set of integral solutions of the corresponding diophantine equation is transitive. Bibliography: 4 items.

In this paper we study the representations Ind(G, P, π) of the group G = GL(n, D), where D is a locally compact nondiscrete division algebra, that are induced by irreducible representations π of an arbitrary parabolic subgroup P⊂G. If D is totally disconnected, π is assumed to be either supercuspidal (in the sense of Harish-Chandra; this is the same as absolutely cuspidal in the sense of Jacquet), or one-dimensional; we also allow combinations of these cases of a specific sort. We give a construction of intertwining operators in this class of representations generalizing the construction of Schiffmann, Knapp and Stein. Using these intertwining operators, we prove that for the "principal series" representation Ind(G, P, π) to be contained in the "complementary series" the necessary formal condition of symmetry on (P, π) turns out to also be sufficient. If π is one-dimensional we estimate the width of the "critical interval". Under certain conditions this estimate is best possible. Bibliography: 28 items.

This paper considers the action of the operator on functions of the form , where and . In particular, when , , one has It is proved that for the differential operators can be obtained from the analogous differential operators for by means of almost analytic extension with respect to the arguments .Bibliography: 12 items.

In this paper the asymptotic behavior is studied of the spectrum of weighted integral operators of the form (1)acting in the space .Bibliography: 10 titles.

A number of authors have introduced ring groups as objects generalizing locally compact groups. An analogue of the Pontrjagin principle of duality holds for ring groups. In this paper we introduce a wider class of ring groups, one including the locally compact groups.A construction is given whereby to each ring group there is defined a dual ring group ; here . By definition a ring group is determined by a -algebra (the space of the ring group) equipped with an additional structure which allows to be considered, in particular, as a Hopf-von Neumann algebra. When is a locally compact group, is the -algebra of bounded measurable functions on , considered in the natural way as operators in .Bibliography: 15 items.

A variety of universal algebras is called a Schreier variety if every subalgebra of any free algebra in that variety is also free in that variety. This paper gives a description of the Schreier varieties of linear Ω-algebras over an associative commutative ring, defined by systems of homogeneous identities. As a corollary to these results one obtains a description of all Schreier varieties of linear Ω-algebras over an infinite field (in particular, over a field of characteristic zero). These algebras include, in particular, nonassociative algebras. Bibliography: 25 items.

In this paper we give a method for constructing formal asymptotic solutions. This method uses in some sense "approximate solutions" of the equation of the characteristics and the transport equation. The construction of approximate solutions is brought abount by means of an analogue of the analytic Hamiltonian formalism in a complex phase space. Bibliography: 19 items.

We examine the selfadjoint operator in . We assume that the potential tends to as . Under these conditions the spectrum of is discrete. In the paper the well-known asymptotic formula (*)for the distribution function of the eigenvalues is justified under very weak assumptions on , namely the following conditions: 1) , where . 2) almost everywhere when . 3) There exist a continuous function , , , and an index such that for any , , .Bibliography: 12 items.

In this paper, the author proves the existence of a versal family of deformations for any projective algebraic variety. Bibliography: 6 items.

In this paper we construct a one-to-one (and quasi-isometric) transformation of a phase space that allows us to pass from a diffusion process with nonzero drift coefficient to a process without drift. Using this transformation we construct strong solutions of stochastic differential equations with a "bad" drift coefficient and give other applications. Bibliography: 21 items.

This paper deals with three objects of the theory of formations: -stable groups of automorphisms, -coradicals and -abnormal maximal subgroups.Bibliography: 26 items.

Suppose is some property of modules. Let denote the class of rings over which all modules possess property . The main theorem of this paper answers the following question for a rather extensive class of properties ; what must the property of modules be in order that if and only if , for any free -module ? Among the corollaries are many well-known theorems relating properties of the ring and the rings , and also a number of new results of similar type.Bibliography: 35 items.

This paper considers the Cauchy problem for a hyperbolic equation with constant operator coefficients in Hilbert space: where and are selfadjoint operators and is semibounded.As an example we consider ultraparabolic systems.Bibliography: 10 items.

In this paper we prove the followingTheorem. Let be a finite simple group, an involution of and the centralizer of in . If where 3$ SRC=http://ej.iop.org/images/0025-5734/22/4/A04/tex_sm_2171_img5.gif/>, then a Sylow 2-subgroup of is an elementary group of order 8.Bibliography: 14 items.

In this paper a p-adic analytic function of two variables is constructed whose values in some "common" domain coincide with the values of the family of Hecke L-series of an imaginary quadratic field. The functional equation for such a function is obtained. The p-adic Mellin integral transform is the main technique. Bibliography: 13 items.

Let be the system of polynomials orthonormal on with weight where , , on and ( is the modulus of continuity in ). Consider the class of functions , where Let denote the partial sums of the Fourier series of a function with respect to the system .In the paper, conditions are obtained on the exponents of the functions and and the exponent that are necessary and sufficient for the boundedness in of each of the operators and . Sufficient conditions for the convergence of the partial sums to in the mean and almost everywhere in are revealed as a consequence. It is proved that these conditions are best possible on the class (for in the case of convergence almost everywhere). Estimates of the polynomials and necessary and sufficient conditions for their boundedness in the mean are also obtained.Bibliography: 26 items.

This paper deals with questions related to the nil radical and the Jacobson radical of the endomorphism rings of torsion-free abelian groups. The most complete results are obtained for groups of finite rank. A characterization is given for the Jacobson radical of the endomorphism ring of a torsion-free abelian group of finite rank. The question of when the Jacobson radical of the endomorphism ring of a torsion-free abelian group of finite rank is nilpotent (equal to zero) is completely settled. Bibliography: 7 items.

A Fourier series of a summable function is defined in the paper for which any sequence of Cesaro means of order α satisfying the inequality -1 < α < 0 diverges on a set of positive measure. A Fourier series of a square summable function is also defined that has the same property for α satisfying the inequality -1 < α < - 1/2. Bibliography: 4 items.

This paper consists of two parts. In the first part, a general theorem is proved on the behavior along chords of functions defined on the unit disk, and this theorem is applied to continuous functions of bounded type. In the second part, boundary uniqueness theorems are proved for functions meromorphic in the unit disk. Bibliography: 12 items.

The main result of the paper is a description of the normal structure of the groups , where is a linearly ordered set satisfying one of the following equivalent conditions: I. is 2-transitive. II. is -transitive. III. does not have a greatest or a least element, and any two intervals , and , , are similar. IV. is a 0-primitive, transitive, nonregular permutation group.Main theorem. Suppose is 2-transitive. Then , and are the only nontrivial normal and subnormal subgroups of . Here is bounded below, is bounded above, , .Bibliography: 21 titles.

In this paper we give a method for solving the periodic Cauchy problem for the Korteweg-de Vries equation: We justify our method with the aid of the theory of inverse spectral problems for Sturm-Liouville operators, considered on a finite interval.Bibliography: 8 items.

Let be a sub-prevariety of a fixed prevariety (residually closed class) . The smallest ordinal number such that an algebra is -step -soluble is called the step of -solubility of . The smallest ordinal number such that there exists an -step -soluble algebra is called the degree of idempotency of relative to . In the paper is taken to be the class of all lattices, and all ordinal numbers that can be degrees of idempotency of prevarieties of lattices are found. Further, a description is given, depending on the degree of idempotency of a prevariety , of the ordinal numbers that can be steps of -solubility of suitable lattices.Bibliography: 11 items.

Let A be a central simple algebra on which an involutory antiautomorphism S is given whose restriction to the center K of A is not the identity. Let ?(A*) be the subgroup of the multiplicative group A* of A generated by the elements x A* such that xS = x, let NrdA/K : ?A?K be the reduced norm mapping of A into K, and let ? '(A*) be the subgroup of A* generated by the elements x A* whose reduced norm is invariant with respect to S. This paper considers the problem of when the groups ? '(A*) and ?(A*) coincide. Bibliography: 15 titles.

The paper deals with closed Hermitian operators and their selfadjoint extensions. New criteria are obtained for Hermitian operators to be regular in the sense of M. A. Krasnosel'skiĭ. For arbitrary closed Hermitian operators an expression is found for the extent to which an extended generalized resolvent fails to be an -function. A new class of operators is found and investigated for which is an -function for all admissible elements .Bibliography: 14 items.

In this paper it is shown that a complete reduced curve which is locally a complete intersection over an algebraically closed field has a smooth deformation over the complete local ring of formal power series over the field of definition of the curve. Bibliography: 10 items.