Locally nilpotent groups with the min−∞−n condition

The study of locally nilpotent groups with the weak minimality condition for normal subgroups, the min^––n condition, is continued. The following results are obtained.THEOREM 1. A locally nilpotent group with the min^––n condition is countable.THEOREM 2. If G is a locally nilpotent group with the min^––n condition whose periodic part is nilpotent and the orders of the elements of the periodic part are bounded in the aggregate, then G=t(G)A, where the subgroup A is minimax.THEOREM 3. Suppose G is a locally nilpotent group with the min^––n condition and T is its periodic part. If T is nilpotent and G/T is Abelian, then G=TA, where the subgroup A is minimax.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 340–346, March, 1990.     

A simplified variant of the method of computation of the Fourier coefficients of a series

Making use of the results of Golubeva and Fomenko (Zapiski Nauchn. Sem. Leningr. Otd. Mat. Inst.,71, 85–95 (1977)), one computes the Fourier coefficients of the nonanalytic Eisenstein series relative to Sp₂() by a method which is simpler than that of Kaufhold (Math. Ann.,137, No. 5, 454–476 (1959)).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 60–64, 1978.     

Application of fields generated by gauss periods to the study of cyclic Diophantine equations. II

This is the continuation of the paper having the same title (Zapiski,67, 201–222, 1977). The results of that paper are generalized to the case when f is the power of an odd prime number.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 149–157, 1979.     

Cobordism of hyperframed manifolds and the stable J-homomorphism

This paper considers the cobordism groups of hyperframed manifolds; that is, manifolds which are framed off a codimension one submanifold. These groups are related to thestable J-homomorphism in the same way that the cobordism groups of almost parallelizable manifolds are related to the J-homomorphism. We compute characteristic numbers and present some low dimensional examples.Research partially supported by NSF Grant MCS76–06973Research partially supported by NSF Grant MPS74–22738     

Abel's identities

We generalize Abel's identities (cf. H. Weber, Lehrbuch der Algebra,2, 205–216 (1908)) for elliptic curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 58–61, 1983.     

Representation of integers by positive-ternary quadratic forms

The discrete ergodic method is applied to the problem mentioned in the title. We obtain estimates (of exact order) of the number of primitive representations in a given cone and a given residue class. The paper is an extension of the work of Peters (M. Peters, Acta Arithm.,34, 57–80 (1977)).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 117–156, 1983.     

Asymptotic density of certain sets of natural numbers

We consider a generalization of a theorem of Ruzsa (I. Z. Ruzsa, Acta Arithm.,32, 313–347) concerning the existence of an asymptotic density connected with problems posed by Ruzsa.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 187–194, 1983.     

Construction of the functional model of B. S.-Nagy-C. Foias

Methodological paper: quick and simple construction of the Nagy—Foias functional model, using only elementary information about Hilbert spaces and operators on them. A formula is derived for a characteristic function. The account is based on an idea of B. S. Pavlov (Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 1, 123–148 (1975)) on symmetrization of the notation for a model space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 16–23, 1977.     

Dense lattice packings of spheres in Euclidean spaces of dimension n⩽16 *

We present a number of lattice packings of equal spheres in ⁿ for n16 For n15, these packings have the same density as the densest known lattice packings. For n=16, the packing described here is denser than the known ones.It should be pointed out that the 16-dimensional lattice described here is equivalent to one found by E. S. Barnes and G. E. Wall, J. Aust. Math. Soc.,1, 47–63 (1959); see also J. Leech and N. J. A. Sloane, Can. J. Math.,23, 718–745 (1971) — Translator.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, 144–146, 1979.     

Differential and difference algebra

In a survey the main results in differential and difference algebra (differential rings and modules; differential fields; differential-algebraic geometry; differential and difference dimension; differential and difference Galois theory; integration in finite terms; differential algebraic groups), compiled from the reviews published inReferativnyl Zhurnal Matematika for the years 1975–1986, are reflected.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 25, pp. 67–139, 1987.     

A. F. Lavrik's truncated equations

By a new method, we obtain some known results of A. F. Lavrik (Dokl. Akad. Nauk SSSR,171, No. 2, 278–280 (1966); Mat. Zametki,2, No. 5, 475–482 (1967); Izv. Akad. Nauk SSSR, Ser. Mat.,30, No. 2, 433–448 (1966)) regarding the truncated functional equations of various L-functions. As an application, we give an estimate of Dedekind's zeta-function of an algebraic number field K of degree n4, _k(1/2+it)t^n/6log^ct,t>1 and a similar estimate for L-series with grössencharacters. The method of the paper allows us to consider fields of degree n12.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 124–158, 1978.     

A class of Dirichlet series

Two theorems are proved on Dirichlet series of a special type related to multiplicative functions f(n), ¦f(n)¦1. These theorems develop well-known results of Halász (G. Halász, Acta Math. Acad. Sci. Hungar.,19, Nos. 3–4, 365#x2013;404 (1968)).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 187–194, 1975.     

Sufficient statistics for families of distributions with densities of variable support. II

This paper is a continuation of the paper from J. Sov. Math.,24, No. 5 (1984).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 136, pp. 27–47, 1984.     

Duality in spectral optimization and numerical ranges of a family of self-adjoint operators

One gives results which complement the author's investigation, published in Dokl. Akad. Nauk SSSR,255, No. 4, 777–780 (1980). One establishes the relationship of the fundamental condition ensuring the duality relation between the direct and the dual problems in spectral optimization problems with the geometry of the numerical ranges of certain families of self-adjoint operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 189–192, 1982.     

Higher-order degenerate elliptic equations: Spaces,operators, boundary-value problems

One gives a survey and one presents the method of the reduction of a general boundary-value problem to a Dirichlet type problem, based on the systematic use of the concept of a weighted pseudodifferential operator. One gives the fundamental results in the theory of weighted spaces of functions. The survey is compiled on the basis of the publications refereed during 1970–1983 in the Referativnyi ZhurnalMatematika and on the basis of the latest results published in the form of reports or preprints or deposited at VINITI.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 23, pp. 125–218, 1985.     

Representation of integers by positive ternary quadratic forms (a new modification of the discrete ergodic method)

One gives a detailed presentation of a new modification of the discrete ergodic method outlined in A. V. Malyshev's note (this Zapiski,50, 179–186 (1975)). One gives new proofs for the asymptotic formulas obtained in Chap VI of A. V. Malyshev's monography (Tr. Mat. Inst. Akad. Nauk SSSR,65 (1962). One obtains estimates for the remainder terms of this formulas under the assumption of some hypotheses about the zeros of Dirichlet's L -functions.Translated from Zapiski Nauchnykh Seminarog Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 33–87, 1979.     

Automorphic scattering matrix for the Hecke group Γ[2 cos (π/q)]

This paper is the second part of the work published in Proc. Steklov Math. Inst.,158 (1981). In this part a new exact analytic formula for the automorphic scattering matrix is presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 109, pp. 34–40, 1981.     

Expansion of automorphic functions on sl3(C>/SU<3)

One constructs the expansion for automorphic functions on SL₃,()/SU(3) using the method presented by the author in a previous paper [Zap. Nauch. Sem. Leningr. Otd. Mat. Inst.,116, 119–141 (1982)].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 144–153, 1983.     

On holomorphic families of subspaces of a banach space

In this paper we study holomorphic families of subspaces of a Banach space, which are parametrized by some analytic space. We consider the question of the existence of a holomorphic complement for a given family. It turns out that such a complement does always exist if the basic space is a Stein space, provided for each fixed value of the parameter the subspaces of the family are complemented. From this follows, in particular, the existence of a holomorphic left inverse of a holomorphic operator function, provided such a left inverse exists for each fixed value of the parameter. The latter result gives a positive answer to a question formulated by I.C. Gohberg in a personal conversation with the author. Note of the Editor: This paper was originally published in "Mat. Issled. (Kishinev) 5, vyp 4 (18) (1970), 153–165. The editor is grateful to K. Clancey, W. Kaballo and G. Ph. A. Thijsse for preparing the translation from Russian. Note of the Editor: After the appearance of the original version of the above paper the author informed the editor of Mat. Issled (see Mat. Issled. (Kishinev) 6, vyp (19) (1971), 180) that Corollaries 2 and 3 of Theorem 2 are contained in the papers of G.R. Allan, J. London Math. Soc. 42 (1967), 463–470 and 509–513.     

Universal covers of finite sets

A recurrent problem in geometry, convexity, and combinatorics is to find universal covers for various classes of sets. (See [2] pp 84–88 for example.) This paper considers that problem for finite sets in the plane. Various covers are given including one which coversall finite sets, is closed, and has empty interior. The problem of covering a finite set or a similar copy of the set is also considered.