On the Constant Type of Almost Hermitian Manifolds

We suggest a most natural generalization of the notion of constant type for nearly Kählerian manifolds introduced by A. Gray to arbitrary almost Hermitian manifolds. We prove that the class of almost Hermitian manifolds of zero constant type coincides with the class of Hermitian manifolds. We show that the class of G ₁-manifolds of zero constant type coincides with the class of 6-dimensional G ₁-manifolds with a non-integrable structure. Finally, we prove that the class of normal G ₂-manifolds of nonzero constant type coincides with the class of 4-dimensional G ₂-manifolds with a nonintegrable structure.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Convergent sequences of linear operators in semiordered spaces

The definition given by P. P. Korovkin of operators of the class S_m and conditions for the convergence of these operators to the identity operator are extended to apply to regular operators from a K-space R₀ with a unit, into a K-space R₁ where R₀ and R₁ are normally contained in the union of the spaces S [a, b] and s.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 475–486, October, 1970.     

On co-semihereditary rings

A ringR is left co-semihereditary (strongly left co-semihereditary) if every finitely cogenerated factor of a finitely cogenerated (arbitrary) injective leftR-module is injective. A left co-semihereditary ring, which is not strongly left co-semihereditary, is given to answer a question of Miller and Tumidge in the negative. If _R U _S defines a Morita duality,R is proved to be left co-semihereditary (left semihereditmy) if and only ifS is right semihereditary (right co-semihereditary). Assuming thatS⩾R is an almost excellent extension,S is shown to be (strongly) right co-semihereditary if and only ifR is (strongly) right co-semihereditary. Project supported by the National Natural Science Foundation of China.     

Maximal Holonomy of Almost Bieberbach Groups for Heis5

Let Heis _2n+1 be the Heisenberg group of dimension 2n + 1 and M an infra-nilmanifold with Heis _2n+1-geometry. The fundamental group of M contains a cocompact lattice of Heis _2n+1 with index bounded above by a universal constant I _n+1, i.e., I _n+1 is the maximal order of the holonomy groups. We prove that I ₃ = 24. As an application we give an estimate for the volumes of finite volume non-compact complex hyperbolic 3-manifolds.     

On the geometry of principalT 1-bundles over Hodge manifolds

We study Sasakian structures induced in principalT ¹-bundles over Kähler manifolds. A natural model of a Sasakian manifold of constant -holomorphic sectional curvature –3 is constructed.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 824–829, December, 1998.The author is greatly indebted to Professor V. F. Kirichenko for setting the problem, as well as for interest and help during the research.     

Normalizer of a group of “diagonal” automorphisms in algebras over a commutative ring

Let S be a faithful algebra over commutative ring R. It is assumed that S is additively generated by its invertible elements. It is shown that the nomalizer of subgroup Aut(S_s) of group Aut(S_R) coincides with the semidirect product Aut(S_S) Aut(S/R),where the second factor is the group of all ring automorphisms of ring S identical on R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 5–8, 1991.     

On the Manifold of Almost Complex Structures

Let (M, g ₀) be a smooth closed Riemannian manifold of even dimension 2_n admitting an almost complex structure. It is shown that the space of all almost complex structures on M determining the same orientation as the one determined by a fixed almost complex structure J ₀ is a smooth locally trivial fiber bundle over the space of almost complex structures orthogonal with respect to g ₀ and determining the same orientation as J ₀.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 66–71.Original Russian Text Copyright © 2005 by N. A. Daurtseva.     

Almost sure central limit theorems for random functions

Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.     

A remark on the essential self-adjointness of a Schrödinger operator with a singular potential

We examine the operators=–+v, v L_{2, loe} (R ⁿ ), where S satisfies a natural additional condition of a local nature. If a condition of Titchmarsh type is fulfilled at infinity, then S is essentially self-adjoint in L₂(Rⁿ).Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 571–580, October, 1976.     

The space of maps from a locally compact space to a Banach space

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel ₂ ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R ⁿ,R^m) is homomorphic to Hilbert spacel ₂. This research is supported by the Science Foundation of Shanxi Province's Scientific Committee     

Reconstruction of 3-submanifolds of large codimension in euclidean spaces from their gauss image

We present necessary and sufficient conditions for a regular three-dimensional manifold in the Grassmannian manifoldG(m, m+3) to be the Gauss image of a regular 3-submanifold in (m+3)-dimensional Euclidean space form>4. Translated fromMatemalicheskie Zametki, Vol. 62, No. 5, pp. 694–699, November, 1997. Translated by S. K. Lando     

Groups with an almost regular involution

An involution j of a group G is said to be almost perfect in G if any two involutions in j^G whose product has infinite order are conjugated by a suitable involution in j^G. Let G contain an almost perfect involution j and |C_G(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |C_G(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite index in G. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 360–368, May–June, 2007.     

Amalgams of clones

The structure of a clone of functions respecting an amalgam is described. Also, for clones R_A and R_B of relations, conditions are specified under which the amalgam of R_A and R_B, restricted to the domains of R_A and R_B, coincides with these.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 97–113, January–February, 2005.     

An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim _{r → ∞} e^2r s(r) = 0, then (M, g) has to be isometric to ℍ ⁿ . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim _{r → ∞} r ² s(r) = 0, then (M, g) is isometric to ℝ ⁿ , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if K ≤ a on a geodesic ball B _p (R) in M and K = a on ∂B _p (R), then K = a on B _p (R).     

General symbolic powers: A homological point of view

Let I be an ideal of a Noetherian ring R and let S be a multiplicatively closed subset of R. We define the n-th (S)-symbolic power of 7 as S(Iⁿ) = IⁿR_s ∩R. The purpose of this paper is to compare the topologies defined by the adic {I_n}_n≤0 and the (S)-symbolic filtration {S(Iⁿ)}n≥o using the direct system {Extⁱ _R(R/Iⁿ,R)}_n≥0     

Non commutative unique factorization rings

Let R and S be two rings. Each category equivalence between a torsion class of left (right) R-modules and a torsion-free class of left (right) S-modules is represented by a left (right) quasi-tilting triple. Suppose we have a pair of equivalences T ? Y and X F between the torsion class T of R-modules and the torsion-free class Y of S-modules and between the torsion class X of S-modules and the torsion-free class F of R-modules. Denote by (R, V, S) and (S, U, R) the quasi-tilting triples representing these equivalences. We say that (R, V, S) and (S, U, R) are complementary if T, F) and X, Y) are torsion theories in R-Mod and S-Mod, respectively. We find necessary and sufficient conditions on the bimodules _RV_S and _SU_R to have the complementarity of (R, V, S) and (S, U, R).     

Almost split sequences for dedekind-like rings. II

Let R be a two-pullback ring, that is, the pullback of two Dedekind domains over a common residue field. This paper contains explicit constructions of all the almost split sequences in the category of finitely generated R-modules, under the assumption that the Dedekind domains are local.     

One-loop effective action for super-QCD

The path integral method with respect to the scalar (quark) superfields is used to obtain the effective action for fields with the quantum numbers of the mesons and their superpartners inN-1 super-QCD. The background fields include gauge superfields of the groupSU _L (N)xSU _R (N)xU _L(1)xU _R(1) and also Higgs superfields in the adjoint representation of the groupSU _L (N)xSU _R (N).St. Petersburg Wood Technology Academy. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 2, pp. 237–246, February, 1995.     

Iterations of homogeneous polynomial transformations with positive coefficients

A study is made of the behavior as k of the iterations T^k(x) of a homogeneous polynomial transformation T acting from Rⁿ to Rⁿ according to the formula (T(x))_i=Q_i (x), i=1, 2,..., n, where Q_i(x) is a homogeneous polynomial of degree m>1 with positive coefficients.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 411–416, October, 1969.In conclusion I should like to thank S. A. Molchanov for proposing the topic and S. M. Natanzon for his help with the paper.