Factorizations of composition formations

It is proved that, if is a singly generated composition formation, where , then is a composition formation. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 389–395, March, 1999.     

Integral-valued rational functions on valued fields

Let K be a field and a non-trivial valuation ring of K withm as its maximal ideal. Denote by and the rings of polynomials f∈K[X] and rational functions f∈K(X) resp. such that . We prove that for one variable X we have if and only if the completion of (K, ) is locally compact or algebraically closed. In the second case—i.e. if K is dense in the algebraic closure of (K, )—we even get for any number of variables X=(X₁,...,X_n). This work contains parts of the second author's thesis [Ri] written under the supervision of the first author.     

On the representation of differentiations by Hamiltonians,operating from an algebra of local observable spin systems

Let and be algebras of local and quasilocal observable spin systems corresponding to the group Zr, be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian formed by the displacements of an elementx ε is considered. It is shown that such a representation exists if the condition holds, where means an element obtained from the elements [T_kX,a] by some r-multiple process of summation. Translated from Matematicheskii Zametki, Vol. 21, No. 1, pp. 93–98, January, 1977.     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

On the word problem and the conjugacy problem for groups of the formF/V(R)

LetF be a free group with at most countable system of free generators, letR be its normal subgroup recursively enumerable with respect to , and let be a variety of groups that differs from and for which the corresponding verbal subgroupV of the free group of countable rank is recursive. It is proved that the word problem inF/V(R) is solvable if and only if this problem is solvable inF/R, and if , then there exists anR such, that the conjugacy problem inF/R is solvable, but this problem is unsolvable inF/V(R) for any Abelian variety (all algorithmic problems are regarded with respect to the images of under the corresponding natural epimorphisms). Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 3–9, January, 1997. Translated by M. I. Anokhin     

On periodic groups of automorphisms of extremal groups

It is proved that if a periodic group has an extremal normal divisor , determining a complete abelian factor group , then the center of the group contains a complete abelian subgroup , satisfying the relation and intersecting on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group is a finite extension of a contained in it subgroup of inner automorphisms of the group .Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 91–96, July, 1968.     

Eine Verallgemeinerung derCW-Komplexe

In the definition ofCW-complexes, the one-point spaceP, respectively the spaceP∪* with basepoint *, play the roll of the only “building-stone”. Let be a family of compact spaces. Then the definition of a generalizedCW-complex over is obtained from the definition of aCW-complex by replacingP by the spaces of and formation of the mapping cone by a slightly modified construction. LetCW * denote the category of all pointed spaces which have the homotopy type of a generalizedCW-complex over . If , thenCW * is the category of all pointedCW-spaces.CW * is closed under the formation of direct sums and of mapping cones, cylinders and tori, and is formally characterized as the smallest such subcategory of Top * containing the spaces W∪*, . Following the methods of E. H. Brown, it is proved, that any half exact homotopy functor onCW * is representable, and any cohomology theory onCW is naturally equivalent to the cohomology theory of an Ω-spectrum; for example, the singular cohomo logy is representable onCW for any family of compact spaces.      

Primitive systems of elements in the variety\mathfrak{A}\mathfrak{N}_2 and some locally finite varieties of groupsand some locally finite varieties of groups

Part of any basis of a relatively free group in the variety is called a primitive system of elements. We provide a criterion of being primitive for , where is a variety of Abelian groups satisfying x^m=1, and a variety generated by a finite group. Let be a variety of nilpotent groups of class ≤c. It is proved that, for the group , the property of being primitive for an element g is stronger than the condition of being unimodular on a vector composed of values of Fox derivatives in the ring . The group is not residually finite whenever a system of elements is primitive. Supported by RFFR grant No. 96-01-01948. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 687–699, November–December, 1998.     

Quasi-uniform completions of partially ordered spaces

In this paper we define partially ordered quasi-uniform spaces (X, , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology of a PO-quasi-uniform space (X, , ≤), the bicompletion of (X, ) is also a PO-quasi-uniform space ( , ⪯) with a partial order ⪯ on that extends ≤ in a natural way.      

Invariant convex sets and functions in Lie algebras

We say that an invariant convex coneW in a Lie algebras is elliptic if its interior consists of elliptic elements of . If such a cone exists, then has a compactly embedded Cartan subalgebra. The first main result, of this paper is a characterization of those Lie algebras, which contain elliptic invariant cones. If is an invariant domain in such a cone, then we characterize the invariant locally convex functions onD by their restrictions to where is a compactly embedded Cartan subalgebra.     

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let and be the valuation rings, and letG be the Galois group. We consider the powers of the maximal ideal of as modules over the group ring . We show that, ifG has orderp ^m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then and are isomorphic iffr≡r′ modp ^m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.     

On some generalizations of the factorization method

The classical factorization method reduces the study of a system of ordinary differential equations U_t=[U₊, U] to solving algebraic equations. Here U(t) belongs to a Lie algebra which is the direct sum of its subalgebras and , where “+” signifies the projection on . We generalize this method to the case . The corresponding quadratic systems are reducible to a linear system with variable coefficients. It is shown that the generalized version of the factorization method can also be applied to Liouville equation-type systems of partial differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 339–350, March, 1997.     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

Elementary properties of free extensions

For any partial groupoid , let Fr be the free extension of to a total groupoid. We show that implies and that the theory of Fr is uniformly recursive in the theory of . These results fail if “groupoid” is replaced by “semigroup”, “commutative semigroup”, “group”, “abelian group”, “semilattice”, “K-lattice” for any nontrivial varietyK of lattices, or “Boolean algebra”. Research supported in part by NSF Grant MCS78-01867. We thank the referee for his valuable comments. Presented by B. Jónsson.     

Asymptotic bounds for estimators without limit distribution

Let be a general family of probability measures,κ : a functional, and the optimal limit distribution for regular estimator sequences of κ. On intervals symmetric about 0, the concentration of this optimal limit distribution can be surpassed by the asymptotic concentration of an arbitrary estimator sequence only forP in a “small” subset of . For asymptotically median unbiased estimator sequences the same is true for arbitrary intervals containing 0. The emphasis of the paper is on “pointwise” conditions for , as opposed to conditions on shrinking neighbourhoods, and on “general” rather than parametric families.     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .     

Almost semirecursive sets

LetA be a subset of , and leta∉A. The setA is said to be almost semirecursive, if there is a two-place general recursive functionf such thatf(x, y)ε{x, y, a}∧({x, y}⊆A⇌f(x, y)εA) for all . Among other facts, it is proved that ifA and are almost semirecursive sets, thenA is a semirecursive set, and that there exists a wsr^*-set that is neither a wsr-nor an almost semirecursive set. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 188–193, August, 1999.     

An operator model for the oscillation problem of liquids on an elastic bottom

This paper deals with the problem of small oscillations in a liquid layer of finite depth under the assumption that the bottom is an elastic medium. The system of equations corresponding to the problem is written out and explained. The main aim of the paper is to recast these equations in the form

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .