A question in the theory of saturated formations of finite soluble groups

This paper examines the following question. If and are saturated formations then is defined to be the class of all soluble groups whose belong to . In general is a formation, but need not be a saturated formation. Here the smallest saturated formation containing is studied.     

Loose block independence

A finite state stationary process is defined to be loosely block independent if long blocks are almost independent in the sense. We show that loose block independence is preserved under Kakutani equivalence and limits. We show directly that any loosely block independent process is the limit of Bernoulli processes and is a factor of a process which is Kakutani equivalent to a Bernoulli shift. The existing equivalence theory then yields that the loosely block independent processes are exactly the loosely Bernoulli (or finitely fixed) processes.     

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.      

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.     

The limits in\bar d of multi-step Markov chainsof multi-step Markov chains

The of the class of multi-step Markov chains is shown to consist of all direct products of Bernoulli processes with processes of rational pure point spectrum. The class of processes that are approached in by their canonical multi-step Markov approximations is also studied. It is found to be strictly smaller than the former class, dense in it, and characterized within it by a certain (noninvariant) property of its rotation factors.     

A new class of contact riemannian manifolds

N. Tanaka ([10]) defined the canonical affine connection on a nondegenerate integrable CR manifold. In the present paper, we introduce a new class of contact Riemannian manifolds satisfying (C) ( for any unit -geodesic ( , where is the generalized Tanaka connection. In particular, when the associated CR structure of a given contact Riemannian manifold is integrable we have a structure theorem and find examples which are neither Sasakian nor locally symmetric but satisfy the condition (C). This work was supported in part by BSRI 98-1425.     

Application des inequations quasi-variationnelles a quelques problemes non lineaires de la physique des plasmas

This paper deals with some problems arising in plasma physics. The typical example is the following: where is the (neither local, nor monotone, nor continuous) operator: . Using a quasi-variational approach, we prove the existence of minimal and maximal solutions for a weak form of this problem, involving a multi-valued operator β. Various generalizations are treated.      

On residue complexes,dualizing sheaves and local cohomology modules

According to Grothendieck Duality Theory [RD], on each varietyV over a fieldk, there is a canonical complex of -modules, theresidue complex . These complexes satisfy (and are characterized by) functorial properties in the categoryV ofk-varieties. In [Ye] a complex is constructed explicitly (when the fieldk is perfect). The main result of this paper is that the two families of complexes, and , which carry certain additional data (such as trace maps…), are uniquely isomorphic. As a corollary we recover Lipman’s canonical dualizing sheaf of [Li], and we obtain formulas for residues of local cohomology classes of differential forms.     

Algebraic number fields elementarily determined by their absolute galois group

LetK be an (infinite) number field. IfK is elementarily determined by its absolute Galois groupG(K) (see (1.1) below), thenK is isomorphic to , R ∩ or some henselian subfield of .     

Some properties of the absolute galois group of a hilbertian field

LetK be a hilbertian field,G(K) its absolute Galois group. IfK is countable, then for a.a. inG(K) ^e , and there is no intermediate field with . Let ∈G(K) ^e . Then for a.a. in .     

An example of pointwise non-convergence of iterated conditional expectation operators

Given ∈, we construct a sequence , … of Borel sub-sigma-algebras on the unit interval with the following property. Suppose the identity functionf(x)=x is transformed by successive conditioning on , then , then , Then the lim sup, with respect ton, will exceed (pointwise almost-everywhere) 1−∈ and its lim inf will be less than ∈. The sequence of functions also will fail to converge in the . This contrasts with the long-open conjecture that if all the come from a finite set of sigma-algebras, then the resulting sequence of functions must converge in . J. L. King was partially supported by NSF grant DMS-9112595.     

On a certain basis inc 0

A basis is constructed inc ₀ such that there exists no bounded linear projection ofc ₀ onto the subspace spanned by a certain subsequence of . This is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the suppervision of Professor A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful advice.     

On simple periodic linear groups— Dense subgroups,permutation representations,and induced modules

We determine the Zariski-dense subgroups of Chevalley groups and their twisted analogues over infinite algebraic extensions of finite fields. It turns out that these are essentially forms of the same group (possibly becoming twisted) over smaller infinite fields. It follows from our classification that if is a simple algebraic group over the algebraic closure of a finite field, then a dense subgroup of can never be maximal, and so the maximal subgroups of are necessarily closed. It follows that Seitz’s determination of the closed maximal subgroups of actually gives all the maximal subgroups. It also enables us to prove that ifG is a simple Chevalley group or twisted type over an infinite algebraic extension of a finite field, then in every non-trivial permutation representation ofG, every finite subgroup has a regular orbit. It follows that every non-trivial permutation module forG over a fieldk iskG-faithful. This is relevant to a programme of studying ideals in group rings of simple locally finite groups. To John Thompson in recognition of his many outstanding contributions to group theory     

Mixing examples in the class of piecewise monotone and continuous maps of the unit interval

A process (T, P) is said to have the “ ” property if there is a uniform, positive lowerbound δ on the separation between theT-P names of (almost) every pair of pointsx≠y. A finite group rotation with partition into distinct points provides a trivial example. Given any process having the property we show that there exists a Bernoulli shiftB so thatT×B is measurably isomorphic to the natural extension of a piecewise monotone, continuous, and expanding map of the unit interval. This construction is applied to produce interval maps which are ergodic but not weak-mixing, weak-mixing but not mixing, and mixing but not exact with respect to their unique absolutely continuous invariant measures, in contrast with the results known for piecewiseC ^1+∈ expansive interval maps. In obtaining these examples we identify a number of nontrivial classes of automorphismsT which admit processes having the property. Supported by NSERC grant OGP0046586 90.     

Counting the relatively finite factors of a Bernoulli shift

LetT acting on (Ω, ℱ, μ) be afinite entropy Bernoulli shift. AT invariant factor is “relatively finite” if a.e. fiber of has a finite, hence constantk, number of points. We say two factors , “sit the same” if there is a measurable measure preserving map ϕ with and . We show here that up to sitting the same there are only finitely many relatively finite factors withk point fibers in a Bernoulli shift, and that they are classified by a certain algebraic structure in the symmetric group onk-points. This work was supported in part by the Miller Institute for Basic Research, U. C. Berkeley.     

Two theorems about projective sets

In this paper we prove two (rather unrelated) theorems about projective sets. The first one asserts that subsets of ℵ₁ which are in the codes are constructible; thus it extends the familiar theorem of Shoenfield that subsets of ω are constructible. The second is concerned with largest countable sets and establishes their existence under the hypothesis of Projective Determinacy and the assumption that there exist only countably many ordinal definable reals. Y. N. Moschovakis is a Sloan Foundation Fellow. During the preparation of this paper, both authors were partially supported by NSF Grant GP-27964.     

The 2-length of the hughes subgroup

For a finite groupG and some prime powerp ⁿ , the -subgroup is defined by . Meixner proved that ifG is a finite solvable group and for somen≧1, then the Fitting length of is bounded by 4n. In the following note it is shown that the 2-length of is at mostn. This result cannot be derived from Meixner’s paper, since his result implies only that the 2-length is bounded by 2n.     

Blaschke-decomposable convex bodies

For a given centred convex bodyK of ℝ,n≥3, let be the class of all convex bodies with the same projection body asK. The question whetherK can be expressed as a Blaschke average of two non-homothetic bodies from is considered. Necessary and sufficient conditions onK to be Blaschke decomposable in are given. The paper provides also a characterization of the bodiesK such that the Blaschke indecomposable bodies in are dense in itself.     

Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.     

Division by a holomorphic matrix in strictly pseudoconvex Domains

LetF andG be respectively a vector- and a matrix-function in a bounded strictly pseudoconvex domainD, with entries holomorphic inD and continuous in . We prove that ifF can be divided locally byG with holomorphic factors in a neighborhood of a given pointw inD, and the rank ofG is maximal at all points of , then the division ofF byG holds globally, with some factors which are holomorphic inD and continuous in . This method applies also to other function algebras in pseudoconvex domains.