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.     

About entire functions with specialL 2-properties on one-dimensional subspaces of ℂ n

In this paper we consider special elements of the Fock space #x2131; _n . That is the space of entire functionsf:ℂ: ⁿ →ℂ, such that the followingL ²- condition is satisfied: . Here we show that there exists an entire functiong:ℂ ⁿ →ℂ such that for every one-dimensional subspace Π⊂ℂ ⁿ and for all 0<∈<2 we have , but in the limit case ∈=0 we have . This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely for spaces of entire functions.     

On the Corona problem

We show that if f₁, f₂ are bounded holomorphic functions in the unit ball of ℂⁿ such that , |f₁(z)|² + |f₂(z)2|² ≥ δ² >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f₁h₁ + f₂h₂ with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.     

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.     

Pseudo algebraically closed fields over rings

We prove that for almost allσ ∈G ℚ the field has the following property: For each absolutely irreducible affine varietyV of dimensionr and each dominating separable rational mapϕ:V→ there exists a point a ∈ such thatϕ(a) ∈ ℤ^r. We then say that is PAC over ℤ. This is a stronger property then being PAC. Indeed we show that beside the fields other fields which are algebraic over ℤ and are known in the literature to be PAC are not PAC over ℤ.     

On Banach spaces with unconditional bases

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR _n:X→X such thatR _nR_m=R_min(_n,m) ifn≠m and lim _n→∞ R _n x=x for allx∈X. We prove that, ifR _n−R_n −1 factors uniformly through somel _p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL _Λ=closed span , where , has an unconditional basis. Examples include the Hardy space .     

H p (Rn) is equidistributed withL p (Rn)

Let 0<p<∞. LetH ^p (R ⁿ) be the real variable Hardy spaces defined by Stein and Weiss. Let L^p(R ⁿ) be the usual Lebesgue space. It is shown that forf∈L ^p there is an with the distribution functions of |f| and identical and . The converse is trivially true. Research partially supported by NSF Grant #MCS77-02213.     

Carleson measures and the heat equation

Let { }, where { } is the open unit disk on the complex plane { }. In G, we consider analytic solutions u(t, z) ({ }, { }) of the heat equation 2u_t=u_zz with initial data f(z)=u(0, z) belonging to the Fock space F, i.e., to the space of entire functions square summable with the weight e−|z|2.Conditions on a nonnegative measure μ on G are described under which for all f ∈ F we have { } Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 146–155. Translated by S. V. Kislyakov.     

Lower bounds at infinity for solutions of differential equations with constant coefficients

Two extensions of a classical theorem of Rellich are proved: (1) LetP=P(−iϖ/ϖx) be a partial differential operator with constant coefficients in , let the manifolds contained in have codimension ≧k>0, and denote by Γ an open cone in intersecting each normal plane of every such manifold. If ,Pu=0 and it follows thatu=0. (2) Assume in addition that each irreducibe lfactor ofP van shes on a real hypersurface and that Γ contains both normal directions at some such point. If andP(D) u has compact support, the same condition withk=1 implies thatu has compact support. In both results the hypotheses on the cone Γ and on the operatorP are minimal.     

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.     

Isometric embeddings of coinvariant subspaces of the shift operator

Let θ be an inner function. The main aim of this paper is to describe all positive measures on the unit circle such that for all continuous functions f∈H²⊖θH². Bibliography: 8 titles. Translated fromZapiski Nauchnykh Serninarov POMI, Vol. 232, 1996, pp. 5‐15. Translated by V. V. Kapustin.     

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.      

The two sides of a fourier-stieltjes transform and almost idempotent measures

For measuresμ on the circleT the quantities , need not be equal; it is shown, however, that they are continuous with respect to each other whenμ varies on bounded subsets ofM(T), the space of measures onT. It is also shown that measuresμ which areɛ-almost idempotent (i.e. ) are the sum of an idempotent measure and of a measureυ satisfying providedɛ is small enough (as a function of ‖μ‖).     

On small zeros of quadratic forms over finite fields (II)

Let be a quadratic form with integer coefficients and letp denote a prime. Cochrane^[1] proved that ifn≥4 then has a solution satisfying , where . The aim of the present paper is to generalize the above result to finite fields.     

A class of multipliers for $$\mathcal{W}^ \bot $$

Let denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let be the class of multipliers for , i.e. ergodic transformations whose all ergodic joinings with any element of are also in . Fix an ergodic rotationT, a mildly mixing actionS of a locally compact second countable groupG and an ergodic cocycle ϕ forT with values inG. The main result of the paper is a sufficient (and also necessary by [LeP] whenG is countable Abelian andS is Bernoullian) condition for the skew product build fromT, ϕ andS to be an element of . Moreover, the self-joinings of such extensions ofT are described with an application to study semisimple extensions of rotations. Dedicated to Hillel Furstenberg on the occasion of his retirement The first-named author was supported in part by CRDF, grant UM1-2546-KH-03. The second-named author was supported in part by KBN grant 1P03A 03826.     

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.     

Local rigidity for certain groups of toral automorphisms

Let Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on ⁿ is locally rigid, i.e., every action of Γ on ⁿ by C^∞ diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a C^∞ diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, ℤ). Partially supported by NSF grant DMS9011749.     

Endomorphisms of matrix semigroups over division rings

Let be an arbitrary division ring and M _n ( ) the multiplicative semigroup of all n × n matrices over . We describe the general form of endomorphisms of M _n ( ). Supported in part by a grant from the Ministry of Science of Slovenia.