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.     

Analytic Hausdorff gaps II: The density zero ideal

We prove two results about the quotient over the asymptotic density zero ideal. First, it is forcing equivalent to % MathType!End!2!1!, where % MathType!End!2!1! is the homogeneous probability measure algebra of characterc. Second, if it has analytic Hausdorff gaps, then they look considerably different from proviously known gaps of this form. Partially supported by NSERC.     

On a cubic lifting

Let % MathType!End!2!1! be the three-fold cover of SL₂. We prove there is a lifting of the genuine automorphic representations of % MathType!End!2!1! to the automorphic representations of SL₂, using a relative trace formula. Partially supported by NSF DMS 9304580. Partially supported by NSF DMS 7209098.     

Branch rings, thinned rings, tree enveloping rings

We develop the theory of “branch algebras”, which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field % MathType!End!2!1! we contruct a % MathType!End!2!1! which

Atomlessr-maximal sets

We focus on % MathType!End!2!1!(A), the filter of supersets ofA in the structure of the computably enumerable sets under the inclusion relation, whereA is an atomlessr-maximal set. We answer a long-standing question by showing that there are infinitely many pairwise non-isomorphic filters of this type. Research partially supported NSF Grants DMS-96-3465 (Cholak) and DMS-95-00983 (Nies). The authors would like to thank Mike Stob for his help and interest.     

Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras

We consider associativePI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of theT-ideal generated by some Amitsur's Capelli-type polynomialsE _M,L ^* [1]. We recall that two sequencesa _n,b _nare asymptotically equal, and we writea _n ≃b _n,if and only if lim _n→∞(a _n/b _n)=1.In this paper we prove that % MathType!End!2!1!, whereG is the Grassmann algebra. These results extend to all verbally primePI-algebras a theorem of A. Giambruno and M. Zaicev [9] giving the asymptotic equality % MathType!End!2!1! between the codimensions of the matrix algebraM _k(F) and the Capelli polynomials. The second author is partially supported by grants RFFI 04-01-00739a, E02-2.0-26.     

Self-similar solutions for a convection-diffusion equation with absorption inR N

We prove the existence of a positive and smooth solution for the following semilinear elliptic problem: % MathType!End!2!1! for anya∈R ^N , 1<p<1+2/N andq=(p+1)/2. This solution decays exponentially as |x|→+∞. Moreover, if |a| is sufficiently small, this positive and rapidly decaying solution is unique. The existence of a positive, self-similar solution % MathType!End!2!1! follows for the following convection-diffusion equation with absorption: % MathType!End!2!1!. It is also a very singular solution. This solution decays as |x|→+∞ for anyt>0 fixed. Because of the nonvariational nature of the elliptic problem, a fixed point method is used for proving the existence result. The uniqueness is proved applying the Implicit Function Theorem. The work of the first author has been partially supported by Grant 1273/00003/88 of the University of the Basque Country. The work of the second author has been supported by Grant PB 86-0112-C02-00 of the Dirección General de Investigación Científica y Técnica.     

Forbidden paths and cycles in ordered graphs and matrices

Assume % MathType!End!2!1! and let Ω⊂R ^N(N≥4) be a smooth bounded domain, 0∈Ω. We study the semilinear elliptic problem: % MathType!End!2!1!. By investigating the effect of the coefficientQ, we establish the existence of nontrivial solutions for any λ>0 and multiple positive solutions with λ,μ>0 small.     

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 .     

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 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.     

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.     

Between p-points and nowhere dense ultrafilters

We discuss the problem of existence and that of generic existence for various classes of ultrafilters onω. For example, we prove it is consistent that there are nowhere dense ultrafilters while there are no measure zero ultrafilters, and that there are measure zero ultrafilters while there are no P-points. We also prove that every filter base of size <c can be extended to a nowhere dense ultrafilter iff cof ( % MathType!End!2!1!)=c, and that every filter base of size <c can be extended to an ordinal ultrafilter iff ∂=c. Along the way we get a few new results on cardinal invariants of the continuum.     

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.     

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.     

Almost orthogonal submatrices of an orthogonal matrix

Lett≥1 and letn, M be natural numbers,n<M. Leta=(a _i,j ) be ann xM matrix whose rows are orthonormal. Suppose that the ℓ₂-norms of the columns ofA are uniformly bounded. Namely, for allj Using majorizing measure estimates we prove that for every ε>0 there exists, a setI ⊃ {1,…,M} of cardinality at most such that the matrix , whereA _I =(a _i,j ) _j∈I , acts as a (1+ε)-isomorphism from ℓ ₂ ⁿ into . Research supported in part by a grant of the US-Israel BSF. Part of this research was performed when the author held a postdoctoral position at MSRI. Research at MSRI was supported in part by NSF grant DMS-9022140.     

Nichtlineares invariantes Differential

Zusammenfassung Das invariante Differential eines Vektors ist selbst ein Vektor, der vom Orte xⁱ, von dxⁱ, und von selbst abh?ngig ist. Es wird nun infolge dieser Definition in einigen F?llen die m?gliche Form des invarianten Differentials bestimmt. Enrico Bompiani zu seinem wissenschaftlichen Jubil?um.     

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.     

Extensions of lipschitz maps into Banach spaces

It is proved that ifY ⊂X are metric spaces withY havingn≧2 points then any mapf fromY into a Banach spaceZ can be extended to a map fromX intoZ so that wherec is an absolute constant. A related result is obtained for the case whereX is assumed to be a finite-dimensional normed space andY is an arbitrary subset ofX. Supported in part by US-Israel Binational Science Foundation and by NSF MCS-7903042. Supported in part by NSF MCS-8102714.     

Binomial sums, moments and invariant subspaces

The main result of this paper is that if a sequence of complex numbers (a _n)_n≥0 satisfies and % MathType!MTEF!2!1!+- for some integerr≥0, thena _n=0 for alln>r. As an application, we deduce a localized form of a theorem of Allan about nilpotent elements in Banach algebras, and this in turn leads to an invariant-subspace theorem. As a further application, we prove a variant of Carleman's theorem on the unique determination of probability distributions by their moments. The paper concludes with a quantitative form of the main result. Research supported by grants from the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR of Québec, and the Ministry of Education of Québec (co-opération Québec-France).