Power closure and the Engel condition

A Liep-algebraL is calledn-power closed if, in every section ofL, any sum ofp ⁱ⁺ⁿ th powers is ap ⁱ th power (i>0). It is easy to see that ifL isp ⁿ -Engel then it isn-power closed. We establish a partial converse to this statement: ifL is residually nilpotent andn-power closed for somen≥0 thenL is (3p ^{n +2} +1)-Engel ifp>2 and (3 · 2 ⁿ⁺³+1)-Engel ifp=2. In particular, thenL is locally nilpotent by a theorem of Zel’manov. We deduce that a finitely generated pro-p group is a Lie group over thep-adic field if and only if its associated Liep-algebra isn-power closed for somen. We also deduce that any associative algebraR generated by nilpotent elements satisfies an identity of the form (x+y) ^{p n} =x ^{p n} +y ^{p n} for somen≥1 if and only ifR satisfies the Engel condition. This project was supported by the CNR in Italy and NSF-EPSCoR in Alabama during the first author’s stay at the Università di Palermo.     

On periodic solvable groups having automorphisms with nilpotent fixed point groups

Letp be a prime,G a periodic solvablep′-group acted on by an elementary groupV of orderp ². We show that ifC _G(v) is abelian for eachv ∈V ^# thenG has nilpotent derived group, and ifp=2 andC _G(v) is nilpotent for eachv ∈V ^# thenG is metanilpotent. Earlier results of this kind were known only for finite groups.     

On a class of groups of prime-power order

LetG be a finitep-group,d(G)=dimH ¹ (G, Z _p) andr(G)=dimH ²(G, Z_p). Thend(G) is the minimal number of generators ofG, and we say thatG is a member of a classG _p of finitep-groups ifG has a presentation withd(G) generators andr(G) relations. We show that ifG is any finitep-group, thenG is the direct factor of a member ofG _p by a member ofG _p .     

A sharpL q-Liouville theorem forp-harmonic functions

We studyL ^q-Liouville properties of nonnegativep-superharmonic and, respectively,p-subharmonic functions on a complete Riemannian manifoldM. In particular, we prove that everyp-harmonic functionu ∈L ^q (M) is constant ifq>p−1. Supported by the Academy of Finland, Project 6355.     

Hölder type quasicontinuity

It is proved that a functionuL ^m,p (R ⁿ ) (which coincides with the Sobolev spaceW ^1,p (R ⁿ ) ifm=1) coincides with a Hölder continuous functionw outside a set of smallm,q-capacity, whereq<p. Moreover, ifm=1, then the functionw can be chosen to be close tou in theW ^1,p -norm.     

Groups with few class sizes and the centraliser equality subgroup

LetG be ap-group whose conjugacy classes have at mostk sizes. We prove thatG is abelian-by-(exponentp ^k−1) (ifp=2, exponent 2 ^k−2). It follows that a 2-group with three class sizes is metabelian. Various other results on class sizes are proved, and some conjectures are formulated.     

Crossed products over prime rings

In this paper we obtain necessary and sufficient conditions for the crossed productR *G to be prime or semiprime under the assumption thatR is prime. The main techniques used are the Δ-methods which reduce these questions to the finite normal subgroups ofG and a study of theX-inner automorphisms ofR which enables us to handle these finite groups. In particular we show thatR *G is semiprime ifR has characteristic 0. Furthermore, ifR has characteristicp>0, thenR *G is semiprime if and only ifR *P is semiprime for all elementary abelianp-subgroupsP of Δ⁺(G) ∩G _inn.     

On the algebraic independence in the selberg class

We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H ^b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF _p. This implies that the equationF ^a=G^b with (a, b)=1 has the unique solutionF=H ^b andG=H ^a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.     

Graph algorithms on a tree-structured parallel computer

Parallel algorithms for some graph-theoretic problems on a tree-structured computer are presented. In particular, ifp denotes the number of processing elements, algorithms that run inO(n ²/p) time for finding connected components, transitive closure and the minimum spanning tree of an undirected graph withn vertices are obtained.     

Decomposability and embeddability of discretely Henselian division algebras

LetF be a discretely Henselian field of rank one, with residue fieldk a number field, and letD/F be anF-division algebra. We conduct an exhaustive study of the decomposability of an arbitraryD. Specifically, we prove the following:D has a semiramified (SR)F-division subalgebra if and only ifD has a totally ramified (TR) subfield. However, there may be TR subfields not contained in any SR subalgebra. IfD has prime-power index, thenD is decomposable if and only ifD properly contains a SR division subalgebra. Equivalently,D has a decomposable Sylow factor if and only if ii(D ^⊗n )≠1/n i(D) for somen dividing the period ofD, that is, if and only if the index fails to mimic the behavior of the period ofD. There exists indecomposableD with prime-power periodp ² and indexp ³. Every proper division subalgebra ofD is indecomposable. Conversely, every indecomposableF-division algebra ofp-power index embeds properly in someD ofp-power index if and only ifk does not have a certain strengthened form of class field theory’s Special Case. Semiramified division algebras and division algebras of odd index always properly embed. Finally, these results apply to an extent overk(t), and we prove that there exist indecomposablek(t)-division algebras of periodp ² and indexp ³, solving an open problem of Saltman. Dedicated to the memory of Amitsur Research supported in part by NSF Grant DMS-9100148.     

On the number of elements of given order in a finitep-group

A. Kulakoff [9] proved that forp>2 the numberN _k =N _k (G) of solutions of the equationx ^{p k} =e in a non-cyclicp-groupG is divisible byp ^k+1. This result is a generalization of the well-known theorem of G. A. Miller asserting that the numberC _k =C _k (G) of cyclic subgroups of orderp ^k >p>2 is divisible byp. In this note we show that, as a rule: (1) ifk>1, thenN _k ≡0(modp ^k+p ); (2) ifk>2, thenC _k ≡0(modp ^p ). These facts are generalizations of many results from [1–5,8,9].     

Separable abelianp-groups having certain prescribed chains

An abelianp-groupG is calledp ^ω+1-projective ifp ^ω+1Ext(G, X)=0 for all groupsX. This class of groups constitutes a natural extension of the well-known class of totally projective groups whose members are precisely those groups classifiable by the Ulm-Kaplansky invariants. Fuchs asked whetherp ^ω+1-projective groupsG can be characterized in terms of filtrations ofG. Our Theorem 1 provides counterexamples.     

Espaces de Banach stables

We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofL ^p(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containsl ^p, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL ¹ containsl ^p, for somep in the interval [1, 2].     

Gleason's problem for harmonic Bergman and Bloch functions on half-spaces

On the setting of the half-spaceR ^n–1×R ₊, we investigate Gleason's problem for harmonic Bergman and Bloch functions. We prove that Gleason's problem for the harmonicL ^p -Bergman space is solvable if and only ifp>n. We also prove that Gleason's problem for the harmonic (little) Bloch space is solvable.     

New necessary conditions on the existence of abelian difference sets

In this paper we develop a new method to obtain identities in a group algebraGF(p)G if an abelian difference set of ordern0 (modp) exists inG. We give an explicit formula ifp ² orp ³ is the exactp-power dividingn. This generalizes the approach of Wilbrink, Arasu and the author. The proof presented here uses some knowledge about field extensions of thep-adic numbers.     

D p spaces on bounded symmetric domains ofC n

In this paper, the spaceD ^p(Ω) of functions holomorphic on bounded symmetric domain ofC ⁿ is defined. We prove thatH ^p(Ω)⊂D ^p(Ω) if 0<p≤2, andD ^p(Ω)⊂H ^p(Ω) ifp≥2, and both the inclusions are proper. Further, we find that some theorems onH ^p(Ω) can be extended to a wider classD ^p(Ω) for 0<p≤2.     

Convergence of linear multistep methods for differential equations with discontinuities

Summary A new stability functional is introduced for analyzing the stability and consistency of linear multistep methods. Using it and the general theory of [1] we prove that a linear multistep method of design orderqp1 which satisfies the weak stability root condition, applied to the differential equationy (t)=f (t, y (t)) wheref is Lipschitz continuous in its second argument, will exhibit actual convergence of ordero(h ^p–1) ify has a (p–1)th derivativey ^(p–1) that is a Riemann integral and ordero(h ^p) ify ^(p–1) is the integral of a function of bounded variation. This result applies for a functiony taking on values in any real vector space, finite or infinite dimensional.This work was supported by Grant GJ-938 from the National Science Foundation     

Interior points of convex hulls

If a setX ⊂E ⁿ has non-emptyk-dimensional interior, or if some point isk-dimensional surrounded, then the classic theorem of E. Steinitz may be extended. For example ifX ⊂E ⁿ has int _k X ≠ 0, (0 ≦k≦n) and ifp ɛ int conX, thenp ɛ int conY for someY ⊂X with cardY≦2n−k+1.     

Admissibility of some preliminary test estimators for the mean of normal distribution

Summary Let a random variableX follow ap-variate normal distributionN _p (θ, I _p ) with an unknownp×1 vector θ andp×p identity matrixI _p . The admissibility of a preliminary test estimator using AIC (Akaike's Information Criterion) procedure will be shown ifp=1 and its inadmissibility will be shown ifp≧3 under the loss function based on Kullback-Leibler information measure. Furthermore the two sample case is also considered.     

Types et suites symetriques dansL p , 1≦p<+∞,p≠2

We prove that every normalized sequence inL ^p , weakly null ifp>2 and equivalent to the unit vector basis ofl ² if 1≦p<2, has for allε>0 a subsequence which is 2(1+ε)-symmetric. This result was known forp=1 (H.P. Rosenthal) andp∈N (W.B. Johnson, B. Maurey, G. Shechtman, L. Tzafriri). Here, we use the techniques of stability which were introduced by J.L. Krivine and B. Maurey: as well as providing new results, this approach unifies and simplifies previous known results.