Projective bases of division algebras and groups of central type

Letk be a field. For each finite groupG and two-cocylef inZ ² (G, k ^x ) (with trivial action), one can form the twisted group algebra wherex _σ x _τ =f(σ,τ)x _στ for all σ, τ∃G. Our main result is a short list ofp-groups containing all thep-groupsG for which there is a fieldk and a cocycle such that the resulting twisted group algebra is ak-central division algebra. We also complete the proof (presented in all but one case in a previous paper by Aljadeff and Haile) that everyk-central division algebra that is a twisted group algebra is isomorphic to a tensor product of cyclic algebras.     

Arnon completion of the Dyer-Lashof algebra

In his PhD thesis, Arnon [1] builds a completion of the Dickson algebras which contains a “free root” algebraD _fin on the top Dickson classes. Hu’ng [5] has shown that this algebra is in fact isomorphic to a similar completion (A _μ)* of the dual of the Steenrod algebraA*. Arnon also completed the Steenrod algebraA with respect to its halving homomorphism to obtainA _μ. Here we study an analogous completion of the Dyer-Lashof algebraR to obtainR _μ with canonical subcoalgebrasR _μ[n]. Unlike the Steenrod algebra, we may further completeR _μ with respect to length to obtain . It turns out, somewhat surprisingly, that the dual ( ) contains (A _μ)* as a dense subalgebra. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada.     

On non-closure of range of values of elliptic operator for a plane angle

Summary Let a plane angle of opening α∈(π, 2π). LetP _D andP _N the Dirichlet and Neumann problems associated to the Poisson equation in . ForP _D andP _N it is proved non existence of solution in L _p ( ) whenp=2/(1±π/α). In other words, the ranges of elliptic operators naturally associated toP _D andP _N are not-closed in L _p ( ) forp=2/(1±π/α).

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Sunto Sia } un angolo piano di apertura α∈(π, 2π). SianoP _D eP _N i problemi di Dirichlet e di Neumann associati all'equazione di Poisson in . PerP _D eP _N si prova non esistenza di soluzioni in L _p ( ) quandop=2/(1±π/α). Vale a dire i ranges degli operatori ellittici naturalmente associati aP _D eP _N sono non-chiusi in π^--AgBrα _K L _p ( ) perp=2/(1±π/α).

     

Projections in the space<Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> and the corona theorem for subdomains of coverings of finite bordered Riemann surfaces

LetM be a non-compact connected Riemann surface of a finite type, andR⋐M be a relatively compact domain such thatH ₁(M,Z)=H ₁(R,Z). Let be a covering. We study the algebraH ^∞(U) of bounded holomorphic functions defined in certain subdomains . Our main result is a Forelli type theorem on projections inH ^∞(D). Research supported in part by NSERC.     

Strongly 1-Bounded Von Neumann Algebras

Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if where is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an with . It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ₀(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ₀ is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II ₁-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of . If M and N are strongly 1-bounded and M ∩ N is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II ₁-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded. Received: November 2005, Revision: March 2006, Accepted: March 2006     

A bound on the <Emphasis Type="Italic">k</Emphasis>-domination number of a graph

Let G be a graph with vertex set V(G), and let k ⩾ 1 be an integer. A subset D ⊆ V(G) is called a k-dominating set if every vertex υ ∈ V(G)-D has at least k neighbors in D. The k-domination number γ _k (G) of G is the minimum cardinality of a k-dominating set in G. If G is a graph with minimum degree δ(G) ⩾ k + 1, then we prove that

Motivic torsors

The torsorP _δ=Hom^⊗ (H _DR,H _σ) under the motivic Galois groupG _σ=Aut^⊗ H _δ of the Tannakian category generated by one-motives related by absolute Hodge cycles over a field κ with an embedding σ:k↪ℂ is shown to be determined by its projectionP _σ→P _σ/G _σ ⁰ to a Gal( )-torsor, and by its localizationsP _σ⊕_k k _ξ at a dense subset of orderings ϕ of the field κ, provided κ has virtual cohomological dimension (vcd) one. This result is an application of a recent local-global principle for connected linear algebraic groups over a field κ of vcd ≤1.     

A division problem for\bar \partial _b - CLOSED forms

LetG ₁,…,G_m be bounded holomorphic functions in a strictly pseudoconvex domainD such that . We prove that for each (0,q)-form ϕ inL ^p(∂D), 1<p<∞, there are formsu ₁, …,u _m inL ^p(∂D) such that ΣG _ju_j=ϕ. This generalizes previous results forq=0. The proof consists in delicate estimates of integral representation formulas of solutions and relies on a certainT1 theorem due to Christ and Journé. For (0,n−1)-forms there is a simpler proof that also gives the result forp=∞. Restricted to one variable this is precisely the corona theorem. The author was partially supported by the Swedish Natural Research Council.     

Construction of Rank 2 Semistable Torsion Free Sheaves Which Are Not Limit of Vector Bundles

We show the following theorem of compensated compactness type: Ifu _n ⇁u weakly in the spaceH ^1,p (Ω, ℝ ^k ) and if also in the sense of distributions then ∂_α(∣∇u∣ ^p-2∂_α u)=0. This result has applications in the partial regularity theory ofp-stationary mappings Ω→S ^k −1.     

A nontrivial product in the stable homotopy groups of spheres

Let A be the mod p Steenrod algebra for p an arbitrary odd prime. In 1962, Li-uleviciusdescribed hi and bk in Ext (A|*,*) (Zp, Zp) having bigrading (1,2pi(p-1))and (2,2pk+1 x(p - 1)), respectively. In this paper we prove that for p ≥ 7,n ≥ 4 and 3 ≤ s < p - 1, (Zp,Zp) survives to E∞ in the Adams spectral sequence, where q = 2(p - 1).     

The orders of nonsingular derivations of Lie algebras of characteristic two

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. A study of the set of positive integers which occur as orders of nonsingular derivations of finite-dimensional nonnilpotent Lie algebras of characteristic p > 0 was initiated by Shalev and continued by the present author. In this paper we continue this study in the case of characteristic two. Among other results, we prove that any divisor n of 2^k − 1 with n ⁴ > (2^k − n)³ belongs to . Our methods consist of elementary arguments with polynomials over finite fields and a little character theory of finite groups. This work was partially supported by Ministero dell’Istruzione e dell’Università, Italy, through PRIN “Graded Lie algebras and pro-p-groups of finite width”.     

A variant of Davenport’s constant

Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of ]n[≔ {1, 2,…, n} such that elements of A are incongruent modulo p and non-zero modulo p. Let k ≥ D(G/|A| be any integer where D(G) denotes the well-known Davenport’s constant. In this article, we prove that for any sequence g ₁, g ₂,…, g _k (not necessarily distinct) in G, one can always extract a subsequence with 1 ≤ ℓ ≤ k such that

Classification of quasifinite\mathcal{W}_\infty -modules-modules

It is proved that an irreducible quasifinite -module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight -module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF _n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)⁽¹⁾ spanned by differential operatorsuD ₁ ^m …D ₁ ^m foru ∈F[Γ] (the group algebra), andm _i≥0 with , whereD _i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)⁽¹⁾-module is a module of the intermediate series if Λ is not isomorphic to ℤ. Supported by NSF grant no. 10471091 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from the Ministry of Education of China.     

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

On the maximum modulus principle for polynomials in a quasidisk

LetG⊂C be a quasidisk,K ⊂ G be a compact set, andp _n be a non-constant complex polynomial of degree at mostn. We establish the inequality whereα _n < 0 depends onn, K, and the geometrical structure of ϖG.     

On Lie derivations of Lie ideals of prime algebras

LetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureA _c and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let be the factor Lie algebra and let δ: be a Lie derivation. Suppose that char(A) ≠ 2 andA does not satisfySt ₁₄, the standard identity of degree 14. We show thatR ΩC =Z(R) and there exists a derivation of algebrasD:A →A _c such that for allx∈R. Our result solves an old problem of Herstein.     

ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.     

Properties of Codes from Difference Sets in 2-Groups

A ( v, k, λ)-difference set D in a group G can be used to create a symmetric 2-( v, k, λ) design, , from which arises a code C, generated by vectors corresponding to the characteristic function of blocks of . This paper examines properties of the code C, and of a subcode, C ^o=JC, where J is the radical of the group algebra of G over . When G is a 2-group, it is shown that C^o is equivalent to the first-order Reed-Muller code, , precisely when the 2-divisor of C^o is maximal. In addition, ifD is a non-trivial difference set in an elementary abelian 2-group, and if D is generated by a quadratic bent function, then C^o is equal to a power of the radical. Finally, an example is given of a difference set whose characteristic function is not quadratic, although the 2-divisor of C^o is maximal.     

The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1| (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α ＜ 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn ∑nj=1|zj|pj ＜ 1} (pj ＞ 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α ＜ 1 and some suitable constants βj,γj,pj.     

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.