Writing Elements of PSL(2, q) as Commutators

We prove that for q ≥ 13, an element A of SL(2, q) is the commutator of a generating pair if and only if A ≠ ?I and the trace of A is not 2. Consequently, when q is odd and q ≥ 13, every nontrivial element of PSL(2, q) is the commutator of a generating pair, and when q is even, an element of PSL(2, q) is the commutator of a generating pair if and only if its trace is not 0. The proof of these results also leads to an improved lower bound on the number of T-systems of generating pairs of PSL(2, q).     

Weighted Boundedness for a Class of Rough Multilinear Operators

In this paper we give the (L ^p (ω ^p ), L ^q (ω ^q )) boundedness for a class of multilinear operators, which is simular to the higher-order commutator for the rough fractional integral. In our results the kernel function is merely assumed on a size condition. Received August 21, 2000, Accepted October 24, 2000     

Necessary and sufficient conditions for boundedness of commutators of strongly singular integral operators with weighted Lipschitz functions

In this paper, we prove the commutator T _b generated by the strongly singular integral operator T and the function b is bounded from L ^p (w) to L ^q (w ^1−q ) if and only if b ∈ Lip _β (w), where w ∈ A ₁, 0 < β < 1, 1 < p < n/β and 1/q = 1/p − β/n. To do this, we first show a maximal function estimate for the commutator.     

Polynomial clones on groups of order <Emphasis Type="Italic">pq</Emphasis>

For two distinct primes p, q, we describe those clones on a set of size pq that contain a given group operation and all constant operations. We show that each such clone is determined by congruences and commutator relations. Thus we obtain that there is only a finite number of such clones on a fixed set. Supported by the Austrian Science Fund (FWF), grant P15691.     

A Steiner 5-Design on 36 Points

Up to now, all known Steiner 5-designs are on q + 1 points where q 3 (mod 4) is a prime power and the design is admitting PSL(2, q) as a group of automorphisms. In this article we present a 5-(36,6,1) design admitting PGL(2, 17) × C ₂ as a group of automorphisms. The design is unique with this automorphism group and even for the commutator group PSL(2, 17) × Id ₂ of this automorphism group there exists no further design with these parameters. We present the incidence matrix of t-orbits and block orbits.     

Dolbeault cohomology of G /( P,P )

Let G be a complex connected semi-simple Lie group, with parabolic subgroup P. Let (P,P) be its commutator subgroup. The generalized Borel-Weil theorem on flag manifolds has an analogous result on the Dolbeault cohomology . Consequently, the dimension of is either 0 or . In this paper, we show that the Dolbeault operator has closed image, and apply the Peter-Weyl theorem to show how q determines the value 0 or . For the case when P is maximal, we apply our result to compute the Dolbeault cohomology of certain examples, such as the punctured determinant bundle over the Grassmannian. Received: September 2, 1997; in final form February 9, 1998     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.     

On Residually Finite Groups with Engel-like Conditions

Let m, n be positive integers. Suppose that G is a residually finite group in which for every element x ∈ G there exists a positive integer q = q(x) ≤ m such that x^q is left n-Engel. We show that G is locally virtually nilpotent. Further, let w be a multilinear commutator and G a residually finite group in which for every product of at most 896 w-values x there exists a positive integer q = q(x) dividing m such that x^q is left n-Engel. Then w(G) is locally virtually nilpotent.     

A new characterization of L 2(q) by its noncommuting graph

Let G be a non-abelian group and associate a non-commuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this short paper we prove that if G is a finite group with ∇(G) ≅ ∇(M), where M = L ₂(q) (q = p ⁿ , p is a prime), then G ≅ M.      

Cbmo estimates for Marcinkiewicz integrals on homogeneous Morrey-Herz spaces

The boundedness of the commutator μΩ,b generalized by Marcinkiewicz integral μΩ and a function b(x) ∈ CBMOq(Rn) on homogeneous Morrey-Herz spaces is established.     

Linear Graininess Time Scales and Ladder Operators of Orthogonal Polynomials

In this paper we introduce linear graininess (LG) time scales. We further study orthogonal polynomials (OPs) with the weight function supported on LG time scales and derive the raising and lowering ladder operators by using the time scales calculus. We also derive a second order dynamic equation satisfied by these polynomials. The notion of an LG time scale encompasses the cases of the reals, the h-equidistant grid, the q-grid and, more general, a mixed (q, h)-grid. This allows a unified treatment of the ladder operators theory for classical OPs on these time scales. Moreover we will explain, why exclusively LG time scales provide the right framework for general OP theory.     

Some estimates of maximal commutators for Bochner-Riesz operator on Morrey spaces

In this note, the authors show the boundedness of the maximal commutators of Bocher-Riesz operator B ^δ and that of maximal commutator B _δ1* ^b generated by B ^δ and a Lipschitz function b mapping from M _p ^q (R ⁿ ) into BMO space and also maps from M _p ^q (R ⁿ ) into L _(β−n/q).     

Commutator subgroups of the power subgroups of?some Hecke groups

Let q??3 be a prime and let H(?? _q ) be the Hecke group associated to q. Let m be a positive integer and H ^m (?? _q ) be the mth power subgroup of H(?? _q ). In this work, we study the commutator subgroups of the power subgroups H ^m (?? _q ) of H(?? _q ). Then, we give the derived series for all triangle groups of the form (0;2,q,n) for n a positive integer, since there is a nice connection between the signatures of the subgroups we studied and the signatures of these derived series.     

ON MULTILINEAR COMMUTATORS OF Θ-TYPE CALDERóN-ZYGMUND OPERATORS

In this paper, we discuss the multilinear commutator of θ-type Calderón- Zygmund operators, and obtain that this kind of multilinear commutators is bounded from Lp(Rn) to Lq(Rn), from Lp(Rn) to Triebel-Lizorkin spaces and on certain Hardy type spaces.     

Compactness of Commutators of Riesz Potential on Morrey Spaces

In this paper, the authors give a characterization of the (L ^{p, λ} , L ^{q, λ} )-compactness for the Riesz potential commutator [b,I _α ]. More precisely, the authors prove that the commutator [b,I _α ] is a compact operator from the Morrey space L ^{p, λ} (ℝ ⁿ ) to L ^{q, λ} (ℝ ⁿ ) if and only if b ∈ VMO(ℝ ⁿ ), the BMO-closure of . The research was supported by NSF of China (Grant: 10571015, 10826046) and SRFDP of China (Grant: 20050027025).     

Coprime commutators in PSL(2, q)

We show that every element of PSL(2, q) is a commutator of elements of coprime orders. This is proved by showing first that in PSL(2, q) any two involutions are conjugate by an element of odd order.     

Little jordan clofford algebras

Unital quadratic Jordan algebras J(q, I) determined by nondegenerate quadratic forms with basepoints over a field axe called full Jordan Clifford algebras. In characteristic 2 they have ample outer ideals which are also simple; they come in 3 sizes, tiny, small, and large, where the large are full Clifford algebras but the tiny and small algebras are lacking some of their parts. The simple algebras played a role in Zelmanov's solution of the Burnside Problem. In this paper we will analyze these in more detail, determining their centroids and their local algebras; this is important in the classification of prime Jordan triples of Clifford type in arbitrary char-acterstics. In addition we make a careful charcterization of the tiny, small, and large Clifford algebras. We use this to straighten one or two missteps in a proof from the classification of simple algebras. An important role in our characterization is played by commutators, and we describe the Jor­dan commutator products and Bergmann formulas for Clifford algebras in general.     

Partition models for the crystal of the basic $U_{q}(\widehat {\mathfrak{sl}}_{n})$-module

For each n ≥3, we construct an uncountable family of models of the crystal of the basic U_q([^(\mathfrak sl)]_n)U_{q}(\widehat {\mathfrak {sl}}_{n})-module. These models are all based on partitions, and include the usual n-regular and n-restricted models, as well as Berg’s ladder crystal, as special cases.     

Tight sets and <Emphasis Type="Italic">m</Emphasis>-ovoids of generalised quadrangles

The concept of a tight set of points of a generalised quadrangle was introduced by S. E. Payne in 1987, and that of an m-ovoid of a generalised quadrangle was introduced by J. A. Thas in 1989, and we unify these two concepts by defining intriguing sets of points. We prove that every intriguing set of points in a generalised quadrangle is an m-ovoid or a tight set, and we state an intersection result concerning these objects. In the classical generalised quadrangles, we construct new m-ovoids and tight sets. In particular, we construct m-ovoids of W(3,q), q odd, for all even m; we construct (q+1)/2-ovoids of W(3,q) for q odd; and we give a lower bound on m for m-ovoids of H(4,q ²).