A general theory of Hankel forms over domains in one or several variables has been set forth in [6]. In [7] the study of Hankel forms over an annulus in the complex plane ℂ was begun. (An extension of the results of [7] to multiply connected domains was given in [4].) The present paper amplifies the results of [7] in various respects. First of all we define and study more general Hankel forms associated with a one parameter family of projective structures on the annulus. This displays several new features. For instance, we are now dealing with quadratic integral metrics which do not correspond to integration of the square of the function with respect to a weight. Furthermore, whereas in [7] essentially only the issue of the boundedness of Hankel forms was studied, we obtain here rather satisfactory Sp-results, even for 0 < p < 1. The question which remains is, of course, to which extent all this extends to multiply connected domains (or more general (open) Riemann surfaces).

Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences ρ on a completely regular semigroup S such that S/ρ is a semilattice of groups. We shall call such a congruence an SG-congruence.

For a lifted nontrivial additive character λ' and a multiplicative character λ of the finite field with q2 elements, the “Gauss” sums Σ λ'(trg) over g ∈SU(2n, q2) and Σ λ (detg)λ'(trg) over g ∈ U(2n, q2) are considered. We show that the first sum is a polynomial in q with coefficients involving averages of “bihyperkloosterman sums” and that the second one is a polynomial in q with coefficients involving powers of the usual twisted Kloosterman sums. As a consequence, we can determine certain “generalized Kloosterman sums over nonsingular Hermitian matrices”, which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.

During the past few years, some papers of P. Deligne and J.-P. Serre (see [2], [9], [10] and other references cited there) have included an investigation of certain properties of coefficients of modular forms, and in particular Serre [10] (see also [11]) obtained the divisibility property (1) below. Let

be a modular form of integral weight k ≧ 1 on a congruence subgroup of SL2(Z), and suppose that each cn belongs to the ring RK of integers of an algebraic number field K finite over Q. For c ∈ RK and m ≧ 1 an integer, write c ≡ 0 (mod m) if c ∈ m RK and c ≢ 0 (mod m) otherwise. Then Serre showed that there exists α > 0 such that

as x → ∞, where throughout this note N(n ≦ x: P) denotes the number of positive integers n ≦ x with the property P.

For each m≥1, u_{m}(G) is defined to be the intersection of the normalizers of all the subnormal subgroups of defect at most m in G. An ascending chain of subgroups u_{m,i}(G) is defined by setting u_{m,i}(G)/u_{m,i−1}(G)=u_{m}(G/u_{m,i−1}(G)). If u_{m,n}(G)=G, for some integer n, the m-Wielandt length of G is the minimal of such n.

In [3], Bryce examined the structure of a finite soluble group with given m-Wielandt length, in terms of its polynilpotent structure. In this paper we extend his results to infinite soluble groups.

1991 Mathematics Subject Classification. 20E15, 20F22.

The exceptional simple Lie algebras of types E7 and E8 are endowed with optimal $\mathsf{SL}_2^n$ -structures, and are thus described in terms of the corresponding coordinate algebras. These are nonassociative algebras which much resemble the so-called code algebras.

Recently several authors have studied dualizing Goldie dimension of a module: spanning dimension in [2], codimension in [13], corank in [16] and also [9,17,12, 5,11, 6, 4, 7] ([13] may be read in comparison with the others). In the present note we prove the equality corank RP = corank SS, where P is a quasi-projective left R-module and S is its endomorphism ring. This result is an answer to the question [12, p. 1898] and an extension of [3, Corollary 4.3] which shows the above equality for a Σ-quasi-projective left R-module P.

Several writers (4), (6), (7), (9) have used orthogonal expansions in discussing properties of Fourier transformations, and Kober (3) has used such expansions to derive fractional Fourier and Hankel transformations. In 1950 Barrucand (1) noted a reciprocity holding between the coefficients in the expansions in Laguerre polynomials of pairs of functions which are transforms with respect to the kernel J0(2x½).

In this paper we shall first show that if T is a class A(k) operator then its operator transform is hyponormal. Secondly we prove some spectral properties of T via . Finally we show that T has property (β).

A group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.

In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub- or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, $\mathcal{D}=\mathcal{J}$ and minimal conditions on ideals.

Let R be a commutative Noetherian ring, is an ideal of R and M is an R-module. We intend to establish the dual of Grothendieck's Vanishing Theorem for local homology modules. We conjecture that =0 for all i>magRM. We prove this in several cases.

In general, a prime ideal P of a prime Noetherian ring need not be classically localisable. Since such a localisation, when it does exist, is a striking property; sufficiency criteria which guarantee it are worthy of careful study. One such condition which ensures localisation is when P is an invertible ideal [5, Theorem 1.3]. The known proofs of this result utilise both the left as well as the right invertiblity of P. Such a requirement is, in practice, somewhat restrictive. There are many occasions such as when a product of prime ideals is invertible [6] or when a non-idempotent maximal ideal is known to be projective only on one side [2], when the assumptions lead to invertibilty also on just one side. Our main purpose here is to show that in the context of Noetherian prime polynomial identity rings, this one-sided assumption is enough to ensure classical localisation [Theorem 3.5]. Consequently, if a maximal ideal in such a ring is invertible on one side then it is invertible on both sides [Proposition 4.1]. This result plays a crucial role in [2]. As a further application we show that for polynomial identity rings the definition of a unique factorisation ring is left-right symmetric [Theorem 4.4].

Elements\alpha\in A\otimes E of the tensor product of a Banach algebra A and a Banach space E induce systems \{\psi(\alpha):\psi\in E^*\} of elements of A indexed by the dual space E^*, whose joint spectrum belongs to the second dual E^{**}. In this note we investigate when the spectrum actually lies in E\subseteq E^{**}, and extend the spectral mapping theorem P\sigma_A(\alpha)=\sigma_AP(\alpha) to polynomial mappings P:E\to F between Banach spaces. When the algebra A is commutative and the Banach space E=B is another algebra we also reach a sort of vector-valued Gelfand theory.

We prove some results on algebras, satisfying many generic relations. As an application we show that there are Golod–Shafarevich algebras which cannot be homomorphically mapped onto infinite dimensional algebras with finite Gelfand–Kirillov dimension. This answers a question of Zelmanov (Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44(5) 2007, 1185–1195).

We have indicated in our tract [9] that several interesting problems in the theory of numbers are related to results about the evenness of the distribution of the roots v of a polynomial congruence

where f(x) = a0xn + … + an is an irreducible polynomial having integral coefficients and degree n≧2. We alluded, for example, to our work on the Chebyshev problem of the greatest prime factor of n2 – D [8], in which an essential component was our earlier demonstration [6] of the uniform distribution, modulo 1, of v/k when f(x) = x2 – D. But, having pointed out that the quantitative descriptions of such uniformity had to be very sharp for substantial applications, we then noted with regret that little more than mere uniform distribution was obtained in our generalization [7] of [6] to congruences of higher degree. Indeed, it has only been for certain cubic polynomials that results have been produced that are comparable in power with those for quadratic polynomials, and even these depend on the assumption of the unproved hypothesis R* regarding the size of incomplete Kloosterman sums [10].

The purpose of this paper is to give an explicit formula of the Łojasiewicz exponent of an isolated weighted homogeneous singularity in terms of its weights.

Suppose that a complex manifold M is locally embedded into a higher-dimensional neighbourhood as a submanifold. We show that, if the local neighbourhood germs are compatible in a suitable sense, then they glue together to give a global neighbourhood of M. As an application, we prove a global version of Hertling–Manin's unfolding theorem for germs of TEP structures; this has applications in the study of quantum cohomology.

We obtain the complex orbifold structure of the moduli space for one parameter equisymmetric Riemann surfaces of genus two. For each family, by using the orbifold structure, we obtain the points in the moduli corresponding to real algebraic curves and a special form for the period matrices of Riemann surfaces that admit an anticonformal involution. We describe the topological type of anti-conformal involutions admitted by surfaces of the families depending on the type of period matrix.

We study the moduli space of pluriregular threefolds of general type with K3X=8, 10, 12 and p_g(X)=5 and whose canonical map is a finite morphism onto a smooth quadric Q⊂ ℙ4[Copf].