The study of special radicals was begun by Andrunakievič [1]. A class of prime rings is called special if it is hereditary and closed under prime extensions. The upper radicals determined by special classes are called special. In later works Andrunakievič and Rjabuhin [2] and [3] defined the concept of a special class of modules.

The most classical sufficient condition for the fixed point property of non-expansive mappings FPP in Banach spaces is the normal structure (see [6] and [10]). (See definitions below). Although the normal structure is preserved under finite lp-product of Banach spaces, (1<p≤∞), (see Landes, [12], [13]), not too many positive results are known about the normal structure of an l1,-product of two Banach spaces with this property. In fact, this question was explicitly raised by T. Landes [12], and M. A. Khamsi [9] and T. Domíinguez Benavides [1] proved partial affirmative answers. Here we give wider conditions yielding normal structure for the product X1⊗1X2.

Suppose that is a newform of weight k on Г1(N). Thus f is in particular a cusp form on Г1(N), satisfying

for all n≥1. Associated with f is a Dirichlet character

such that

for all, .

A generalized tensor product of groups was introduced by R. Brown and J.-L. Loday [6], and has led to a substantial algebraic theory contained essentially in the following papers: [6, 7, 1, 5, 11, 12, 13, 14, 18, 19, 20, 23, 24] ([9, 27, 28] also contain results related to the theory, but are independent of Brown and Loday's work). It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as Lie algebras or commutative algebras. However to do so, many non-obvious algebraic identities need to be verified, and various topological proofs (which exist only in the group case) have to be replaced by purely algebraic ones. The work involved is sufficiently non-trivial to make it interesting.

By a representation of the extended Dynkin diagram we shall mean a list of 5 vector spaces P, E1, E2, E3, E4 over an algebraically closed field K, and 4 linear maps a1, a2, a3, a4 as shown.

The spaces need not be of finite dimension.

In their solution of the 4-subspace problem [6], Gelfand and Ponomarev have classified such representations when the spaces are finite dimensional. A representation like (1) can also be viewed as a module over the K-algebra R4 consisting of all 5 × 5 matrices having zeros off the first row and off the main diagonal.

A Banach space E is said to have Mazur's property if every weak* sequentially continuous functional in E” is weak* continuous, i.e. belongs to E. Such spaces were investigated in [5] and [9] where they were called d-complete and μB-spaces respectively. The class of Banach spaces with Mazur's property includes the WCG spaces and, more generally, the Banach spaces with weak* angelic dual balls [4, p. 564]. Also, it is easy to see that Mazur's property is inherited by closed subspaces. The main goal of this paper is to present generalizations of some results of [5] concerning the stability of Mazur's property with respect to forming some tensor products of Banach spaces. In particular, we show in Sections 2 and 3 that the spaces E ⊗εF and Lp(μ, E) inherit Mazur's property from E andF under some conditions. In Section 4, we will also show the stability of Mazur's property under the formation of Schauder decompositions and some unconditional sums of Banach spaces.

For any positive integers n and v let

where d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,

as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

Let M be a compact connected boundaryless surface and f: M → ℝ3 a smooth immersion transverse to a straight line L. Thus there is an even number p of points xεM such that f(x)εL. Under further transversality assumptions on f (see §3) there is a finite number q of points x of M such that the plane containing f(x) and L touches f(M) at f(x). These assumptions are mild in the sense that they hold for any f in an open dense subset of the space of smooth immersions under consideration. Suppose that the Gaussian curvature of f(M) is positive at q+ of these points and negative at q−, with q = q++ q−. Then

where e(M) denotes the Euler number of M.

Suppose that m0 is an integer, m0≥3, ρ = exp(2πi/m0), K = ℚ(ρ, i), v denotes the degree of K, ξ∈K has degree N over ℚ. The length, where is the (irreducible) minimal polynomial for with ξ relatively prime integer coefficients. Feldman [2, p. 49] proved that there is an absolute constant c0>0 such that

From [2, p. 49, Notes 1 and 2] we know that v = φ(m0) or v = 2φ(m0), and φ(m0)≥ c1m0(log log m0)−1 (c1 > an absolute constant), where φ(m0) denotes Euler's function.

In a previous paper [5] the equivalence of randomisation and normal theory distributions of linear combinations was discussed. In the present paper we discuss the asymptotic randomisation distributions of statistics used in analysis of variance and in a closely related problem which includes, in particular, the “problem of m-rankings“. Kruskal [4] has studied the first of these questions in the case where observations are replaced by ranks.

We denote by ‖…‖ the distance to the nearest integer. Let α and β be real. W. M. Schmidt [5] proved that for ε > 0 and N>c1(ε) there is a natural number n such that

This extends a theorem of H. Heilbronn [4] and also sharpens a theorem of H. Davenport [3].

Several properties are known to be inherited from the derived factor group of a group G by other factors Γi = γi(G)/γi + 1(G) of the lower central series. Derek Robinson proved [1] that the denning property of any class of groups which is closed under the forming of homomorphic images of tensor products is so inherited. Possibilities for here include the following classes.

Number fields such as described in the title play a rôle in the study of Artin L-functions and automorphic forms for the groups SL2 over rings of integers in quadratic extensions of ℚ. They are also of some interest on their own. We have not found many examples in the literature. Lang [4] mentions an unramified A5-extension of a real quadratic number field which is due to E. Artin.

In Muir's Theory of Determinants, Vol. III, pp. 232–237, there will be found accounts of papers by H. Nägelsbach, J. Hammond and J. W. L. Glaisher, in which expressions for the Bernoulli numbers are obtained in terms of determinants. In the present paper an expression for Bn will be derived which appears to be new, but which is very like some of those mentioned by Muir.