We show in a direct and elementary way that the spherical building at infinity of every rank 3 affine building which satisfies Tits' Moufang condition, is itself a Moufang building. This result is also true for higher rank affine buildings by Tits' classification [4].

The fractional part of the sequence {αnk}, where α is an irrational real number and k is an integer, was first studied early this century, initiated by the work of Hardy, Littlewood and Weyl. It seems very natural to consider the subsequence {αpk}, where p denotes a prime variable. The pioneering work in this direction was conducted by Vinogradov [13,14]. Improvements have since been made by Vaughan [12], Ghosh [4], Harman [6,7,8] and Jia [11]. The best results to date have been obtained by Harman for k = 1 [9], by Baker and Harman for 2 ≤ k ≤ 12 [1], and by Harman for larger k [8]. In the following work, we shall adopt a sieve technique developed by Harman in [6] to show the following.

Let U(RG) be the group of units of a group ring RG over a commutative ring R with 1. We say that a group is an SIT-group if it is an extension of a group which satisfies a semigroup identity by a torsion group. It is a consequence of the main result that if G is torsion and R = Z, then U(RG) is an SIT-group if and only if G is either abelian or a Hamiltonian 2-group. If R is a local ring of characteristic 0 only the first alternative can occur.

B. Rouxel [7] and S. J. Li and C. S. Houh [6] have generalised the notion of an -submanifold (Chen submanifold) to an k-submanifold. In [1] we have studied the relation between their definitions for the Euclidean case.

In [18] Shioda proved that the space of holomorphic 2-forms on a certain type of elliptic surface is canonically isomorphic to the space of modular forms of weight three for the associated Fuchsian group. Later, Hunt and Meyer [6] made an observation that the holomorphic 2-forms on a more general elliptic surface should in fact be identified with mixed automorphic forms associated to an automorphy factor of the form

for z in the Poincaré upper half plane ℋ, g = and χ(g) = , where g is an element of the fundamental group Γ⊂PSL(2, R) of the base space of the elliptic fibration, χ-Γ→SL(2, R) the monodromy representation, and w: ℋ→ℋ the lifting of the period map of the elliptic surface.

On a semigroup S let the relation ℛ*, sometimes denoted by ℛ, be defined by xℛ*y[(sx = txsy = ty]. A semigroup S is called left type-A, iff the set Es of idempotents of S forms a semilattice under multiplication, each element x of Sis ℛ* related to a (necessarily unique) idempotent x+, and xe = (xe)+x for all x ∈ S, е ∈ Es.

In this paper we give an algorithm for computing the 2-Selmer group of an elliptic curve

which has complexity O(LD(0·5),c1)), where D is the absolute discriminant of the curve. Our algorithm is unconditional but the complexity estimate assumes the GRH and a standard conjecture on the distribution of smooth reduced ideals. This improves on the corresponding algorithm of Birch and Swinnerton-Dyer, which has complexity of O(√D).

The aim of this paper is to give certain conditions characterizing ruled affine surfaces in terms of the Blaschke structure (∇, h, S) induced on a surface (M, f) in ℝ3. The investigation of affine ruled surfaces was started by W. Blaschke in the beginning of our century (see [1]). The description of affine ruled surfaces can be also found in the book [11], [3] and [7]. Ruled extremal surfaces are described in [9]. We show in the present paper that a shape operator S is a Codazzi tensor with respect to the Levi-Civita connection ∇ of affine metric h if and only if (M, f) is an affine sphere or a ruled surface. Affine surfaces with ∇S = 0 are described in [2] (see also [4]). We also show that a surface which is not an affine sphere is ruled iff im(S - HI) =ker(S - HI) and ket(S - HI) ⊂ ker dH. Finally we prove that an affine surface with indefinite affine metric is a ruled affine sphere if and only if the difference tensor K is a Codazzi tensor with respect to ∇.

Let G be a non-nilpotent group in which all proper subgroups are nilpotent. If G is finite then G is soluble [18], and a classification of such groups is given in [14]. The paper [12]. of Newman and Wiegold discusses infinite groups with this property. Clearly such a group is either finitely generated or locally nilpotent. Many interesting results concerning the finitely generated case are established in [12]. Since the publication of that paper there have appeared the examples due to Ol'shanskii and Rips (see [13]) of finitely generated infinite simple p-groups all of whose proper nontrivial subgroups have order p, a prime. Following [12], let us say that a group G is an AN-group if it is locally nilpotent and non-nilpotent with all proper subgroups nilpotent. A complete description is given in Section 4 of [12] of AN-groups having maximal subgroups. Every soluble AN-gvoup has locally cyclic derived factor group and is a p-group for some prime p ([12; Lemma 4.2]). The only further information provided in [12] on AN-groups without maximal subgroups is that they are countable. Four years or so after the publication of [12], there appeared the examples of Heineken and Mohamed [5]: for every prime p there exists a metabelian, non-nilpotent p-group G having all proper subgroups nilpotent and subnormal; further, G has no maximal subgroups and so G/G' is a Prüfer p-group in each case.

Let k be an algebraically closed field. By an algebra is meant an associative finite dimensional k-algebra A with an identity. We are interested in studying the representation theory of Λ, that is, in describing the category mod Λ of finitely generated right Λ-modules. Thus we may, without loss of generality, assume that Λ is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7],[8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.

A finite group G is efficient if it has a presentation on n generators and n + m relations, where m is the minimal number of generators of the Schur multiplier M (G)of G. The deficiency of a presentation of G is r–n, where r is the number of relations and n the number of generators. The deficiency of G, def G, is the minimum deficiency over all finite presentations of G. Thus a group is efficient if def G = m. Both the problem of efficiency and the converse problem of inefficiency have received considerable attention recently; see for example [1], [3], [14] and [15].

In the classical geometry of Banach spaces the notions of smoothness, uniform smoothness, strict and uniform convexity introduced by Day [1] and Clarkson [2] play a very important role and are used in many branches of functional analysis ([3,4,5], for example). In recent years a lot of papers have appeared containing interesting generalizations of these notions in terms of a measure of noncompactness. These new concepts investigated in this paper as near uniform smoothness, local near uniform smoothness and modulus of near smoothness have been introduced by Stachura and Sekowski [6] and Banaś [7] (see also [8,9]).

The punctured neighbourhood theorem an be interpreted as saying that if 0 ∈ C is on the boundary of the spectrum of a Fredholm operator then it must be an isolated point of that spectrum. This extends to semi-Fredholm operators, in particular to operators with closed range and finite dimensional null space. In this note we generalise both the finite dimensionality of the null space and the scalars involved in the definition of an isolated point of the spectrum.

Simons [5] has proved a pinching theorem for compact minimal submanifolds in a unit sphere, which led to an intrinsic rigidity result. Sakaki [4] improved this result of Simons for arbitrary codimension and has proved that if the scalar curvature S of the minimal submanifold Mn of Sn+P satisfies

then either Mn is totally geodesic or S= 2/3 in which case n = 2 and M2 is the Veronese surface in a totally geodesic 4-sphere. This result of Sakaki was further improved by Shen [6] but only for dimension n=3, where it is shown that if S>4, then M3 is totally geodesic (cf. Theorem 3, p. 791).

We extend to Lie algebroids the notion variously known as a double Lie algebra (Lu and Weinstein), matched pair of Lie algebras (Majid), or twilled extension of Lie algebras (Kosmann-Schwarzbach and Magri). It is proved that a matched pair of Lie groupoids induces a matched pair of Lie algebroids. Conversely, we show that under certain conditions a matched pair of Lie algebroids integrates to a matched pair of Lie groupoids. The importance of matched pairs of Lie algebroids has been recently demonstrated by Lu.

Two-sided bounds are obtained for the maximal eigenvalue of a positive matrix by iterating computations of row sums. The result provides an algorithm for approximating the maximal eigenvalue of a nonnegative matrix.

Given a probability space (X, ℱ, μ) and a σ-algebra A ⊂ ℱ, arguably the most powerful tool in gaining information about an ℱ-measurable function f from restricted knowledge of -measurability is that of the conditional expectation E(f | ); written throughout the remainder of this note. Two properties of conditional expectation that may be exploited to gain information, but which also limit conditional expectation's use are the following.

A ring R is called radical if it coincides with its Jacobson radical, which means that Rforms a group under the operation a ° b = a + b + ab for all a and b in R. This group is called the adjoint group R° of R. The relation between the adjoint group R° and the additive group R+ of a radical rin R is an interesting topic to study. It has been shown in [1] that the finiteness conditions “minimax”, “finite Prufer rank”, “finite abelian subgroup rank” and “finite torsionfree rank” carry over from the adjoint group to the additive group of a radical ring. The converse is true for the minimax condition, while it fails for all the other above finiteness conditions by an example due to Sysak [6] (see also [2, Theorem 6.1.2]). However, we will show that the converse holds if we restrict to the class of nil rings, i.e. the rings R such that for any a є R there exists an n = n(a) with an = 0.

In [4] Dipper and James investigated the representation theory of Hecke algebras of type Bn, H(Bn). Using the results in [4] and exploiting the fact that the Hecke algebra of type F4, denoted by H(W), contains two copies of H(B3) certain right ideals of H(W) will be constructed in this paper. These right ideals will be proved to be irreducible on the assumption that H(W) is semisimple.

In this paper, we show that a commutative Noetherian ring which satisfies the radical formula must be of dimension at most one. From this we give a characterization of commutative Noetherian rings that satisfy the radical formula.