A theorem of Macbeath asserts that μ(A+B)[ges ]min(1, μ(A)+μ(B)) for any subsets A and B of a finite-dimensional torus. We conjecture that, when the obvious exceptions are excluded, a stronger inequality

formula here

holds, and we prove this conjecture under some technical restrictions.

We work in set theory without the axiom of choice: ZF. We show that the axiom BC: Compact Hausdorff spaces are Baire, is equivalent to the following axiom: Every tree has a subtree whose levels are finite, which was introduced by Blass (cf. [4]). This settles a question raised by Brunner (cf. [9, p. 438]). We also show that the axiom of Dependent Choices is equivalent to the axiom: In a Hausdorff locally convex topological vector space, convex-compact convex sets are Baire. Here convex-compact is the notion which was introduced by Luxemburg (cf. [16]).

In [9], Shelah introduced a class of first order theories, which he called simple, properly containing the class of stable theories. Here we prove for simple theories, (i) the equivalence of forking and dividing, (ii) the symmetry and transivity of forking.

The first person to consider the discrepancy of sequences of the type (αnσ)n[ges ]1 (where 0<σ<1) was H. Behnke. The subject was taken up again by one of the authors of this paper, who gave a detailed description of the discrepancy's behaviour if either 0<σ<½ or σ=½ and α2∉ℚ or σ=½ and α−2∈ℕ. In this paper, we study the case of sequences (α√n)n[ges ]1 where α>0 and α2∈ℚ. Both

formula here

are expressed as maxima of finitely many numbers which involve class numbers of imaginary quadratic fields in many cases.

We consider digital expansions with respect to complex integer bases. We derive precise information about the length of these expansions and the corresponding sum-of-digits function. Furthermore we give an asymptotic formula for the sum-of-digits function in large circles and prove that this function is uniformly distributed with respect to the argument. Finally the summatory function of the sum-of-digits function along the real axis is analyzed.

In this paper, we prove the following theorems. (i) Let G be a graph of minimum degree δ[ges ]5. If G is embeddable in a surface σ and satisfies (δ−5)[mid ]V(G)[mid ]+6χ(σ)[ges ]0, then G is edge reconstructible. (ii) Any graph of minimum degree 4 that triangulates a surface is edge reconstructible. (iii) Any graph which triangulates a surface of characteristic χ[ges ]0 is edge reconstructible.

In an attempt to prove all the identities given in Ramanujan's first two letters to G. H. Hardy, G. N. Watson devoted a paper to the evaluation of G1353. At the end of his paper, Watson remarked that his proof was not rigorous as he had assumed certain identities (see (1.1) and (1.2)) which he found empirically. In this paper, we use class field theory, Galois theory and Kronecker's limit formula to justify Watson's assumptions. We shall then use our results to compute some new values of Gn.

By an algebra Λ we mean an associative k-algebra with identity, where k is an algebraically closed field. All algebras are assumed to be finite dimensional over k (except the path algebra kQ). An algebra is said to be biserial if every indecomposable projective left or right Λ-module P contains uniserial submodules U and V such that U+V=Rad(P) and U∩V is either zero or simple. (Recall that a module is uniserial if it has a unique composition series, and the radical Rad(M) of a module M is the intersection of its maximal submodules.) Biserial algebras arose as a natural generalization of Nakayama's generalized uniserial algebras [2]. The condition first appeared in the work of Tachikawa [6, Proposition 2.7], and it was formalized by Fuller [1]. Examples include blocks of group algebras with cyclic defect group; finite dimensional quotients of the algebras (1)–(4) and (7)–(9) in Ringel's list of tame local algebras [4]; the special biserial algebras of [5, 8] and the regularly biserial algebras of [3]. An algebra Λ is basic if Λ/Rad(Λ) is a product of copies of k. This paper contains a natural alternative characterization of basic biserial algebras, the concept of a bisected presentation. Using this characterization we can prove a number of results about biserial algebras which were inaccessible before. In particular we can describe basic biserial algebras by means of quivers with relations.

In this paper we study the L1 mean

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of the exponential sum M(α)= [sum ]n[les ]Xμ(n)e(nα), where μ(n) is the Möbius function and e(x)=e2πix. From the Cauchy–Schwarz inequality and Parseval's identity, we have

formula here

and it is an interesting problem to investigate whether (2) reflects the true order of magnitude of (1).

The Schur algebra S(n, r) has a basis (described in [6, §2.3]) consisting of certain elements ζi,j, where i, j∈I(n, r), the set of all ordered r-tuples of elements from the set n={1, 2, …, n}. The multiplication of two such basis elements is given by a formula known as Schur's product rule. In recent years, a q-analogue Sq(n, r) of the Schur algebra has been investigated by a number of authors, particularly Dipper and James [3, 4]. The main result of the present paper, Theorem 3.6, shows how to embed the q-Schur algebra in the rth tensor power Tr(Mn) of the n×n matrix ring. This embedding allows products in the q-Schur algebra to be computed in a straightforward manner, and gives a method for generalising results on S(n, r) to Sq(n, r). In particular we shall make use of this embedding in subsequent work to prove a straightening formula in Sq(n, r) which generalises the straightening formula for codeterminants due to Woodcock [12].

We shall be working mainly with three types of algebra: the quantized enveloping algebra U(gln) corresponding to the Lie algebra gln, the q-Schur algebra Sq(n, r), and the Hecke algebra, [Hscr ](Ar−1). It is often convenient, in the case of the q-Schur algebra and the Hecke algebra, to introduce a square root of the usual parameter q which will be denoted by v, as in [5]. This corresponds to the parameter v in U(gln). We shall denote this ‘extended’ version of the q-Schur algebra by Sv(n, r), and we shall usually refer to it as the v-Schur algebra. All three algebras are associative and have multiplicative identities, and the base field will be the field of rational functions, ℚ(v), unless otherwise stated. The symbols n and r shall be reserved for the integers given in the definitions of these three algebras.

For normalised generalized matrix functions f and g, we say that f dominates g if f(A)[ges ]g(A) for every M-matrix A. We first demonstrate a finite set of test matrices for any such inequality. Then, using results from group representation theory, all comparisons among immanants in certain classes are determined. This work parallels ongoing research into gmf inequalities on positive semidefinite matrices, for which no finite set of test matrices is available. However, the inequalities for the two classes are quite different, and the test matrices permit more rapid progress in the M-matrix case. Just as in the positive semidefinite case, the gmf inequalities we prove may be used to verify previously unknown determinantal inequalities for M-matrices, such as the symmetrized Fischer inequalities recently proved in the positive semidefinite case.

A primitive permutation group is said to be of twisted wreath type if its socle is both non-abelian and regular. We reduce the study of such primitive permutation groups to the study of ‘maximal non-abelian simple sections’ of non-abelian simple groups.

Hammocks have been considered by Brenner [1], who gave a numerical criterion for a finite translation quiver to be the Auslander–Reiten quiver of some representation-finite algebra. Ringel and Vossieck [11] gave a combinatorial definition of left hammocks which generalised the concept of hammocks in the sense of Brenner, as a translation quiver H and an additive function h on H (called the hammock function) satisfying some conditions. They showed that a thin left hammock with finitely many projective vertices is just the preprojective component of the Auslander–Reiten quiver of the category of [Sscr ]-spaces, where [Sscr ] is a finite partially ordered set (abbreviated as ‘poset’). An important role in the representation theory of posets is played by two differentiation algorithms. One of the algorithms was developed by Nazarova and Roiter [8], and it reduces a poset [Sscr ] with a maximal element a to a new poset [Sscr ]′=a∂[Sscr ]. The second algorithm was developed by Zavadskij [13], and it reduces a poset [Sscr ] with a suitable pair (a, b) of elements a, b to a new poset [Sscr ]′=∂(a,b)[Sscr ]. The main purpose of this paper is to construct new left hammocks from a given one, and to show the relationship between these new left hammocks and the Nazarova–Roiter algorithm. In a later paper [5], we discuss the relationship between hammocks and the Zavadskij algorithm.

The 2-modular and 3-modular decomposition numbers for the sporadic simple O'Nan group O'N and its triple cover are determined. The results are obtained with the help of the computer algebra packages MeatAxe and GAP. Permutation modules and tensor products of modules have been condensed. The result relies on the construction of a 153-dimensional representation for 3·O'N over GF(4) and a 154-dimensional representation for O'N over GF(3).

It is well known that the Horrocks–Mumford bundle [Fscr ] encodes a lot of very interesting geometric information. This is essentially the reason for the fact that much work has been done in order to find other rank-2 bundles on ℙ4. The only nonsplit vector bundles of rank 2 on ℙ4, known up to now, are twists of pullbacks of [Fscr ] by finite coverings f[ratio ]ℙ4→ℙ4. So it seems to be a natural question to consider, instead of ℙ4, other Fano 4-folds. It is the aim of this note to give an example of a rank-2 vector bundle on ℙ1×ℙ3 and to show that it also admits very interesting geometric properties.

Recently J. Coates and R. Greenberg introduced a new important class of extensions of local number fields with finite residue field which they call deeply ramified fields. These extensions play an essential role in their study of the arithmetics of abelian varieties over local fields with finite residue fields [1].

The first aim of this paper is to provide an ‘elementary’ treatment of deeply ramified extensions of local fields with arbitrary perfect residue fields using a method different from the original approach of Coates and Greenberg. Equivalent properties (1), (3)–(8) of deeply ramified extensions in Section 1 are due to them, and for their proofs a presentation of the different as an integral was involved. We translate the most important constructions into the language of the Hasse–Herbrand function (Section 1) and then apply the methods of [3, Chapter 3], where the Hasse–Herbrand function is defined in terms of the norm map. Some of the properties of deeply ramified extensions (two implications in the language of this paper) have been already studied by M. Matignon [7] and J. Fresnel, M. Matignon [5] for different purposes (see Remark 1.6 and the beginning of Section 2).

The second aim of this text is to expose relations among classes of deeply ramified extensions, that of arithmetically profinite extensions (Subsections 2.1–2.4) and that of p-adic Lie extensions (Subsections 2.5–2.7). For local fields with finite residue field we give in Subsection 2.2 an example of a Galois deeply ramified extension with infinite residue extension in which every Galois deeply ramified subextension is not arithmetically profinite; and in Subsection 2.4 we give an example of a Galois deeply ramified extension with finite residue field extension and a nondiscrete set of breaks (that means that this extension is not arithmetically profinite). The main result is that for local fields with finite residue field the class of Galois deeply ramified extensions with finite residue extension and a discrete set of breaks coincides with the class of infinite Galois arithmetically profinite extensions (Proposition 2.3). However, in the case of the fields with infinite residue fields Proposition 2.3 does not hold (Subsection 2.1). In Subsection 2.5 we construct an example that shows that the class of infinite Galois totally ramified arithmetically profinite extensions is strictly larger than the class of the most natural arithmetic origin – the class of totally ramified p-adic Lie extensions. Subsection 2.6 contains an example of a Galois deeply and totally ramified extension L of a local field F with finite residue field such that the norm group of L/E is of finite index in E*.

The main result is that a finitely generated group that is isomorphic to all of its finite index subgroups has free Abelian first homology, and that its commutator subgroup is a perfect group. A number of corollaries on the structure of such groups are obtained, including a method of constructing all such groups for which the commutator subgroup has a trivial centralizer. As an application, conditions are presented for the covering spaces of compact manifolds that determine when the fundamental groups of the base spaces are free Abelian.

The purpose of this note is to generalise, and give a more illuminating proof, of a theorem of [13] (Theorem 1.1 below). Before stating it, we provide some introductory information. Consider the following two sequences of pictures: in each we see a 1-parameter family Xℝ,t of real algebraic hypersurfaces, which undergoes a bifurcation when the parameter t is equal to 0. Note that in Figure 1, both (i) (a) and (i) (b), and in (ii) (b), the surface Xℝ,t has a purely 1-dimensional part, which we have indicated with a dotted line, and that in (i) (b) we have drawn a curve vertically along the middle of the surface to make clearer the way it passes through itself. The reader will observe that in (a) the surface Xℝ,t is homotopically a 2-sphere when t>0 and a 0-sphere when t<0, while in (b) Xℝ,t is a homotopy 1-sphere both for t<0 and t>0.

Such sequences are typical in singularity theory; each is in fact the family of algebraic closures of images of a versal deformation of a codimension 1 singularity of mapping.

Now suppose that the complexification X[Copf ],t is a homotopy n-sphere. In [13] the second author pointed out that it follows that Xℝ,t is a homotopy sphere for t≠0 (allowing the empty set as a −1-sphere). Indeed, in the local situation, or globally in the weighted homogeneous case, there are well-defined integers k+ and k− between −1 and n such that Xℝ,t≃Sk+ for t>0 and Xℝ,t≃Sk− for t<0.

We describe Xℝ,t for t∈ℝ−0 as ‘good’ if the homotopy dimension of Xℝ,t is equal to n. In this case the inclusion Xℝ,t[rarrhk ]Xt is a homotopy equivalence [13, 1.1].

A group G is conjugacy separable if whenever x and y are non-conjugate elements of G, there exists some finite quotient of G in which the images of x and y are non- conjugate. It is known that free products of conjugacy separable groups are again conjugacy separable [19, 12]. The property is not preserved in general by the formation of free products with amalgamation; but in [15] a method was introduced for showing that under certain circumstances, the free product of two conjugacy separable groups G1 and G2 amalgamating a cyclic subgroup is again conjugacy separable. The main result of [15] states that this is the case if G1 and G2 are free-by-finite or finitely generated and nilpotent-by-finite. We show here that the same conclusion holds for groups G1 and G2 in a considerably wider class, including, in particular, all polycyclic-by-finite groups. (This answers a question posed by C. Y. Tang, Problem 8.70 of the Kourovka Notebook [7], as well as two questions recently asked by Kim, MacCarron and Tang in G. Kim, J. MacCarron and C. Y. Tang, ‘On generalised free products of conjugacy separable groups’, J. Algebra 180 (1996) 121–135.)

The quantum group Uq(SLn) introduced by Drinfel'd [2] and Jimbo [5] is a Hopf algebra which is naturally paired with Oq(sln), the coordinate ring of quantum SLn. When q is not a root of unity, the finite dimensional representation theory of Uq(sln) is essentially the same as that of U(sln). Furthermore, it is known that Uq(sln) is essentially a quasitriangular Hopf algebra [2]. When q is a root of unity the situation changes dramatically, and the representation theory of U(sln) is no longer effective. Moreover, Uq(sln) is not quasitriangular. In this case one can consider the quotient Hopf algebra Uq(sln)′, introduced by Lusztig [7], which is a finite dimensional Hopf algebra with a nice representation theory. It is well known that Uq(sl2)′ is quasitriangular. Finite dimensional quasitriangular Hopf algebras are important for the study of knot invariants [11, 12]. Thus, a natural question is: when is Uq(sln)′ quasitriangular? The somewhat unexpected answer is given in Theorem 3.7: it depends sharply on the greatest common divisor of n and the order of q1/2. For these Hopf algebras we classify all the possible R-matrices and give necessary and sufficient conditions for them to be minimal quasitriangular. These conditions depend again on n and the order of q1/2. In the process we describe the groups of Hopf automorphisms of Uq(sln)′ and Oq(SLn)′, where Oq(SLn)′ is a finite dimensional quotient Hopf algebra of Qq(SLn) proved to be the dual of Uq(sln)′ in [15].