We show that the analytic rank, as defined by Murphy, of a unital graph C*-algebra is either 1 or 0, depending on whether or not the underlying graph possesses an initial loop. As a corollary, we show that the analytic rank of certain quantum spaces (including some quantum spheres, projective spaces and lens spaces) is 1.

We consider the equation

a1x1 + ··· + anxn = 1, (1.1)

where the coefficients a1, …, an are non-zero elements of a field K, to be solved for (x1, …, xn) in a finitely generated multiplicative subgroup G of (K*)n. Evertse [2] and van der Poorten and Schlickewei [6] showed, independently, for the most important case in which K is a number field and G is a group of S-units in K, that (1.1) has only finitely many solutions (x1, …, xn) for which no subsum in (1.1) vanishes. Such solutions are called non-degenerate.

In 1990, Schlickewei [7] was the first to obtain an explicit upper bound for the number of non-degenerate solutions of (1.1) when G is a group of S-units of K. His result went through various successive improvements, and the best result to date is the bound (235n2)n3s by Evertse [3, Theorem 3], where s is the cardinality of S. Very recently, Evertse, Schlickewei and Schmidt [4, 5] obtained a remarkable result. They showed, again for K a number field, that the number of non-degenerate solutions of (1.1) with (x1, …, xn) in a finitely generated subgroup of (K*)n of rank r is at most c(n)r+2, with c(n) = exp((6n)4n). The importance of this result lies with its uniformity with respect to the rank r of the group and its independence of the field K. The proofs of the above-mentioned results are all quite difficult and depend on deep tools from Schmidt's Subspace Theorem and diophantine approximation. In this paper we consider equation (1.1) in the rational function field K = k(t) where k is an algebraically closed field of characteristic 0. Of course, results of this type will follow rather easily from the above-mentioned results by a specialization argument. However, our object is to show that our approach to the function field case is quite elementary and certainly very different, being a simple and direct consequence of the powerful abc-theorem.

Before stating our result, we first define proportional solutions of (1.1). We say that (x1, …, xn) and (x′1, …, x′n) are proportional if xi, /, x′i ∈ k, 1 [les ] i [les ] n. This determines equivalence classes of solutions.

Let G be a finite group acting faithfully and transitively on a set Ω of size m, and let E = {x1, …, xk} be a generating set for G with x1x2…xk = 1. If x∈G has cycles of length r1, …, rl in its action on Ω, define ind (x) = [sum ]l1(ri−1). Then the genusg = g(G, Ω, E) is defined by

formula here

We prove that if n>66 and (n, 30) = 1, then there exist uncountably many infinite simple (2, 3, n)- groups, that is, groups generated by a pair of elements x, y, say, where the orders of x, y and xy are 2, 3 and n, respectively. This extends previous results of Schupp and the authors.

These results are used to prove the existence of subgroups of the modular group with special arithmetic properties.

We prove that Schanuel's conjecture for the reals is equivalent to a uniform version of itself.

In this paper, the strongly Cohen–Macaulay ideals of second analytic deviation one are characterized in terms of the depth properties of the powers of the ideal in the ‘standard range’. This provides an explanation of the behaviour of certain ideals that have appeared in the literature.

The purpose of this note is initially to present an elementary but surprising connectedness principle pertaining to the intersection of a fixed subvariety X of some ambient space Z with another subvariety Y which is ‘mobile’ (in the sense of being movable, rather than actually moving). It is via this mobility that monodromy enters the picture, permitting the crucial passage from ‘relative’ or total-space irreducibility to ‘absolute’ or fibrewise connectedness (and sometimes irreducibility). A general form of this principle is given in Theorem 2 below.

a structure theorem is given for $n$-dimensional smooth subvarieties of the grassmannian $g(1,n)$, with $n\geq n+3$, that can be isomorphically projected to $g(1,n+1)$. a complete classification in the cases $n=2n+1$ and $n=2n$ follows, as a corollary.

The ‘isomorphism problem for integral group rings’ (IP) is the question whether for two finite groups G and H, the existence of an isomorphism of the integral group rings ℤG and ℤH implies that G and H are isomorphic. Though (IP) is not true in general [4], it is still an interesting question for which classes of finite groups (IP) has a positive solution. In this note, we want to show that (IP) holds for finite groups of Lie type associated to Chevalley groups of universal type.

Note that if U(ℤG) denotes the units of ℤG, and V(ℤG) stands for the units with augmentation 1, then U(ℤG)≅V(ℤG)×U(ℤ), and we can always assume that H[les ]V(ℤG).

In a recent paper, Külshammer, Olsson, and Robinson proved a deep generalization of the Nakayama conjecture for symmetric groups. We provide a similar but a shorter and relatively elementary proof of their result. Our method enables us to obtain a more general $H$-analogue of the Nakayama conjecture where $H$ is a set of positive integers.

1. Ordered spaces

Risking semantic incoherence, we propose in this paper to use the term ordered space to mean any totally ordered set X equipped with a topology [Tscr ], while retaining the term linearly ordered space for its usual role of denoting the special case in which [Tscr ] is the order topology: the topology generated by the order-open initial and final segments of X.

The categories that we shall consider are full subcategories of the category OrdTop of ordered spaces and order-preserving continuous maps. It is easy to see that — in contrast to the situation in the full subcategory LinOrdTop of linearly ordered spaces — every inverse spectrum in OrdTop has a limit and that OrdTop admits subspaces and quotient spaces. These limits, subspaces and quotient spaces are all constructed precisely as in the category Top and are given their naturally induced orderings: where, however, we note that, for the ordered space X to induce an ordering on a quotient space, the quotient space (construed as a decomposition) must partition X into intervals rather than into arbitrary subsets. (In the terminology of [1], these constructions define subspaces and quotient spaces in OrdTop as, respectively, the domains and codomains of extremal monomorphisms and epimorphisms.) The purpose of the paper is to identify the closures of LinOrdTop under these constructions.

Let $(W,S)$ be a finite Coxeter system, and let $J\,{\subseteq}\, S$. Any $w\,{\in}\,W$ has a unique factorization $w\,{=}\,w^Jw_J$, where $w_J$ belongs to the parabolic subgroup $W_J$ generated by $J$, and $w^J$ is of minimal length in the coset $wW_J$. It is shown here that $w_I\,{=}\,w^J$ if and only if $w^I\,{=}\,w_J$, for all $I,J\,{\subseteq}\,S$. Furthermore, a similar symmetry property in arbitrary $(W_I,W_J)$-double cosets is conjectured, which links this result to the Solomon descent algebra of $W$.

Let f be an orientation-preserving univalent harmonic mapping of the unit disk U. Then f=h+[gmacron], where h and g are analytic in U. Furthermore, f satisfies the equation

formula here

in U, where a(z)=g′(z)/h′(z), and [mid ]a(z)[mid ]<1 in U. The function a(z) is the analytic dilatation of f.

In [2], Hengartner and Schober proved the following version of the Riemann mapping theorem for univalent harmonic mappings.

In his curriculum vitae, John Hawkes lists his research interests as geometric measure theory, probability (Lévy processes), and potential theory (probabilistic). In fact, he made significant contributions to all three areas, and there are strong relationships between them. He used both geometric measure theory and potential theory as tools for his study of the trajectories of particular Lévy processes, but in many cases he needed to develop the tool before it was ready to be used. We will summarise his research later, but first we discuss what is known of his life history.

We work in the smooth category.

An (oriented) (ordered) m-component n-(dimensional) link is a smooth oriented submanifold L = {K1, …, Km} of Sn+2 which is the ordered disjoint union of m manifolds, each PL-homeomorphic to the standard n-sphere. If m = 1, then L is called a knot.

We say that m-component n-dimensional links L0 and L1 are (link-)concordant or (link-)cobordant if there is a smooth oriented submanifold C˜ = {C1, …, Cm} of Sn+2 × [0, 1] which meets the boundary transversely in ∂C˜, is PL-homeomorphic to L0 × [0, 1], and meets Sn+2 × {l} in Ll (l = 0, 1). If m = 1, then we say that n-knots L0 and Ll are (knot-)concordant or (knot-)cobordant. Then we call C a concordance-cylinder of the two n-knots L0 and Ll.

If an n-link L is concordant to the trivial link, then we call L a slice link.

If an n-link L = {K1, …, Km} ⊂ Sn+2 = ∂Bn+3 ⊂ Bn+3 is slice, then there is a disjoint union of (n + 1)-discs Dn+11 [amalg ] … [amalg ] Dn+1m in Bn+3 such that Dn+1i ∩ Sn+2 = ∂Dn+1i = Ki. (Dn+11, …, Dn+1m) is called a set of slice discs for L. If m = 1, then Dn+11 is called a slice disc for the knot L.

In [8, 6] it was shown that for each k and n such that 2k>n, there exists a contractible k-dimensional complex Y and a continuous map ϕ[ratio ][ ]n→Y without the antipodal coincidence property, that is, ϕ(x)≠ϕ(−x) for all x∈[ ]n. In this paper it is shown that for each k and n such that 2k>n, and for each fixed-point free homeomorphism f of an n-dimensional paracompact Hausdorff space X onto itself, there is a contractible k-dimensional complex Y and a continuous map ϕ[ratio ]X→Y such that ϕ(x)≠ϕ(f(x)) for all x∈X. Various results along these lines are obtained.

As well as the arc-sine laws for the time spent positive before time 1 and the time at which the maximum over [0, 1] occurs, Paul Lévy also established a third arc-sine law for Brownian motion; this relates to the time of the last visit to zero before time 1. For random walks, analogues of the first two of the these results are well known. Here it is shown that for any random walk satisfying Spitzer's condition there is an analogue of the third, but care has to be taken in its formulation. An associated conditional uniform law and some applications to stable processes are also given.

This paper provides a new, simpler proof of the fact that the smallest volume hyperbolic 3-manifold has betti number at least 3. In the process, the best known lower bound on the volume of such a manifold is improved.