The motivation for the theory of Euler characteristics of groups, which was introduced by C. T. C. Wall [21], was topology, but it has interesting connections to other branches of mathematics such as group theory and number theory. This paper investigates Euler characteristics of Coxeter groups and their applications. In his paper [20], J.-P. Serre obtained several fundamental results concerning the Euler characteristics of Coxeter groups. In particular, he obtained a recursive formula for the Euler characteristic of a Coxeter group, as well as its relation to the Poincaré series (see §3). Later, I. M. Chiswell obtained in [10] a formula expressing the Euler characteristic of a Coxeter group in terms of orders of finite parabolic subgroups (Theorem 1). These formulae enable us to compute Euler characteristics of arbitrary Coxeter groups.

On the other hand, the Euler characteristics of Coxeter groups W happen to be intimately related to their associated complexes [Fscr ]W, which are defined by means of the posets of nontrivial parabolic subgroups of finite order (see §2.1 for the precise definition). In particular, it follows from the recent result of M. W. Davis [13] that if [Fscr ]W is a product of a simplex and a generalized homology 2n-sphere, then the Euler characteristic of W is zero (Corollary 3.1). The first objective of this paper is to generalize the previously mentioned result to the case when [Fscr ]W is a PL-triangulation of a closed 2n-manifold which is not necessarily a homology 2n-sphere. In other words (as given below in Theorem 3), if W is a Coxeter group such that [Fscr ]W is a PL- triangulation of a closed 2n-manifold, then the Euler characteristic of W is equal to 1−χ([Fscr ]W)/2.

It is proved that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof is that the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F).

Let [Bscr ]2 denote the family of all circular discs in the plane. It is proved that the discrepancy for the family {B1 × B2 [ratio ] B1, B2 ∈ [Bscr ]2} in R4 is O(n1/4+ε) for an arbitrarily small constant ε > 0, that is, it is essentially the same as that for the family [Bscr ]2 itself. The result is established for the combinatorial discrepancy, and consequently it holds for the discrepancy with respect to the Lebesgue measure as well. This answers a question of Beck and Chen. More generally, we prove an upper bound for the discrepancy for a family {Πki=1Ai[ratio ] Ai∈Ai, i = 1, 2, …, k}, where each Ai is a family in Rdi, each of whose sets is described by a bounded number of polynomial inequalities of bounded degree. The resulting discrepancy bound is determined by the ‘worst’ of the families Ai , and it depends on the existence of certain decompositions into constant-complexity cells for arrangements of surfaces bounding the sets of Ai. The proof uses Beck's partial coloring method and decomposition techniques developed for the range-searching problem in computational geometry.

In this paper we define three elements of a certain generalised cohomology ring BP〈m, n〉* BVk. Here m, n and k are non-negative integers with k+m[les ]n+1, there is a fixed prime p not exhibited in the notation, and Vk is an elementary Abelian p-group of rank k. We show that these elements are equal; this is striking, because the three definitions are very different. The significance of our equation is not yet entirely clear, but it makes contact with other work in the literature in a number of fascinating ways.

A Coxeter system is a pair (W, S) where W is a group and where S is a set of involutions in W such that W has a presentation of the form

formula here

Here m(s, t) denotes the order of st in W and in the presentation for W, (s, t) ranges over all pairs in S × S such that m(s, t) ≠ ∞. We further require the set S to be finite. W is a Coxeter group and S is a fundamental set of generators for W.

Obviously, if S is a fundamental set of generators, then so is wSw−1, for any w∈W. Our main result is that, under certain circumstances, this is the only way in which two fundamental sets of generators can differ. In Section 3, we will prove the following result as Theorem 3.1.

Let G be a locally compact Abelian group and consider the group algebra L1(G) of all complex-valued absolutely integrable functions on G. Some invariants are found that are necessarily shared by groups with isomorphic group algebras, such as dimension, ranks or minimal divisible extensions in the case of torsion-free groups. This sheds some light on the question of finding out to what extent the algebra structure of L1(G) reflects the topological group structure of G. It is shown in particular that some classes of torsion- free locally compact Abelian groups are characterized by their group algebra.

An effective method is given for computing the Hausdorff dimension of the boundary of a self-similar digit tile T in n-dimensional Euclidean space:

formula here

where 1/c is the contraction factor and λ is the largest eigenvalue of a certain contact matrix first defined by Gröchenig and Haas.

The commonly known properties of continuous measures such as the symmetry of the A∞ condition, its equivalence to the reverse Hölder inequality, the left-openness of the Ap condition, etc., are no longer necessarily true when the underlying measure is allowed to have atoms. The measures that preserve these properties are called good measures. The class of good measures is investigated and various criteria for a measure to belong to this class are presented.

For a sequence (cn) of complex numbers, the quadratic polynomials fcn(z) := z2 + cn and the sequence (Fn) of iterates Fn := fcn∘…∘fc1 are considered. The Fatou set [Fscr ](cn) is by definition the set of all z ∈ [Copf ]ˆ such that (Fn) is normal in some neighbourhood of z, while the complement of [Fscr ](cn) is called the Julia set [Jscr ](cn). The aim of this article is to study the connectedness and stability of the Julia set [Jscr ](cn) provided that the sequence (cn) is bounded.

Throughout this paper, we adopt the same notations as in [1, 6, 8] such as the Möbius group M(ℝn), the Clifford algebra Cn−1, the Clifford matrix group SL(2, Γn), the Clifford norm of

formula here

and the Clifford metric of SL(2, Γn) or of the Möbius group M(ℝn)

formula here

where [mid ]·[mid ] is the norm of a Clifford number and

formula here

represents fi ∈ M(ℝn), i = 1, 2, and so on. In addition, we adopt some notions in [6, 12]: the elementary group, the uniformly bounded torsion, and so on. For example, the definition of the uniformly bounded torsion is as follows.

It is known that, for a transcendental entire function f, the Hausdorff dimension of J(f) satisfies 1 [les ] dimJ(f) [les ] 2. For each d ∈ (1, 2), an example of a transcendental entire function f with dimJ(f) = d is given. It is then indicated how this function can be modified to produce a transcendental meromorphic function F with one pole with dimJ(F) = d. These appear to be the first examples of Julia sets with non-integer dimensions whose dimensions have been calculated exactly.

The paper proves the existence and multiplicity of periodic solutions of the equation

formula here

where g[ratio ]R→R is of class C1, h[ratio ]R→R is continuous and 2π-periodic, and s is a parameter.

Let T be an ergodic and free ℤdrotation on the d-dimensional torus [ ]d given by

formula here

where (m1, …, md) ∈ ℤd, (z1, …, zd) ∈ [ ]d and [αjk]j,k=1 …, d ∈ Md(ℝ). For a continuous circle cocycle ϕ[ratio ]ℤd × [ ]d → [ ](ϕm+n(z) = ϕm(Tnz)ϕn(z) for any m, n ∈ ℤd), the winding matrix W(ϕ) of a cocycle ϕ, which is a generalization of the topological degree, is defined. Spectral properties of extensions given by

formula here

are studied. It is shown that if ϕ is smooth (for example ϕ is of class C1) and det W(ϕ) ≠ 0, then Tϕ is mixing on the orthocomplement of the eigenfunctions of T. For d = 2 it is shown that if ϕ is smooth (for example ϕ is of class C4), det W(ϕ) ≠ 0 and T is a ℤ2-rotation of finite type, then Tϕ has countable Lebesgue spectrum on the orthocomplement of the eigenfunctions of T. If rank W(ϕ) = 1, then Tϕ has singular spectrum.

Troels Jørgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in ‘most’ cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty (see Theorem 3.1). We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian groups which are not (non-trivial) free products of surface groups and infinite cyclic groups (see Corollary 3.3). These results are extensions of similar results for purely loxodromic groups which can be found in [4]. Thurston [32] previously established these results in the case when the Kleinian groups are freely indecomposable (see also Ohshika [24, 25, 27]). Using different techniques from ours, Ohshika [26] has proven versions of these results for purely loxodromic function groups.

Certain results which have been proved or conjectured for the subclass consisting of those in L2(ℝ) are generalized to entire functions of exponential type bounded on the real axis.

It is important and interesting to study harmonic functions on a Riemannian manifold. In an earlier work of Li and Tam [21] it was demonstrated that the dimensions of various spaces of bounded and positive harmonic functions are closely related to the number of ends of a manifold. For the linear space consisting of all harmonic functions of polynomial growth of degree at most d on a complete Riemannian manifold Mn of dimension n, denoted by [Hscr ]d(Mn), it was proved by Li and Tam [20] that the dimension of the space [Hscr ]1(M) always satisfies dim[Hscr ]1(M) [les ] dim[Hscr ]1(ℝn) when M has non-negative Ricci curvature. They went on to ask as a refinement of a conjecture of Yau [32] whether in general dim [Hscr ]d(Mn) [les ] dim[Hscr ]d(ℝn) for all d. Colding and Minicozzi made an important contribution to this question in a sequence of papers [5–11] by showing among other things that dim[Hscr ]d(M) is finite when M has non-negative Ricci curvature. On the other hand, in a very remarkable paper [16], Li produced an elegant and powerful argument to prove the following. Recall that M satisfies a weak volume growth condition if, for some constant A and ν,

formula here

for all x ∈ M and r [les ] R, where Vx(r) is the volume of the geodesic ball Bx(r) in M; M has mean value property if there exists a constant B such that, for any non- negative subharmonic function f on M,

formula here

for all p ∈ M and r > 0.

Central to the Fourier development of automorphic forms on GL(n, ℝ) are the class 1 principal series Whittaker functions Wn,a(z), which were first studied systematically by Jacquet [13]. (See Section 2 below for the definition of Wn,a(z).)

Of particular interest are the Mellin transforms Mn,a(s) (s ∈ [Copf ]n−1) of Wn,a(z). (See equation (2.11) below.) For example, such transforms, and analogous Mellin transforms of products of Whittaker functions, arise as archimedean Euler factors for certain automorphic L-functions (see [5, 6, 8, 21, 22] for discussions and examples.) Moreover, Mn,a(s) has relevance to the problem of special values of Whittaker functions; cf. [7] in the case n = 3.

Friedberg and Goldfeld [11] have shown Mn,a(s), for general n, to have analytic continuation and to satisfy certain recurrence relations. However, explicit formulae for these transforms have, until the present work, been deduced only for n [les ] 4. In particular, both M2,a(s) (cf. [4, 25]) and M3,a(s) (cf. [7, 9, 19]) are expressible as (some powers of 2 and π times) ratios of gamma functions. On the other hand, M4,a(s) (cf. [22]) may be realized essentially as a hypergeometric series of type 7F6(1), or equivalently as a sum of two series of type 4F3(1). (See Section 2 below for a brief general discussion of hypergeometric series.)

Asymptotics on the diagonal of the Green function and, as a consequence, asymptotic distribution of the spectrum for the semibounded Sturm–Liouville operator are obtained.

Simple singularities of functions on space curves are classified. It is shown that their bifurcation sets have properties very similar to those of functions on smooth manifolds and complete intersections: the k(π, 1)-theorem for the bifurcation diagram of functions is true, and both this diagram and the discriminant are Saito's free divisors.

Monotonicity with respect to the order v of the magnitude of general Bessel functions [Cscr ]v(x) = aJv(x)+bYv(x) at positive stationary points of associated functions is derived. In particular, the magnitude of [Cscr ]v at its positive stationary points is strictly decreasing in v for all positive v. It follows that supx[mid ]Jv(x)[mid ] strictly decreases from 1 to 0 as v increases from 0 to ∞. The magnitude of x1/2[Cscr ]v(x) at its positive stationary points is strictly increasing in v. It follows that supx[mid ]x1/2[Cscr ]v(x)[mid ] equals √2/π for 0 [les ] v [les ] 1/2 and strictly increases to ∞ as v increases from 1/2 to ∞.

It is shown that v1/3supx[mid ]Jv(x)[mid ] strictly increases from 0 to b = 0.674885… as v increases from 0 to ∞. Hence for all positive v and real x,

formula here

where b is the best possible such constant. Furthermore, for all positive v and real x,

formula here

where c = 0.7857468704… is the best possible such constant.

Additionally, errors in work by Abramowitz and Stegun and by Watson are pointed out.