Four methods for constructing anti-Pasch Steiner triple systems are developed. The first generalises a construction of Stinson and Wei to obtain a general singular direct product construction. The second generalises the Bose construction. The third employs a construction due to Lu. The fourth employs Wilson-type inflation techniques using Latin squares having no subsquares of order 2. As a consequence of these constructions we are able to produce anti-Pasch systems of order v for v 31 or 7 (mod 18), for v ≡ 49 (mod 72), and for many other values of v.

The classical canonical Ramsey theorem of Erdős and Rado states that, for any integer q [ges ] 1, any edge colouring of a large enough complete graph contains one of three canonically coloured complete subgraphs of order q. Of these canonical subgraphs, one is coloured monochromatically while each of the other two has its edge set coloured with many different colours. The paper presents a condition on colourings that, roughly speaking, requires them to make effective use of many colours (‘essential infiniteness’); this condition is then shown to imply that the colourings in question must contain large refinements of one of two ‘unavoidable’ colourings that are rich in colours. As it turns out, one of these unavoidable colourings is one of the canonical colourings of Erdős and Rado, and the other is a ‘bipartite variant’ of this colouring.

Let f(x) be an integer-valued polynomial with no fixed integer divisor [ges ] 2, that is, for no integer d [ges ] 2 does d[mid ]f(x) for all integers x. One generalization of the famous Waring problem is to determine whether, for large enough s, the equation

formula here

is solvable in positive integers x1, …, xs for sufficiently large integers n. The existence of such s for every f was established by Kamke [5] in 1921. Subsequent authors (Pillai, Hua [2–4], Vinogradov, Načaev [7], and others) have studied the problem of bounding G(f), the least s for which (1.1) is solvable for all large n.

Questions of local solubility of (1.1), that is, solubility of the congruence

formula here

play a more important and complicated role in this problem than in the classical Waring problem. Let Γ0(f) denote the least number s so that (1.2) is solvable for every pair n, q. It is well known that Γ0(xk) [les ] 4k for every k, but Hua [4] found that for every k, the polynomial

formula here

satisfies Γ0(fk) [ges ] 2k − 1 (take s = 2k−2, q = 2k and n = (−1)k in (1.2)). Clearly G(f) [ges ] Γ0(f), but one can say more by restricting the values of n under consideration, as has been done by several authors in the case f(x) = x4 (for example [1, 6]).

The singular series

formula here

where e(z) = e2πiz, encapsulates the local solubility information. In particular, [Sfr ]s, f(n) [ges ] 0 for every n and [Sfr ]s, f(n) > 0 if and only if (1.2) is soluble for every q. Define G(f) to be the least number s so that for every δ >0 and every n>n0(δ) with [Sfr ]s, f(n) [ges ] δ, (1.1) is soluble. The reason for taking [Sfr ]s, f(n) [ges ] δ instead of [Sfr ]s, f(n) > 0 is that we wish to exclude from consideration certain n lying in sparse sequences for which (1.1) is insoluble but [Sfr ]s, f(n) > 0. For example, taking f(x) = x4, s = 15 and nj = 79·16j (j = 0, 1, …), it can be shown that (1.1) is not soluble for n = nj, that [Sfr ]s, f(nj) > 0 for all j, and that [Sfr ]s, f(nj) → 0 as j → ∞. It is known that G(x4) = 16 (see [1]) and that G(x4) [ges ] 11 almost holds (see [6]).

Let d be a square-free number and let CL(−d) denote the ideal class group of the imaginary quadratic number field ℚ(√−d). Further let h(−d) = #CL(−d) denote the class number. For integers g [ges ] 2, we define [Nscr ]g(X) to be the number of square-free d [ges ] X such that CL(−d) contains an element of order g. Gauss' genus theory demonstrates that if d has at least two odd prime factors (in particular, for almost all d) then CL(−d) contains ℤ2 as a subgroup. Thus N2(X) ∼ 6X/π2. The behaviour of [Nscr ]g(X) is not understood for any other value of g. It is believed that [Nscr ]g(X) ∼ CgX for some positive constant Cg. For odd primes g, H. Cohen and H. Lenstra [3] conjectured that

formula here

N. Ankeny and S. Chowla [1] first showed that [Nscr ]g(X) → ∞ as X → ∞. Although they did not point this out, their method demonstrates that [Nscr ]g(X) [Gt ] X1/2. Recently, M. R. Murty [11] improved this to [Nscr ]g(X) [Gt ] X1/2+1/g. Hitherto this represented the best known lower bounds for [Nscr ]g(X) except in the cases g = 4 and g = 8. In the cases g = 4 or 8, P. Morton [9] used class field theory techniques to show that [Nscr ]g(X) [Gt ] X1−ε. In fact, he demonstrated the elegant result that given any non-negative integers r, s and t, there are ‘many’ d with CL(−d)/CL(−d)8 = ℤr2 × ℤs4 × ℤt8 (see [9] for a precise statement). The complementary question of finding d with p [nmid ] h(−d) has also attracted a lot of attention. H. Davenport and H. Heilbronn [5] proved the striking result that the proportion of d with 3 [nmid ] h(−d) is at least 1/2. For larger primes p, recently W. Kohnen and K. Ono [7] have shown that there are [Gt ] √X/log X square- free integers d [les ] X such that p [nmid ] h(−d).

In this paper, we sharpen Murty's lower bounds on [Nscr ]g(X) for all values of g; see Theorem 1 below. We also offer a simple proof that [Nscr ]4(X) [Gt ] X/√log X; see Proposition 2 below. In §5 we express the hope that these methods may lead to [Nscr ]3(X) [Gt ] X1−ε.

The paper gives the best quantitative forms of Kronecker's theorem.

Let q be a prime and χ be a non-principal character modulo q. Let

formula here

where 1 [les ] t [les ] q is the character polynomial associated to χ (cyclically permuted t places). The principal result is that for any non-principal and non-real character χ modulo q and 1 [les ] t [les ] q,

formula here

where the implicit constant is independent of t and q. Here ∥·∥4 denotes the L4 norm on the unit circle.

It follows from this that all cyclically permuted character polynomials associated with non-principal and non-real characters have merit factors that approach 3. This complements and completes results of Golay, Høholdt and Jensen, and Turyn (and others). These results show that the merit factors of cyclically permuted character polynomials associated with non-principal real characters vary asymptotically between 3/2 and 6.

The averages of the L4 norms are also computed. Let q be a prime number. Then

formula here

where the summation is over all characters modulo q.

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.

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.

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.

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.

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.

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.

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.

A relation is described between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M. Kobayashi. It is shown that this relation continues to hold for the extension of Arnold's strange duality found by C. T. C. Wall and the author. By a method of Ehlers–Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. This method is generalized to the isolated complete intersection singularities embraced in the extended duality. This is used to explain the duality of characteristic polynomials of the monodromy discovered by K. Saito for Arnold's original strange duality and extended by the author to the other cases.

Le but de cet article est de démontrer une version locale de certains résultats classiques concernant l'espace de Hardy H1 et l'action sur cet espace des opérateurs de Calderón–Zygmund. Plus précisément, voici deux exemples de questions étudiées.

La première concerne l'invariance de l'espace H1 relativement aux fonctions maximales utilisées pour le définir. On sait que, si une fonction f est telle qu'une certaine fonction maximale raisonnable M1f soit intégrable dans ℝn, alors toute autre fonction maximale raisonnable M2f est également intégrable dans ℝn. Si on suppose seulement que M1f est intégrable sur une boule de ℝn, que peut-on dire de l'intégrabilitéde M2f?

La seconde concerne l'action sur l'espace H1 des opérateurs de Calderón–Zygmund. Un résultat classique de cette théorie affirme que, si T est un opérateur de Calderón–Zygmund (respectivement un opérateur de Calderón–Zygmund vérifiant la condition T*() = 0) et si f ∈ H1, alors T(f) ∈ L1 (respectivement T(f) ∈ H1). Que peut-on dire de T(f), si on suppose seulement qu'une certaine fonction maximale associée à f est intégrable sur une boule de ℝn?

La deuxième question s'est naturellement posée au cours de notre travail [3], dans lequel nous avons eu besoin de certains résultats de la Section 4 du présent article. A notre connaissance, ces résultats ne se trouvent pas dans la littérature et pour les obtenir, on doit “localiser” convenablement les méthodes utilisées classiquement pour établir les résultats globaux correspondants; nous en rappelons les grandes lignes, qui fixent le plan de notre article. La description de l'action des opérateurs de Calderón–Zygmund sur H1 est basée sur la décomposition atomique de cet espace; de façon un peu similaire, notre réponse à la deuxième question passe par le résultat de localisation de la Section 3 qui est, sans être aussi précis, dans l'esprit des décompositions atomiques. A son tour, la décomposition atomique de H1 est une conséquence du fait que cet espace puisse être défini au moyen de “grandes” fonctions maximales; parallèlement, notre résultat de localisation de la Section 3 utilise une version locale de ce résultat, qui fait l'objet de la Section 2.

The weak type 1 for the Mehler maximal function is studied via a precise estimate for the ‘maximal kernel’. This, in turn, allows the geometry involved in this setting to be described.

Some of the arguments and techniques developed by the authors in a previous paper are applied to the study of the boundedness of certain operators associated with the Ornstein–Uhlenbeck semigroup. In particular, a simple proof is given of the weak type 1 with respect to the Gaussian measure of the Riesz transforms of order 1 and the Littlewood–Paley g-function which then is extended to show the same property for the Riesz transforms of order 2. For Riesz transforms of higher order boundedness is shown in appropriate spaces close to L1.

The essential norm of weighted composition operators on weighted Banach spaces of analytic functions is computed in terms of the weights and the inducing symbols. As a consequence the boundedness and compactness of these operators is characterized. As another consequence the essential norm of composition operators on weighted Bloch spaces is obtained and, consequently, the boundedness and compactness of composition operators on these spaces is also characterized. Particular instances of weighted Bloch spaces are the Lipschitz spaces. The method used allows a unified treatment of the problem of boundedness and compactness on these spaces.

A classical result of Kadison concerning the extension, via the Hahn–Banach theorem, of extremal states on unital self-adjoint linear manifolds (that is, operator systems) in C*-algebras is generalised to the setting of noncommutative convexity, where one has matrix states (that is, unital completely positive linear maps) and matrix convexity. It is shown that if ϕ is a matrix extreme point of the matrix state space of an operator system R in a unital C*-algebra A, then ϕ has a completely positive extension to a matrix extreme point Φ of the matrix state space of A. This result leads to a characterisation of extremal matrix states as pure completely positive maps and to a new proof of a decomposition of C*-extreme points.

When S is a discrete subsemigroup of a discrete group G such that G = S−1S, it is possible to extend circle-valued multipliers from S to G, to dilate (projective) isometric representations of S to (projective) unitary representations of G, and to dilate/extend actions of S by injective endomorphisms of a C*-algebra to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilated action of G. This shows that crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making powerful tools available to study their ideal structure and representation theory. The dilation of the system giving the Bost–Connes Hecke C*-algebra from number theory is constructed explicitly as an application: it is the crossed product C0([Aopf ]f)[rtimes ]ℚ*+, corresponding to the multiplicative action of the positive rationals on the additive group [Aopf ]f of finite adeles.