The exact best possible range of p is determined such that any dyadic $A_{1}$ weight w on $\mathbb{R}^{n}$ satisfies a reverse Hölder inequality for p, which depends on the dimension n and the corresponding $A_{1}$ constant of w. The proof is based on an effective linearization of the dyadic maximal operator applied to dyadic step functions.

A proof is given that for each non-negative integer $g$, there is an infinite family of knotted surfaces of genus $g$, none of which is ambient isotopic to itself with the orientation reversed.

Let $\Cal F$ be a holomorphic foliation (possibly with singularities) on a non-singular manifold $M$, and let $V$ be a complex analytic subset of $M$. Usual residue theorems along $V$ in the theory of complex foliations require that $V$ be tangent to the foliation (that is, a union of leaves and singular points of $V$ and $\Cal F$); this is the case for instance for the blow-up of a non-dicritical isolated singularity. In this paper, residue theorems are introduced along subvarieties that are not necessarily tangent to the foliation, including the blow-up of the dicritical situation.

functional identities for the classical rogers dilogarithm associated to simply laced dynkin diagrams are proved uniformly in the setting of $y$-systems related to clusters.

Hyperplane arrangements in a three-dimensional vector space are considered in this paper. A characterization of the freeness of such an arrangement is given in terms of the characteristic polynomial and a restricted multiarrangement. As an application, the freeness of cones over certain two-dimensional affine arrangements is proved.

The Hofer–Zehnder theorem states that almost every hypersurface in a thickening of a hypersurface $S$ in a symplectic manifold $(M,\omega)$ carries a closed characteristic, provided that $S$ bounds a compact submanifold and $(M,\omega)$ has finite capacity. It is shown that it is enough to assume that the thickening of $S$ has finite capacity.

Symplectic induction was first introduced by Weinstein as the symplectic analogue of induced representations, and was further developed by Guillemin and Sternberg. This paper deals with the case where the symplectic manifold in question is a semisimple coadjoint orbit of a Lie group. In this case, the construction is generalized by adding a smooth mapping, in order to obtain various symplectic forms. In particular, when the orbit is elliptic, a study of the complex geometry shows that quantization commutes with induction.

The relationship between two natural definitions of the relative pressure function, for a locally constant potential function and a factor map from a shift of finite type, is clarified by showing that they coincide almost everywhere with respect to every invariant measure. With a suitable extension of one of the definitions, the same holds true for any continuous potential function.

let $e$ be a subset of the boundary of the open unit disc $d$, and let $a$ be the algebra of bounded holomorphic functions on $d$ that extend continuously to $d \cup e$. it is shown that if $f$ is a bounded harmonic function on $d$ that extends continuously to $d \cup e$ and is not holomorphic, then the uniformly closed algebra $a[f]$ generated by $a$ and $f$ contains $c({\overline{d}})$. this result contains as special cases a result on the disc algebra due to čirka and a result on $h^{\infty}(d)$ due to axler and shields. a stronger form of the result, in which $f$ is allowed to have discontinuities on a small subset of $e$, is also established.

It is shown that if Ω is an infinite, metrizable, compact space such that some finite order iterated derived set of Ω is empty, then $\dg C(\om)=\db C(\om)=2$.

Thomas conjectured that there is no such thing as an almost prime element in a commutative radical Banach algebra. This paper shows that there is such an element. However, it still leaves open the question of whether there is an element that is almost prime, but not prime.

Noel Baker was born on 10 August 1932 and died, of a heart attack, on 20 May 2001. He was the only child of a farming family living near the township of Virginia north of Adelaide in South Australia, and a fourth-generation Australian. From 1938 to 1944 he attended the local school, winning a scholarship to King's College, Adelaide, and from there another scholarship to Prince Alfred College, Adelaide. Here he was inspired by an enthusiastic mathematics teacher, Mr Williams, who expected pupils to practise mathematics assiduously, even on Saturday mornings. Prince Alfred College still aims to foster a love of mathematics today, and it has recently launched a centre for excellence in mathematics, named after Noel.

Cyril Offord was one of the most distinguished and certainly the most senior of British mathematical analysts. With J. E. Littlewood, he pioneered the subject of probabilistic analysis by studying the typical behaviour of the zeros of a random polynomial with real coefficients, or an entire function.

Let $G$ be a compact Lie group, and let $\pi$ be any prime or set of primes. A ‘$\pi$-perfection map’ is constructed: that is, a continuous function from the space of conjugacy classes of all closed subgroups of $G$ to the space of conjugacy classes of $\pi$-perfect subgroups with finite index in their normalizer. This is used to show that the idempotent elements of the Burnside ring of $G$ localized at $\pi$ are in bijective correspondence with the open and closed subsets of the space of conjugacy classes of $\pi$-perfect subgroups of $G$ with finite index in their normalizer.

Given a graph of $\delta$-hyperbolic spaces, this paper gives sufficient conditions that ensure that the graph itself is $\delta$-hyperbolic. As an application, a simple proof is given to show that limit groups are relatively hyperbolic.

a general, albeit simple, approach to obtaining rearrangement inequalities of maximal operators is given. being applied to specific operators it leads not only to new proofs of well-known relations in a simpler and shorter way, but also to new, profitable results.

the degree defined by p. benevieri and m. furi (see ‘a simple notion of orientability for fredholm maps of index zero between banach manifolds and degree theory’, ann. sci. math. québec 22 (1998) 131–148) is used here to obtain some sharp lower bounds for the number of solutions of the $\lambda$-sections of the compact components of the set of non-trivial solutions of $\mf{f}(\lambda,x)=0$, where $\mf{f}$ is a $\cal{c}^1$ fredholm map of index 1 such that $\mf{f}(\lambda,0)=0$ for all $\lambda\in\mathbb{r}$. these bounds are given in terms of the parity of the linearized fredholm family $d_2\mf{f}(\cdot,0)$. the parity is a local invariant measuring the change of the orientation of $d_2\mf{f}(\lambda,0)$ as $\lambda$ crosses an interval. the set of eigenvalues of $d_2\mf{f}(\cdot,0)$ is not assumed to be discrete. therefore, even in the classical situation when $\mf{f}$ is a compact perturbation of the identity, the results presented here are completely new.

In this paper, some identities in laws involving ladder processes for random walks and Lévy processes are extended and unified.

The London Mathematical Society has elected Professor Isadore M. Singer of the Massachusetts Institute of Technology to Honorary Membership of the Society. Professor Singer has been a leading figure in geometry and analysis for over fifty years. He has received many honours and prizes, including the American Mathematical Society's Prize for Distinguished Public Service and the Steele Prize for Lifetime Achievement, as well as the US National Medal of Science. Together with Sir Michael Atiyah, he was awarded the Abel Prize for 2004. Professor Singer is best known for his share in the Atiyah–Singer index theorem for elliptic operators on manifolds, which was proved in the mid-1960s. This powerful tool not only gave an explanation for a host of earlier results in geometry and topology, but also provided entirely new connections with other subjects such as number theory and theoretical physics. Professor Singer was one of those responsible for establishing the very successful dialogue that has taken place in recent decades between geometers and physicists, introducing into mathematics viewpoints from quantum field theory that have radically changed the agenda in geometry and topology.