A planar set $G \subset {\bb R}^2$ is constructed that is bilipschitz equivalent to ( $G, d^z$ ), where ( $G, d$ ) is not bilipschitz embeddable to any uniformly convex Banach space. Here, $z \in (0, 1)$ and $d^z$ denotes the $z$ th power of the metric $d$ . This proves the existence of a strong $A_{\infty}$ weight in ${\bb R}^2$ , such that the corresponding deformed geometry admits no bilipschitz mappings to any uniformly convex Banach space. Such a weight cannot be comparable to the Jacobian of a quasiconformal self-mapping of ${\bb R}^2$ .

Let $G\subseteq \mathbb{C}^{2}$ be the open symmetrized bidisc, namely $G= \{(\lambda_{1}+\lambda_{2},\lambda_{1}\lambda_{2}):|\lambda_{1}|<1,|\lambda_{2}|<1\}$. In this paper, a proof is given that $G$ is not biholomorphic to any convex domain in $\mathbb{C}^{2}$. By combining this result with earlier work of Agler and Young, the author shows that $G$ is a bounded domain on which the Carathéodory distance and the Kobayashi distance coincide, but which is not biholomorphic to a convex set.

In 1933 Henri Cartan proved a fundamental theorem in Nevanlinna theory, namely a generalization of Nevanlinna's second fundamental theorem. Cartan's theorem works very well for certain kinds of problems. Unfortunately, it seems that Cartan's theorem, its proof, and its usefulness, are not as widely known as they deserve to be. To help give wider exposure to Cartan's theorem, the simple and general forms of the theorem are stated here. A proof of the general form is given, as well as several applications of the theorem.

Let $S={\mathop{\rm Sym}(\Omega)$ be the group of all permutations of an infinite set $\Omega$. Extending an argument of Macpherson and Neumann, it is shown that if $U$ is a generating set for $S$ as a group, then there exists a positive integer $n$ such that every element of $S$ may be written as a group word of length at most $n$ in the elements of $U$. Likewise, if $U$ is a generating set for $S$ as a monoid, then there exists a positive integer $n$ such that every element of $S$ may be written as a monoid word of length at most $n$ in the elements of $U$. Some related questions and recent results are noted, and a brief proof is given of a result of Ore's on commutators, which is used in the proof of the above result.

There are many references showing that a classical solution to the Black–Scholes equation is a stochastic solution. However, it is the converse of this theorem that is most relevant in applications, and the converse is also more mathematically interesting. In this paper we establish such a converse. We find a Feynman–Kac-type theorem showing that the stochastic representation yields a classical solution to the corresponding Black–Scholes equation with appropriate boundary conditions under very general conditions on the coefficients. We also obtain additional regularity results in the one-dimensional case.

The authors study the algebra of uniformly continuous holomorphic symmetric functions on the ball of ${\cal L}_p$ , investigating in particular the spectrum of such algebras. To do so, they examine the algebra of symmetric polynomials on ${\cal L}_p$ -spaces, as well as finitely generated symmetric algebras of holomorphic functions. Such symmetric polynomials determine the points in ${\cal L}_p$ up to a permutation.

The explicit solution of a general three-term bilinear recurrence relation of fourth order is constructed here in terms of the Weierstrass sigma function. The construction of the elliptic curve associated to the Somos 4 sequence is presented as an example. An interpretation via the theory of integrable systems is provided, leading to a conjecture relating certain higher-order recurrences with hyperelliptic curves of higher genus.

Star complements and associated quadratic functions are used to obtain a sharp upper bound for the order of a graph with an eigenspace of prescribed codimension. It is shown that for regular graphs the bound can be reduced by 1, and that this reduced bound is attained by a regular graph $G$ if and only if $G$ is an extremal strongly regular graph.

We obtain operator-valued analogues of Bohr's inequality involving both the absolute values of operators and their norms, that when restricted to the scalar case imply the classical Bohr inequality. In the scalar case we extend Bohr's inequality to the case where one function is majorized by another function, to the Hardy space, $H^2(\mathbb{D})$, and to bounded analytic functions on the annulus.

We study the extent to which sets $A \subseteq \mathbb{Z}/N\mathbb{Z}, N$ prime, resemble sets of integers from the additive point of view (‘up to Freiman isomorphism’). We give a direct proof of a result of Freiman, namely that if $|A + A| \leq K|A|$ and $|A| < c(K)N$, then $A$ is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we obtain a reasonable bound: we can take $\;c(K) \geq (32K)^{-12K^2}$. As a byproduct of our argument we obtain a sharpening of the second author's result on sets with small sumset in torsion groups. For example, if $A \subseteq \mathbb{F}_2^n$, and if $|A + A| \leq K|A|$, then $A$ is contained in a coset of a subspace of size no more than $K^22^{2K^2 - 2}|A|$.

For an irreducible, crystallographic root system $\Phi$ in a Euclidean space $V$ and a positive integer $m$, the arrangement of hyperplanes in $V$ given by the affine equations $(\alpha, x)\,{=}\,k$, for $\alpha\,{\in}\,\Phi$ and $k\,{=}\,0, 1,\dots,m$, is denoted here by ${\mathcal A}_{\Phi}^m$. The characteristic polynomial of ${\mathcal A}_{\Phi}^m$ is related in the paper to that of the Coxeter arrangement ${\mathcal A}_{\Phi}$ (corresponding to $m\,{=}\,0$), and the number of regions into which the fundamental chamber of ${\mathcal A}_{\Phi}$ is dissected by the hyperplanes of ${\mathcal A}_{\Phi}^m$ is deduced to be equal to the product $\prod_{i=1}^{\ell} ({e_i\,{+}\,m h\,{+}\,1})/({e_i\,{+}\,1})$, where $e_1, e_2,\dots,e_\ell$ are the exponents of $\Phi$ and $h$ is the Coxeter number. A similar formula for the number of bounded regions follows. Applications to the enumeration of antichains in the root poset of $\Phi$ are included.

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.

A natural topology on the space of left orderings of an arbitrary semi-group is introduced here. This space is proved to be compact, and for free abelian groups it is shown to be homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases.

The integration-by-parts methods introduced in this paper improve upon the $L^p$ estimates on transport densities given in the recent paper by L. De Pascale and A. Pratelli (Calc. Var. Partial Differential Equations 14 (2002) 249–274).

a tits alternative theorem is proved in this paper for groups acting on cat(0) cubical complexes. that is, a proof is given to show that if $g$ is assumed to be a group for which there is a bound on the orders of its finite subgroups, and if $g$ acts properly on a finite-dimensional cat(0) cubical complex, then either $g$ contains a free subgroup of rank 2, or $g$ is finitely generated and virtually abelian. in particular, the above conclusion holds for any group $g$ with a free action on a finite-dimensional cat(0) cubical complex.

We show that the norms of the sequence of spectral projections of certain non-self-adjoint ordinary differential operators cannot be polynomially bounded. The proof involves the investigation of ‘tame’ bi-orthogonal systems in Banach spaces.

The paper proves a result on the existence of finite flat scheme covers of Deligne–Mumford stacks. This result is used to prove that a large class of smooth Deligne–Mumford stacks with affine moduli space are quotient stacks, and in the case of quasi-projective moduli space, to reduce the question to a classical question on Brauer groups of schemes.Research of the first author supported in part by the Wolfgang Paul program of the Humboldt Foundation; research of the second author supported in part by the University of Bologna, funds for selected research topics.

A series of complexes Qp indexed by all primes p is constructed with catQp=2 and catQp×Sn=2 for either n[ges ]2 or n=1 and p=2. This disproves Ganea's conjecture on Lusternik–Schnirelmann (LS) category.

The main result in this paper is a representation theorem for homogeneous orthogonally additive polynomials on Banach lattices. The representation theorem is used to study the linear span of the set of zeros of homogeneous real-valued orthogonally additive polynomials. It is shown that in certain lattices every element can be represented as the sum of two or three zeros or, at least, can be approximated by such sums. It is also indicated how these results can be used to study weak topologies induced by orthogonally additive polynomials on Banach lattices.

The dual of a reflexive separable hereditarily indecomposable complex Banach space of Gowers and Maurey admits no chaotic continuous linear operator in the sense of Devaney defined on it.