Let X be a reflexive Banach space, and let C ⊂ X be a closed, convex and bounded set with empty interior. Then, for every δ > 0, there is a nonempty finite set F ⊂ X with an arbitrarily small diameter, such that C contains at most δ · |F| points of any translation of F. As a corollary, a separable Banach space X is reflexive if and only if every closed convex subset of X with empty interior is Haar null.

It is proved that in groups with all subgroups subnormal every nilpotent subgroup is contained in a normal nilpotent subgroup.

We construct an infinite family of simply connected, pairwise nondiffeomorphic 4-manifolds, all homeomorphic to $3{\mathbb {CP}}^2 \# 9 {\overline {{\mathbb C}{\mathbb P}^2}}$. Similar ideas provide examples of 4-manifolds with $b_2^+=3$, $b_2^-=8$, vanishing first homology and nontrivial Seiberg–Witten invariants.

A potential for the one-dimensional Dirac operator is constructed such that its essential spectrum does not cover the whole real line, whereas the potential q(x) tends to ∞ as [mid ]x[mid ] → ∞. Furthermore, a criterion by Hartman and Wintner for points of the essential spectrum of Sturm–Liouville operators is generalised to a purely operator-theoretical setting, and a simplified proof is given.

Any 2-block of a finite group G with a quaternion defect group Q8 is Morita equivalent to the corresponding block of the centraliser H of the unique involution of Q8 in G; this answers positively an earlier question raised by M. Broué.

In this paper, what is already known about defect 2 blocks of symmetric groups is used to deduce information about the corresponding blocks of Schur algebras. This information includes Ext-quivers and decomposition numbers, as well as Loewy structures of the Weyl modules, principal indecomposable modules and tilting modules.

A Banach algebra [afr ] is AMNM if whenever a linear functional ϕ on [afr ] and a positive number δ satisfy [mid ]ϕ(ab)−ϕ(a)ϕ(b)[mid ] [les ]δ|a|·|b| for all a, b∈[afr ], there is a multiplicative linear functional ψ on [afr ] such that |ϕ−ψ|=o(1) as δ→0. K. Jarosz [1] asked whether every Banach algebra, or every uniform algebra, is AMNM. B. E. Johnson [3] studied the AMNM property and constructed a commutative semisimple Banach algebra that is not AMNM. In this note we construct uniform algebras that are not AMNM.

In this article, the authors collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. They start with a sketch of Milne's proof of two of these conjectures, and they also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of, respectively, $4s^2$ or $4s(s+1)$ triangular numbers. This approach is inspired by Zagier's proof of the Kac–Wakimoto formulas via modular forms. The survey ends with recent conjectures of the first author and Chua.

This paper concerns involutions and anti-automorphisms on finite-dimensional associative algebras over a field. It is based on a survey talk given at the workshop on ‘Quadratic forms, algebras with involution and algebraic K-theory’ at the University of Nottingham, 13–15 May 2004. This workshop was part of the EU network HPRN-CT-2002-00287 on ‘Algebraic $K$-theory, linear algebraic groups and related structures’ and followed the London Mathematical Society Midlands Regional Meeting in Nottingham on 12 May, 2004.

The contents of the article are as follows:

1. Introduction and historical remarks;

2. Some general results;

3. Central simple algebras;

4. Group algebras.

Suppose that $T(S_0)$ is the Teichmüller space of a compact Riemann surface $S_0$ of genus $g>1$. Let $d_T(\cdot,\cdot)$ be the Teichmüller metric of $T(S_0)$ and let $d_S(\cdot,\cdot)$ be a metric of $T(S_0)$ defined by the length spectrums of Riemann surfaces. The author showed in a previous paper that $d_T$ and $d_S$ are topologically equivalent and $d_S(\tau_1,\tau_2)\leq d_T(\tau_1,\tau_2) \leq 2d_S(\tau_1,\tau_2)+C(\tau_1)$, where $C(\tau_1)$ is a constant depending on $\tau_1$. In this paper, it is shown that $d_T$ and $d_S$ are not metrically equivalent; that is, there is no constant $C>0$ such that $d_S(\tau_1,\tau_2)\leq d_T(\tau_1,\tau_2)\leq Cd_S(\tau_1,\tau_2)$ for all $\tau_1 {\rm and} \tau_2 {\rm in} T(S_0)$.

Let a be an element in a complex Banach algebra with unit, and let r be a nonnegative number. The Gelfand spectral radius formula implies that the spectrum of a is included in the disk {z∈C:[mid ]z[mid ][les ]r} if and only if

formula here

Let G be a finite group, and let IG be the augmentation ideal of ℤG. We denote by d(G) the minimum number of generators for the group G, and by d(IG) the minimum number of elements of IG needed to generate IG as a G-module. The connection between d(G) and d(IG) is given by the following result due to Roggenkamp ]14&]:

formula here

where pr(G) is a non-negative integer, called the presentation rank of G, whose definition comes from the study of relation modules (see [4&] for more details).

We prove that the fundamental group of the double of the figure-eight knot exterior admits a faithful discrete representation into SO(4,1; R), for which the image group is separable on its geometrically finite subgroups.

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

Simply connected closed symplectic 4-manifolds with $b_{2}^+ \,{=}\,1$ and $K^2 \,{=}\,0$ are investigated. As a result, it is confirmed that most of homotopy elliptic surfaces $\{E(1)_{K} \,{|}\,K\, \mathrm{is\, a\, fibred\, knot}$ in $S^3\}$ constructed by R. Fintushel and R. Stern in Invent. Math. 134 (1998) 363–400 are simply connected closed minimal symplectic 4-manifolds that do not admit a complex structure.This work was supported by grant No. R14-2002-007-01002-0 from KOSEF.

Let Γ be a lattice in a connected Lie group. We show that, besides a few exceptional cases, the deficiency of Γ is nonpositive.

This paper provides a survey of both old and new results about minimal surfaces and submanifolds.

In [1], Beardon introduced the Apollonian metric α defined for any domain D in ℝ by

formula here

This metric is Möbius invariant, and for simply connected plane domains it satisfies the inequality αD[les ]2ρD, where ρD denotes the hyperbolic distance in D, and so gives a lower bound on the hyperbolic distance. Furthermore, it is shown in [1, Theorem 6.1] that for convex plane domains, the Apollonian metric satisfies ρD[les ]4 sinh [½αD], and, by considering the example of the infinite strip {x+iy[ratio ][mid ]y[mid ]<1}, that the best possible constant in this inequality is at least π. In this paper we make the following improvements.

Two new families of commutative semifields of order q4 with middle nucleus GF(q2), where q is an odd prime power, are defined. One family occurs when q ≡ 3 (mod 4) and the other when q ≡ 1 (mod 4).

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator $T(a) \in \mathcal{L}(\ell^{p,\mu}_N)$, $1 < p < \infty$ and $\mu \in \mathbb{R}$, with a continuous matrix-valued generating function $a$. We prove that the approximation numbers of the finite sections $T_n(a) = P_n T(a) P_n$ have the $k$-splitting property, provided $T(a)$ is a Fredholm operator on $\ell^{p,\mu}_N$.