To form Riemann sums for generalized Riemann integrals, the domain of integration must be partitioned in a suitable manner. The existence of the required partitions is usually proved by a simple method of repeated bisection of the domain of integration. However, when the domain is the Cartesian product of infinitely many copies of the set of real numbers, this simple method of proof has frequently failed. A proof which works for infinite-dimensional spaces is provided here.

Results of the Landesman–Lazer type provide necessary and sufficient conditions for the existence of periodic solutions of certain nonlinear differential equations with forcing. Typically, they deal with scalar problems. This paper presents a discussion of possible extensions to systems. The emphasis is placed on the new phenomena produced by the increase of the dimension.

We prove weak laws of large numbers for freely independent, uniformly bounded and non-identically distributed random variables belonging to non-commutative Ciach and, in particular, Lorentz Lp-spaces.

This paper solves a problem posed by Colin Adams, showing that any link $L$ in ${\bb R}^3$ is isotopic to a smooth link $\hat{L}$ that has no parallel or antiparallel tangents.

It is proved here that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler-manifold into an Euclidean space must be totally geodesic. As an application, it is shown that an open subset of the real hyperbolic plane ${\mathbb R}H^2$ cannot be minimally immersed into the Euclidean space. As another application, a proof is given that if an irreducible Kähler manifold is minimally immersed in a Euclidean space, then its restricted holonomy group must be $U(n)$, where $n = \dim_{\mathbb C}M$.Supported by an EPSRC Grant GR/R69174.

It is proved that if a Schur superalgebra is not semisimple, then it is neither cellular nor quasi-hereditary (Theorem 2), and it has infinite global dimension (Corollary 18). The algebra $S(m|n,r)$ with $m,n \ge 1$ is semisimple if and only if $p$, the characteristic of the ground field, is zero or greater than $r$, or when $m=n=1$ and $p$ does not divide $r$.

Suppose that $R$ goes to infinity through a second-order lacunary set. Let $S_R$ denote the $R$th spherical partial inverse Fourier integral on ${\rm I\!R}^d$. Then $S_R f$ converges almost everywhere to $f$, provided that $f$ satisfies \[ \int \widehat{f}(\xi)\log\log(8+|\xi|)^2\,d\xi < \infty. \]

We show that all derivations from the group algebra [lscr ]1(G) of a free group into its nth dual, where n is a positive even integer, are inner. Combined with the known result for odd values of n, this shows that these algebras are permanently weakly amenable.

1. Introduction

In 1978, Atiyah, Hitchin and Singer [1] introduced the twistor space as a ℙ1-fibration over a half-conformally flat 4-manifold, and thereby established a beautiful correspondence between Yang–Mills fields on 4-manifolds and holomorphic vector bundles on complex 3-folds. Hitchin gave the Penrose correspondence between solutions of anti-self-dual zero rest-mass field equations on a half-conformally flat manifold M and the sheaf cohomology groups H1(Z, F(k)), k[les ]−2, on its twistor space Z [4]. Here the holomorphic bundle F is the pull-back bundle of an anti-self-dual bundle over M, and we denote by [Oscr ](−1) the tautological line bundle associated to the half-spin bundle. Hitchin further showed how H1(Z, F(−1)) corresponds to solutions of the self-dual Dirac equations, and interpreted H1(Z, F(k)), k[ges ]0, in terms of the cohomology of an elliptic complex on M.

Let Z be the centre of the group algebra of a symmetric group [Sscr ](n) over a field F characteristic p. One of the principal results of this paper is that the image of the Frobenius map z → zp, for z ∈ Z, lies in span Zp′ of the p-regular class sums. When p = 2, the image even coincides with Z2′. Furthermore, in all cases Zp′ forms a subalgebra of Z. Let pt be the p-exponent of [Sscr ](n). Then jpt = 0, for each element j of the Jacobson radical J of Z. It is shown that there exists j ∈ J such that jpt−1 ≠ 0. Most of the results are formulated in terms of the p-blocks of [Sscr ](n).

This paper provides sufficient conditions that guarantee the global attractivity of the zero solution of a delay population model under impulsive perturbations.

The structural stability of constrained polynomial differential systems of the form a(x, y)x′+b(x, y)y′=f(x, y), c(x, y)x′+d(x, y)y′=g(x, y), under small perturbations of the coefficients of the polynomial functions a, b, c, d, f and g is studied. These systems differ from ordinary differential equations at ‘impasse points’ defined by ad−bc=0. Extensions to this case of results for smooth constrained differential systems [7] and for ordinary polynomial differential systems [5] are achieved here.

In this paper, the authors give sharp estimates on the norms in the trace class of wavelet multipliers $P_{\sigma\varphi}:L^2({\bb R}^n) \longrightarrow L^2({\bb R}^n)$ in terms of the symbols $\sigma$ and the admissible wavelets $\varphi$ .

In the Kourovka notebook, Deaconescu asks whether $\gpord{\Aut G}\ge \phi(\gpord{G})$ for all finite groups $G$, where $\phi$ denotes the Euler totient function, and whether $G$ is cyclic whenever $\gpord{\Aut G}= \phi(\gpord{G})$. Both questions are answered in the negative in this paper. Moreover, $\gpord{\Aut G}/ \phi(\gpord{G)$ can be made arbitrarily small.

Let Tt be the semigroup of linear operators generated by a Schrödinger operator − A = Δ − V, where V is a non-negative polynomial, and let ∫∞0λdEA(λ) be the spectral resolution of A. We say that f is an element of HpA if the maximal function [Mscr ]f(x) = supt>0[mid ]Ttf(x)[mid ] belongs to Lp. We prove a criterion of Mihlin type on functions F which implies boundedness of the operators F(A) = ∫∞0F(λ)dEA(λ) on HpA, 0 < p [les ] 1.

An element u of a norm-unital Banach algebra A is said to be unitary if u is invertible in A and satisfies $\Vert u\Vert =\Vert u^{-1}\Vert =1$. The norm-unital Banach algebra A is called unitary if the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A. If X is a complex Hilbert space, then the algebra $\BL(X)$ of all bounded linear operators on X is unitary by the Russo–Dye theorem. The question of whether this property characterizes complex Hilbert spaces among complex Banach spaces seems to be open. Some partial affirmative answers to this question are proved here. In particular, a complex Banach space X is a Hilbert space if (and only if) $\BL(X)$ is unitary and, for Y equal to $X,$ $X^*$ or $X^{**},$ there exists a biholomorphic automorphism of the open unit ball of Y that cannot be extended to a surjective linear isometry on Y.

In this paper, it is proved that if $B$ is a Brauer $p$-block of a $p$-solvable group, for some odd prime $p$, then the height of any ordinary character in $B$ is at most $2b$, where $p^b$ is the largest degree of the irreducible characters of the defect group of $B$. Some other results that relate the heights of characters with properties of the defect group are obtained.

Let $S$ be a compact Riemann surface of genus ${\rm g}\,{>}\,1$. By using Teichmüller theory, a new approach is obtained to the existence of a Jenkins–Strebel differential with prescribed type and heights on $S$. Moreover, the geometric structures of horizontal trajectories of certain classes of quadratic differentials are discussed.

This paper provides examples of closed manifolds having a Morse number different from the stable Morse number.

It is proved that every topologically pure extension of Fréchet algebras 0 → I → A → A/I → 0 such that I is strongly H-unital has the excision property in continuous (co)homology of the following types: bar, naive-Hochschild, Hochschild, cyclic, and periodic cyclic. In particular, the property holds for every extension of Fréchet algebras such that I has a left or right bounded approximate identity.