We prove that the spectrum of a suitably projected Dirac operator — sometimes called the no-pair operator — introduced by Brown and Ravenhall is bounded from below by some positive constant when the nuclear charge is below some critical charge. This bound improves a result obtained recently by Evans et al., and disproves a conjecture made by Hardekopf and Sucher.

The purpose of this note is to give a short proof of the Gross logarithmic Sobolev inequality using the asymptotics of the sharp L2 Sobolev constant and the product structure of Euclidean space. Let F∈Lr(IRn) for some positive r with ∥F∥r=1.

We consider the sequences of fractional parts $\{\xi \alpha^n\}, n \,{=}\,1,2,3,\dots,$ and of integer parts $[\xi \alpha^n], n \,{=}\,1,2,3,\dots,$ where $\xi$ is an arbitrary positive number and $\alpha\,{>}\,1$ is an algebraic number. We obtain an inequality for the difference between the largest and the smallest limit points of the first sequence. Such an inequality was earlier known for rational $\alpha$ only. It is also shown that for roots of some irreducible trinomials the sequence of integer parts contains infinitely many numbers divisible by either 2 or 3. This is proved, for instance, for $[\xi((\sqrt {13}-1)/2)^n], n\,{=}\,1,2,3,\dots.$ The fact that there are infinitely many composite numbers in the sequence of integer parts of powers was proved earlier for Pisot numbers, Salem numbers and the three rational numbers 3/2, 4/3, 5/4, but no such algebraic number having several conjugates outside the unit circle was known.

This paper is concerned with the oscillatory behaviour of first-order delay differential equations of the form \begin{equation} x^{\prime}(t) + p(t)x(\tau(t)) = 0, \quad t \geqslant t_0, \end{equation} where $p, \tau \in C([t_0,\infty),{\rm I\!R}^+),{\rm I\!R}^+ = [0,\infty),\tau(t)$ is non-decreasing, $\tau(t) < t$ for $t \geqslant t_0$ and $\lim_{t\rightarrow\infty}\tau(t) = \infty$. Let the numbers $k$ and $L$ be defined by \[ k = \mathop{{\lim\inf}}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds \quad\hbox{and}\quad L = \mathop{\lim\sup}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds. \]

It is proved here that when $L < 1$ and $0 < k \leqslant 1/e$ all solutions of equation (1) oscillate in several cases in which the condition \[ L > \frac{\ln\lambda_1 - 1 + \sqrt{5 - 2\lambda_1 + 2k\lambda_1}}{\lambda_1} \] holds, where $\lambda_1$ is the smaller root of the equation $\lambda = e^{k\lambda}$.

We precisely evaluate the operator norm of the uncentred Hardy–Littlewood maximal function on Lp(ℝ1). Consequently, we compute the operator norm of the ‘strong’ maximal function on Lp(ℝn), and we observe that the operator norm of the uncentred Hardy–Littlewood maximal function over balls on Lp(ℝn) grows exponentially as n[xrarr ]∞.

Every ε-isometry between real normed spaces of the same finite dimension which maps the origin to the origin may be uniformly approximated to within 2ε by a linear isometry. Under a smoothness hypothesis, necessary and sufficient conditions are obtained for the same conclusion to hold for a given ε-isometry between infinite-dimensional Banach spaces.

This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which non-specialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below.

Contents

1 Historical background 452

2 Differential operators and integral Steenrod squares 460

3 Symmetric functions and differential operators 471

4 Bases, excess and conjugates 476

5 The stripping technique 483

6 Iteration theory and nilpotence 490

7 The hit problem and invariant theory 495

8 The dual of [Ascr ](n) and graph theory 504

9 The Steenrod group 507

10 Computing in the Steenrod algebra 508

References 511

In Section 1 the scene is set with a few remarks on the early history of the Steenrod algebra [Ascr ] at the prime 2 from a topologist's point of view, which puts into context some of the problems posed later. In Section 2 the subject is recast in an algebraic framework, by citing recent work on integral versions of the Steenrod algebra defined in terms of differential operators. In Section 3 there is an explanation of how the divided differential operator algebra [Dscr ] relates to the classical theory of symmetric functions. In Section 4 some comments are made on a few of the recently discovered bases for the Steenrod algebra. The stripping technique in Section 5 refers to a standard action of a Hopf algebra on its dual, which is particularly useful in the case of the Steenrod algebra for deriving relations from relations when implemented on suitable bases. In Section 6 a parallel is drawn between certain elementary aspects of the iteration theory of quadratic polynomials and problems about the nilpotence height of families of elements in the Steenrod algebra. The hit problem in Section 7 refers to the general question in algebra of finding necessary and sufficient conditions for an element in a graded module over a graded ring to be decomposable into elements of lower grading. Equivariant versions of this problem with respect to general linear groups over finite fields have attracted attention in the case of the Steenrod algebra acting on polynomials. Similar problems arise with respect to the symmetric groups and the algebra [Dscr ]. This subject relates to topics in classical invariant theory and modular representation theory. In Section 8 a number of statements about the dual Steenrod algebra are transcribed into the language of graph theory. In Section 9 a standard method is employed for passing from a nilpotent algebra over a finite field of characteristic p to a p-group, and questions are raised about the locally finite 2-groups that arise in this way from the Steenrod algebra. Finally, in Section 10 there are a few comments on the use of a computer in evaluating expressions and testing relations in the Steenrod algebra.

Ramanujan presented (without proof) the following double inequality for the gamma function:\begin{equation}\sqrt{\pi}\Bigl(\frac{x}{e}\Bigr)^x\Bigl[8x^3+4x^2+x+\frac{1}{100}\Bigr]^{1/6} < \Gamma(x+1)< \sqrt{\pi}\Bigl(\frac{x}{e}\Bigr)^x\Bigl[8x^3+4x^2+x+\frac{1}{30}\Bigr]^{1/6} \quad{(x\geq 0)}.\end{equation}Recently, Karatsuba established that these inequalities hold for $x\geq 1$. We show that this can be slightly improved: the inequalities hold for all $x\geq 0$, even if we replace $1/100$ by \[\alpha= \min_{0.6\leq x\leq 0.7}f(x)=0.0100450\dots,\]where $f(x)=(1/\pi)^3[\Gamma(x+1)(e/x)^x]^6-8x^3-4x^2-x$. Moreover, $\alpha$ and $1/30$ are the best possible constant terms.

A new lower bound on $\max \{|A+A|,|A\cdot A|\}$ is given, where $A$ is a finite set of complex numbers.

In his celebrated memoir, Morgan Ward's definition of elliptic divisibility sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist nontrivial examples of such sequences. Even then, Ward's proof of the coherence of his definition relies on displaying a sequence of values of quotients of Weierstraß $\sigma$-functions. We give a direct proof of coherence and show, rather more generally, that a sequence defined by a so-called Somos relation of width 4 is always also given by three-term Somos relations of all larger widths $5, 6, 7, \ldots.$

It is well known that for a Noetherian ring R, an ideal I of R, and M a finitely generated R-module, the local cohomology modules HiI(M) are not always finitely generated. On the other hand, if R is local and m is its maximal ideal, then Him(M) are Artinian modules, which is equivalent to the following two properties:

(i) SuppR(Him(M)) ⊆{m};

(ii) the vector space HomR(k, Him(M)) has finite dimension over k, where k = R/m.

Taking these facts into account, Grothendieck [9] made the following conjecture.

We show that the halting times of infinite time Turing machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This gives a clarification of the nature of ‘supertasks’ or infinite time computations. The proof further yields that the class of sets coded by outputs of halting computations coincides with a level of Gödel's constructible hierarchy: namely that of Lλ where λ is the supremum of halting times. A number of other open questions are thereby answered.

In this note we show that if an ideal I of a noetherian ring is principal, up to radical, then the local cohomology modules with support in V(I) are I-cofinite.

An s-arc in a graph is a vertex sequence (α0,α1,…,αs) such that {αi−1,αi} ∈ EΓ for 1 [les ] i [les ] s and αi−1 ≠ αi+1 for 1 [les ] i [les ] s − 1. This paper gives a characterization of a class of s-transitive graphs; that is, graphs for which the automorphism group is transitive on s-arcs but not on (s + 1)-arcs. It is proved that if Γ is a finite connected s-transitive graph (where s [ges ] 2) of order a p-power with p prime, then s = 2 or 3; further, either s = 3 and Γ is a normal cover of the complete bipartite graph K2m,2m, or s = 2 and Γ is a normal cover of one of the following 2-transitive graphs: Kpm+1 (the complete graph of order pm+1), K2m,2m − 2mK2 (the complete bipartite graph of order 2m+1 minus a 1-factor), a primitive affine graph, or a biprimitive affine graph. (Finite primitive and biprimitive affine 2-arc transitive graphs were classified by Ivanov and Praeger in 1993.)

We determine all primitive groups which do not have a regular orbit on the power set of the permutation domain. As a corollary, we also determine all families of orbit equivalent primitive permutation groups.

This paper proves a conjecture of G. A. Jones, D. Singerman and K. Wicks, that a suborbital graph for the modular group is a forest if and only if it contains no triangles.

The optimal orders are determined for the best constants in the non-commutative Burkholder–Gundy, Doob and Stein inequalities obtained recently in non-commutative martingale theory.

Let p be a point of a Lorentzian manifold M. We show that if M is spacelike Osserman at p, then M has constant sectional curvature at p; similarly, if M is timelike Osserman at p, then M has constant sectional curvature at p. The reverse implications are immediate. The timelike case and 4-dimensional spacelike case were first studied in [3]; we use a different approach to this case.

The aim of this paper is to show the existence of solutions with an arbitrarily large number of bubbles for the slightly super-critical elliptic problem $-\Delta u=u^{{(N+2)/(N-2)} +\ve }$ in $\Omega$, subject to the conditions that $u>0$ in $\Omega$, and $u=0$ on $\partial \Omega$, where $\ve >0$ is a small parameter and $\Omega \subset \RR^N$ is a bounded domain with certain symmetries, for instance an annulus or a torus in $\RR^3$.

A survey is given of several key themes that have characterised mathematics in the 20th century. The impact of physics is also discussed, and some speculations are made about possible developments in the 21st century.