Soit $f$ une fonction arithmétique. Posons $$m(n;f) \,{\coleq}\, \min_{d|n}\!{\Vert}f(d){\Vert}\qquad(n\,{\geq}\,1),$$ oú ${\Vert} u {\Vert}$ désigne la distance du nombre réel $u$ à l'ensemble des entiers. Le comportement normal de $m$($n$;$f$) est une mesure de la nature probabiliste de l'ensemble des diviseurs d'un entier [Lt ]statistique[Gt ]. Une hypothèse standard d'équirépartition conduit à l'évaluation $$m(n;f) = 1/\tau(n)^{1+o(1)}\quad {\rm pp}$$, oú $\tau$($n$) désigne le nombre total des diviseurs d'un entier naturel $n$, et oú, ici et dans la suite, nous utilisons la mention pp (presque partout) pour indiquer qu'une propriété relative à un entier générique est valable sur une suite de densité naturelle unité.

The aim of the paper is to provide a criterion for the stability of plane curves using log canonical thresholds. As an application, we compare stable pairs of degree $d$ with stable plane curves with degree $d$.

We give a new proof of Dehn's lemma and the loop theorem. This is a fundamental tool in the topology of 3-manifolds. Dehn's lemma was originally formulated by Dehn, where an incorrect proof was given. A proof was finally given by Papakyriakopolous in his famous 1957 paper where the fundamental idea of towers of coverings was introduced. This was later extended to the loop theorem, and the version used most frequently was given by Stallings.

We have shown that hierarchies for Haken 3-manifolds could be understood by a ‘local version’ of Dehn's lemma and the loop theorem. Developing the idea further enables us here to give a new proof of the classical theorems of Papakyriakopoulos, which do not use towers of coverings. A similar result was obtained by Johannson, with the added assumption that the 3-manifold in question was Haken. Our approach means that no extra hypotheses are necessary. Our method uses the concept of boundary patterns of hierarchies, as developed by Johannson. Marc Lackenby has independently produced a very similar proof in his lecture notes.

Note that the more difficult sphere theorem [4, 10, 25] can then be deduced using Dehn's lemma and the loop theorem, plus the PL theory of minimal surfaces. Other applications, like the important result that a covering of an irreducible 3-manifold is irreducible, then follow also by PL minimal surface theory.

It is known that if every prime power branched cyclic cover of a knot in $S^3$ is a homology sphere, then the knot has vanishing Casson–Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in $S^3$ whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of $\scrf_{(1.0)}/\scrf_{(1.5)}$ for which Casson–Gordon invariants vanish in Cochran–Orr–Teichner's filtration of the classical knot concordance group. As a corollary, it follows that Casson–Gordon invariants are not a complete set of obstructions to a second layer of Whitney disks.

Let $M$ be a compact, connected surface without boundary and different from $\rp$, and let $B_n(M)$ and $P_n(M)$ denote its braid group and pure braid group on $n$ strings respectively. In this paper, we study the roots of the ‘full twist’ braid in $P_n(M)$ and $B_n(M){\setminus} P_n(M)$. Our main results may be summarised as follows: first, the full twist has no non-trivial root in $P_n(M)$. Further, if $M\ne \St$ and $k\geq 2$, it has a $k\th$ root in $B_n(M){\setminus} P_n(M)$ if and only if $k$ divides either $n$ or $n-1$. This generalises results concerning the sphere of Gillette, Van Buskirk and Murasugi. We also show that the Artin pure braid groups and the pure braid groups of the sphere admit a (non-trivial) splitting as a direct product of which one of the factors is the cyclic group generated by the full twist.

We prove the Martino–Priddy conjecture for an odd prime $p$: the $p$-completions of the classifying spaces of two groups $G$ and $G^\prime$ are homotopy equivalent if and only if there is an isomorphism between their Sylow $p$-subgroups which preserves fusion. A second theorem is a description for odd $p$ of the group of homotopy classes of self homotopy equivalences of the $p$-completion of $BG$, in terms of automorphisms of a Sylow $p$-subgroup of $G$ which preserve fusion in $G$. These are both consequences of a technical algebraic result, which says that for an odd prime $p$ and a finite group $G$, all higher derived functors of the inverse limit vanish for a certain functor $\calz_G$ on the $p$-subgroup orbit category of $G$.

In this paper we consider quaternionic Möbius transformations preserving the unit ball in the quaternions $\bh$. In other words, maps of the form $g(z)=(az+b)(cz+d)^{-1}$ where $a$, $b$, $c$ and $d$ all lie in $\bh$ with the property that $|g(z)|<1$ for all $|z|<1$. We give an explicit expression for the fixed points of $g$ in terms of $a$, $b$, $c$ and $d$ and we use this to classify quaternionic Möbius transformations into six categories determined by their dynamics.

An integral inequality for a compact Lorentzian manifold which admits a timelike conformal vector field and has no conjugate points along its null geodesics is given. Moreover, equality holds if and only if the manifold has nonpositive constant sectional curvature. The inequality can be improved if the timelike vector field is assumed to be Killing and, in this case, the equality characterizes (up to a finite covering) flat Lorentzian $n(\geq3)$-dimensional tori. As an indirect application of our technique, it is proved that a Lorentzian $2-$torus with no conjugate points along its timelike geodesics and admitting a timelike Killing vector field must be flat.

We denote by $B_Y$ the unit ball of a normed linear space $Y$.

Definition. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P{:}\,Y\,{\to}\,X$ such that $P(B_Y)\,{\subset}\,A$.

Minimal-volume sufficient enlargements are determined for two-dimensional spaces. The main results are:

(i) Each minimal-volume sufficient enlargement for a two-dimensional space is a parallelogram or a hexagon.

(ii) If a two-dimensional normed space $X$ has a minimal-volume sufficient enlargement that is not a parallelogram, then $B_X$ is linearly equivalent to the regular hexagon.

In this paper we consider topological properties of ordinary as well as graph directed iterated function systems. First we give the basic definitions.

We are interested in the dynamics that can arise close to rotational group orbits after forced symmetry-breaking to discrete symmetries. In particular we ask how simple or complicated the dynamics induced by symmetric linear vector fields can be. We look in detail at related tetrahedral and dihedral examples, and there we find precise conditions for a linear field to exhibit homoclinic orbits.

We prove that every Lie derivation on a symmetrically amenable semisimple Banach algebra can be uniquely decomposed into the sum of a derivation and a centre-valued trace.

The main theorem states that all Lie and Jordan derivations from a von Neumann algebra $\frak{A}$ into any Banach $\frak{A}$-bimodule are standard. Moreover, this statement is proved for some other classes of algebras. Our approach is based on the algebraic theory of functional identities and the strong degree, and is combined with analytic tools.

It is proved that the countably infinite power of complete Erdős space $\Ec$ is not homeomorphic to $\Ec$. The method by which this result is obtained consists of showing that $\Ec$ does not contain arbitrarily small closed subsets that are one-dimensional at every point. This observation also produces solutions to several problems that were posed by Aarts, Kawamura, Oversteegen and Tymchatyn. In addition, we show that the original (rational) Erdős space does contain arbitrarily small closed sets that are one-dimensional at every point.

By using Mawhin's continuation theorem of coincidence degree theory, we derive criteria for the existence of a positive periodic solution of a generalized prey-predator system with time delay.

We consider the existence of unbounded solutions for the asymmetric oscillator $$(\vp_p(x'))'+(p-1)[\al \vp_p(x^+)-\be \vp_p(x^-)]=f(t)\eqno(1)$$where $\vp_p(u)=|u|^{p-2}u, p>1$, $\al$ and $\be$ are positive constants satisfying $$\al^{{-}\frac 1p}+\be^{{-}\frac 1p}=2m/n\eqno(2)$$ with ($m,n)=1,\,\, m, n\in N$ and $x^\pm=\max\{\pm x,0\}$, $f \in L^\infty[0,2\pi_p]$ is $2\pi_p$-periodic, $\pi_p={2\pi}/({p\sin (\pi/p)})$.

We study locally spatially homogeneous solutions of the Einstein–Vlasov system of all Bianchi types except IX with a positive cosmological constant. First the global existence of solutions of this system and the casual geodesic completeness are shown. Then the asymptotic behaviour of solutions in future time is investigated in various aspects.