An asymptotically perturbed cylinder is a manifold which, to the exterior of a compact set, is of the form R×M, where M is a compact manifold (with or without boundary), and for which the metric approaches the product metric as the axial variable tends to infinity. The properties of eigenfunctions of the Laplacian on asymptotically perturbed cylinders, with either Dirichlet or Neumann boundary conditions if the boundary is non-empty, are studied. For a large class of asymptotic perturbations, the eigenfunctions are proved to decay faster than the reciprocal of any polynomial as the axial variable tends to infinity. The decay estimates are then used to prove that the eigenvalues are of finite multiplicity, and can accumulate at the thresholds only from below. The results are shown to apply to a large class of acoustic and quantum waveguides.

In this paper we study the metric geometry of the space $\Sigma$ of positive invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $\tau$. The trace induces an incomplete Riemannian metric $\langle x,y\rangle_a=\tau (ya^{-1}xa^{-1})$, and, though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the $n\times n$ matrices when they are regarded as a symmetric space. For instance, we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (algebraic) characterization of the geodesically convex submanifolds $M$ of $\Sigma$; and under a suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained via a nonlinear orthogonal projection $\Pi_M:\Sigma\to M$, a map which turns out to be contractive for the geodesic distance.

The main result provides mild conditions under which a closed, orientable, PL 4-manifold $N = N_1\,\#\,N_2$ with $\pi_1(N_e)$ residually finite ($e=1,2$) is a codimension-5 PL fibrator. The paper also presents a rich variety of conditions on a closed 4-manifold $N^4$ under which every PL map between manifolds, where the domain is orientable and all point inverses are copies of $N^4$, must be an approximate fibration.

Let M1 and M2 be two simply connected closed manifolds of the same dimension. It is proved that

(1) if k is a coefficient field such that neither M1 nor M2 has the same cohomology as a sphere, then the sequence (bk)k[ges ]1 of Betti numbers of the free loop space on M1 #M2 is unbounded;

(2) if, moreover, the cohomology H*(M1;k) is not generated as algebra by only one element, then the sequence (bk)k[ges ]1 has an exponential growth.

Thanks to theorems of Gromoll and Meyer and of Gromov, this implies, in case 1, that there exist infinitely many closed geodesics on M1#M2 for each Riemannian metric, and, in case 2, that for a generic metric, the number of closed geodesics of length [les ]t grows exponentially with t.

The paper proves two univalence criteria involving harmonic mappings in multiply connected domains. These generalise a classical result of Radó, Kneser and Choquet and a recent result of Bshouty and Hengartner.

We define a sequence of real functions which coincide with Li's coefficients at one and which allow us to extend Li's criterion for the Riemann Hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips inside the critical strip $0<\Re(z)<1$. We study some of the properties of these functions, including their oscillatory behaviour.

An approach of Swinnerton-Dyer is extended to obtain new upper bounds for the number of lattice points close to a smooth curve. One consequence of these bounds is a new asymptotic result for the distribution of squarefull numbers in short intervals.

For $(\l,a)\in \matbb{C}^*\times \mathbb{C}$, let $f_{\lambda,a}$ be the rational map defined by $f_{\lambda,a}(z) \,{=}\, \lambda z^2 {(az+1)/(z+a)}$. If $\alpha\in \mathbb{R}/\mathbb{Z}$ is a Brjuno number, we let ${\cal D}_\alpha$ be the set of parameters $(\lambda,a)$ such that $f_{\lambda,a}$ has a fixed Herman ring with rotation number $\alpha$ (we consider that $({\it e}^{2i\pi\alpha}{,}0)\,{\in}\, {\cal D}_\alpha$). Results obtained by McMullen and Sullivan imply that, for any $g\in {\cal D}_\alpha$, the connected component of ${\cal D}_\alpha\cap (\mathbb{C}^*\times(\mathbb{C}\setminus \{0,1\}))$ that contains g is isomorphic to a punctured disk.

We show that there is a holomorphic injection $\cal{F}_\alpha\,{:}\,\mathbb{D}\,{\longrightarrow}\, {\cal D}_\alpha$ such that $\cal{F}_\alpha(0) = ({\it e}^{2i\pi \alpha},0)$ and $\cal{F}_\alpha'(0)=(0,r_\alpha),$ where $r_\alpha$ is the conformal radius at 0 of the Siegel disk of the quadratic polynomial $z\longmapsto {\it e}^{2i\pi \alpha}z(1+z)$.

As a consequence, we show that for $a\in (0,1/3)$, if $f_{\l,a}$ has a fixed Herman ring with rotation number $\alpha$ and if $m_a$ is the modulus of the Herman ring, then, as $a\,{\rightarrow}\,0$, we have ${\it e}^{\pi m_a} \,{=} ({r_\alpha}/{a}) + {\cal O}(a).$

We finally explain how to adapt the results to the complex standard family $z\,{\longmapsto} \lambda z {\it e}^{({a}/{2})(z-1/z)}$.

It is shown that each group is the outer automorphism group of a simple group. Surprisingly, the proof is mainly based on the theory of ordered or relational structures and their symmetry groups. By a recent result of Droste and Shelah, any group is the outer automorphism group Out (Aut T) of the automorphism group Aut T of a doubly homogeneous chain (T, [les ]). However, Aut T is never simple. Following recent investigations on automorphism groups of circles, it is possible to turn (T, [les ]) into a circle C such that Out (Aut T) [bcong ] Out (Aut C). The unavoidable normal subgroups in Aut T evaporate in Aut C, which is now simple, and the result follows.

It is proved that the duality map $\langle\,,\rangle:(\ell^\infty,\hbox{weak})\times((\ell^\infty)^*,\hbox{weak}^* )\longrightarrow {\bf R}$ is not Borel. More generally, the evaluation $e:(C(K),\wk)\times K\longrightarrow{\bf R}$, $e(f,x) = f(x)$, is not Borel for any function space $C(K)$ on a compact $F$-space. It is also shown that a non-coincidence of norm-Borel and weak-Borel sets in a function space does not imply that the duality map is non-Borel.

Let M be a smooth, compact, oriented, odd-dimensional Riemannian manifold and let Γ → Mˆ → M be a normal covering of M. It is proved that the relative von Neumann eta-invariant ρ(2)(Mˆ) of Cheeger and Gromov is a homotopy invariant when Γ is torsion-free, discrete and the Baum–Connes assembly map μmax[ratio ]K0(BΓ) → K0(C*Γ) is an isomorphism.

A Blaschke product $B$ with zero-sequence $(a_n)$ is called almost interpolating if the inequality $\liminf_n(1-|a_n|^2)|B'(a_n)|\geqslant \d>0$ holds. The sets $U$ for which there exists a Blaschke product $B$ such that $(a-B)/(1-\ov a B)$ is almost interpolating if and only if $a \in U$ are studied. Examples of such sets include open sets, containing the origin, and whose complement is the closure of an arbitrary set of concentric open arcs around the origin or open sets whose complement is of zero logarithmic capacity. Results on the range of interpolating Blaschke product s on the set of trivial points in the spectrum of $H^{\infty}$ are deduced.

In [2] Bieri and Strebel introduced a geometric invariant for finitely generated abstract metabelian groups that determines which groups are finitely presented. For a valuable survey of their results, see [6]; we recall the definition briefly in Section 4. We shall introduce a similar invariant for pro-p groups.

Let [ ] be the algebraic closure of [ ]p and U be the formal power series algebra [ ][lobrk ]T[robrk ], with group of units U×. Let Q be a finitely generated abelian pro-p group. We write ℤp[lobrk ]Q[robrk ] for the completed group algebra of Q over ℤp. Let T(Q) be the abelian group Hom(Q, U×) of continuous homomorphisms from Q to U×. We write 1 for the trivial homomorphism. Each v∈T(Q) extends to a unique continuous algebra homomorphism vmacr; from ℤp[lobrk ]Q[robrk ] to U.

Topological properties of dynamical systems admitting a local product structure are studied. The results obtained are applied to hyperbolic systems on covers. The main theorem characterizes transitivity on solvable covers in terms of Frobenius elements of periodic orbits.

A complex valued function $g$ , defined on the positive integers, is multiplicative if it satisfies $g(ab) = g(a)g(b)$ whenever the integers $a$ and $b$ are mutually prime.

THEOREM 1. Let $D$ be an integer, $2 \le D \le x, \varepsilon > 0$ . Let $g$ be a multiplicative function with values in the complex unit disc.

There is a character $\chi_1({\rm mod}\, D)$ , real if $g$ is real, such that when $0 < \gamma < 1$ , \[ \sum_{\overset{n \le y}{n\equiv a({\rm mod}\, D)}} g(n)- \frac{1}{\phi(D)} \sum_{\overset{n\le y}{(n,D)=1)}} g(n)-\frac{\overline{\chi_1(a)}}{\phi(D)} \sum_{n\le y} g(n)\chi_1(n) \ll \frac{y}{\phi(D)} \left(\frac{\log D}{\log y}\right)^{1/4-\varepsilon} \] uniformly for $(a, D) = 1, D \le y, x^\gamma \le y \le x$ , the implied constant depending at most upon $\varepsilon, \gamma$ .

The problem of domination for positive compact operators between Banach lattices was solved by Dodds and Fremlin in [6]: given a Banach lattice $E$ with order continuous dual norm, an order continuous Banach lattice $F$ and two positive operators $0 \leqslant S \leqslant T : E \longrightarrow F$ , the operator $S$ is compact if $T$ is. A similar problem has been considered in the class of weakly compact operators by Abramovich [1] and in a general form by Wickstead [26]. Precisely, Wickstead's result shows that the operator $S$ is weakly compact if $T$ is whenever one of the following two conditions holds: either $E^{\prime}$ is order continuous or $F$ is order continuous. When it comes to Dunford–Pettis operators, Kalton and Saab [15] have proved that the operator $S$ is Dunford–Pettis if $T$ is, provided that the Banach lattice $F$ is order continuous. On the other hand, Aliprantis and Burkinshaw settled the problem of domination for compact [2] and weakly compact [3] endomorphisms (that is, the case when $E = F$ ). For example, they proved that if either the norm on $E$ or the norm on $E^{\prime}$ is order continuous, then the compactness of $T$ is inherited by the power operator $S^2$ . Also, they showed that, for $E$ an arbitrary Banach lattice, $T$ being compact always implies that $S^3$ is compact. More recently, Wickstead studied converses for the Dodds–Fremlin and Kalton–Saab theorems in [27] and for the Aliprantis–Burkinshaw theorems in [28].

The non-existence set forth in the title is proved. It is known that for numerical Campedelli surfaces the algebraic fundamental group is of order [les ]9 and that the dihedral group of order 8 cannot occur. Therefore the quaternionic group is the only non-abelian algebraic fundamental group in this range.

Let R be a fully bounded Noetherian ring of finite global dimension. Then we prove that K dim (R) [les ] gldim (R). If, in addition, R is local, in the sense that R/J(R) is simple Artinian, then we prove that R is Auslander-regular and satisfies a version of the Cohen–Macaulay property. As a consequence, we show that a local fully bounded Noetherian ring of finite global dimension is isomorphic to a matrix ring over a local domain, and a maximal order in its simple Artinian quotient ring.

The paper studies the dual algebras of localization Roe algebras over proper metric spaces and develops a localization version of Paschke duality for $K$ -homology. It is shown that the localization $K$ -homology groups are isomorphic to Kasparov's $K$ -homology groups for the Rips complex of proper metric spaces with bounded geometry. It follows that the obstruction groups to the coarse Baum–Connes conjecture can also be derived from the dual localization algebras.

Let R be a commutative Noetherian ring. Let [Pscr ](R) (respectively, [Iscr ](R)) be the category of all finite R-modules of finite projective (respectively, injective) dimension. Sharp [9] constructed a category equivalence between [Iscr ](R) and [Pscr ](R) for certain Cohen–Macaulay local rings R. Thus many properties about finite modules of finite projective dimension can be connected with those of finite injective dimension through this equivalence.