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.

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].

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 $\varepsilon, \gamma$ .

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.

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.

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.

We first establish a combinatorial result on deterministic real chains. This is then applied to prove a path transformation for chains with exchangeable increments. From this transformation we derive an identity on order statistics due to Port, together with some extensions. Then we give an interpretation of these results in continuous time. We extend some identities involving quantiles and occupation times for processes with exchangeable increments. In particular, this yields an extension of the uniform law for bridges obtained by Knight.

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.

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.

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.

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 $z\,{\longmapsto} \lambda z {\it e}^{({a}/{2})(z-1/z)}$.

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.

Almkvist proved that for a commutative ring A the characteristic polynomial of an endomorphism α : P → P of a finitely generated projective A-module determines (P, α) up to extensions. For a non-commutative ring A the generalized characteristic polynomial of an endomorphism of an endomorphism α : P → P of a finitely generated projective A-module is defined to be the Whitehead torsion [1 − xα] ∈ K1(A[[x]]), which is an equivalence class of formal power series with constant coefficient 1.

The paper gives an example of a non-commutative ring A and an endomorphism α : P → P for which the generalized characteristic polynomial does not determine (P, α) up to extensions. The phenomenon is traced back to the non-injectivity of the natural map [sum ]−1A[x] → A[[x]] where [sum ]−1A[x] is the Cohn localization of A[x] inverting the set [sum ] of matrices in A[x] sent to an invertible matrix by A[x] → A;x [map ] 0.

let $l$ denote a right-invariant sub-laplacian on an exponential, hence solvable lie group $g$, endowed with a left-invariant haar measure. depending on the structure of $g$, and possibly also that of $l, l$ may admit differentiable $l^p$-functional calculi, or may be of holomorphic $l^p$-type for a given $p{\ne} 2$. ‘holomorphic $l^p$-type’ means that every $l^p$-spectral multiplier for $l$ is necessarily holomorphic in a complex neighbourhood of some non-isolated point of the $l^2$-spectrum of $l$. this can in fact only arise if the group algebra $l^1(g)$ is non-symmetric.

assume that $p{\ne} 2$. for a point $\ell$ in the dual $\frak{g}^*$ of the lie algebra $\frak{g}$ of $g$, denote by $\omega(\ell){=}\ad^*(g)\ell$ the corresponding coadjoint orbit. it is proved that every sub-laplacian on $g$ is of holomorphic $l^p$-type, provided that there exists a point $\ell{\in} \frak{g}^*$ satisfying boidol's condition (which is equivalent to the non-symmetry of $l^1(g)$), such that the restriction of $\omega(\ell)$ to the nilradical of $\frak{g}$ is closed. this work improves on results in previous work by christ and müller and ludwig and müller in twofold ways: on the one hand, no restriction is imposed on the structure of the exponential group $g$, and on the other hand, for the case $p{>}1$, the conditions need to hold for a single coadjoint orbit only, and not for an open set of orbits.

it seems likely that the condition that the restriction of $\omega(\ell)$ to the nilradical of $\frak{g}$ is closed could be replaced by the weaker condition that the orbit $\omega(\ell)$ itself is closed. this would then prove one implication of a conjecture by ludwig and müller, according to which there exists a sub-laplacian of holomorphic $l^1$ (or, more generally, $l^p$) type on $g$ if and only if there exists a point $\ell{\in} \frak{g}^*$ whose orbit is closed and which satisfies boidol's condition.

The compression theorem is used to prove results for equivariant configuration spaces that are analogous to the well-known non-equivariant results of May, Milgram and Segal.

We study concentration phenomena for the system

\[\varepsilon^2 \Delta v - v - \delta \phi v + \gamma v^{p} =0,\quad \Delta\phi+ \delta v^2=0 \]

in the unit ball $B_1$ of $\mathbb{R}^3$ with Dirichlet boundary conditions. Here $\varepsilon,\ \delta,\ \gamma >0$ and $p>1$. We prove the existence of positive radial solutions $(v_{\varepsilon}, \phi_{\varepsilon})$ such that $v_{\varepsilon}$ concentrates at a distance $({\varepsilon}/{2}) |{\rm log}\, {\varepsilon} |$ away from the boundary $\partial B_1$ as the parameter $\varepsilon$ tends to 0. The approach is based on a combination of Lyapunov–Schmidt reduction procedure together with a variational method.

Let [Escr ] denote the space of all entire functions, equipped with the topology of local uniform convergence (the compact-open topology). MacLane [15] constructed an entire function f whose sequence of derivatives (f, f′, f'', …) is dense in [Escr ]; his construction is succinctly presented in a much later note by Blair and Rubel [2], who unwittingly rederived it (see also [3]). We shall call such a function f a universal entire function. In this note we show that analogous universal functions exist in the space [Hscr ]N of functions harmonic on ℝN, where N[ges ]2. We also study the permissible growth rates of universal functions in [Hscr ]N and show that the set of all such functions is very large.

For purposes of comparison, we first review relevant facts about universal entire functions. The function constructed by MacLane is of exponential type 1. Duyos Ruiz [7] observed that a universal entire function cannot be of exponential type less than 1. G. Herzog [11] refined MacLane's growth estimate by proving the existence of a universal entire function f such that [mid ]f(z)[mid ]=O(rer) as [mid ]z[mid ]=r→∞. Finally, Grosse–Erdmann [10] proved the following sharp result.

The 2-modular and 3-modular decomposition numbers for the sporadic simple O'Nan group O'N and its triple cover are determined. The results are obtained with the help of the computer algebra packages MeatAxe and GAP. Permutation modules and tensor products of modules have been condensed. The result relies on the construction of a 153-dimensional representation for 3·O'N over GF(4) and a 154-dimensional representation for O'N over GF(3).

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.