Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra $S$ in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adjoin inverses to some of the arrows in the quiver, we show that the path algebra of the new quiver with relations is $M_n(S)$ where $n$ is the number of vertices in our quiver. The slogan would be that every finitely presented algebra is Morita equivalent to a universal localization of a finite dimensional algebra.

We discuss several methods which yield spanning sets for the irreducible polynomial $GL(n,K)$-modules, where $K$ is an infinite field. We show that the spanning set which arises from our first method coincides, up to sign, with the spanning set produced by a method due to Pittaluga and Strickland. We prove that these spanning sets are related to the spanning set produced by a third method via the Désarménien matrix.

In this paper we give an asymptotic expansion including error terms for the number of closed orbits in the homology classes for homological full Anosov flows. In particular we obtain formulae concerning the coefficients of error terms which depend on the homology classes.

The moduli space ${\frak N}^{c}_{g,1}$ of non-orientable surfaces of genus $g \geq 0$ with $c \geq 0$ distinguished points and one boundary curve is described via a model ${\frak P}(h,c)$: the space of configurations of $h=g+c+1$ pairs of parallel slits in $\mathbb{C}$. Based on this model, we prove that ${\frak N}^{c}_{g,1}$ is a non-orientable manifold, and we compute its homology for $h \leq 3$ with ${\mathbb Z}_2$-coefficients.

We give an elementary proof of the rational Hurewicz theorem and compute the rational cohomology groups of Eilenberg–MacLane spaces and the rational homotopy groups of spheres. Instead of using the Serre spectral sequence, we only assume the classical Hurewicz theorem, and give a short proof of the rational Gysin and Wang long exact sequences, which are applied inductively to the path fibration of Eilenberg–MacLane spaces.

In this paper we apply the techniques and results from the theory of multifractal divergence points to give a systematic and detailed account of the Hausdorff dimensions of sets of numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their $N$-adic expansion. Using earlier methods and results we investigate and compute the Hausdorff dimension of several new sets of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of numbers for which the limiting frequencies of the digits in their $N$-adic expansion do not exist. Such sets have only very rarely been studied. In addition, our techniques provide simple proofs of (substantial generalizations of) known results, by Cajar and Drobot and Turner and others, on the Hausdorff dimension of sets of normal and non-normal numbers.

We study functions $f$ of one complex variable with values in a complete locally convex space $E$. In [13] we had shown that for $f$ to be holomorphic it suffices that it is locally bounded and that there is a separating subset $H$ of $E'$ such that $x'\circ f$ is holomorphic for each functional $x'\in H$. We give here a new and considerably shorter proof of this result, and we extend it to a weak criterion for holomorphic continuability.

We compute the cobordism rings of the classifying spaces of a certain class of Chevalley groups. In the particular case of the general linear group, we prove that it is isomorphic to the Chow ring.

We prove that the boundary extension of a quasi-isometry of hyperbolic space which conjugates two Kleinian groups maps the Myrberg limit set of the first group bijectively onto the Myrberg limit set of the second group. As a first application of this observation we obtain new geometric proofs of Mostow's Rigidity Theorem. Furthermore, we recover the classical result that motions in the Teichmüller space of a compact hyperbolic surface induce singular boundary maps.

We examine the multifractal spectra of one-sided local dimensions of Ahlfors regular measures on ${\bf R}$. This brings into a natural context a curious property that has been observed in a number of instances, namely that the Hausdorff dimension of the set of points of non-differentiability of a self-affine ‘devil's staircase’ function is the square of the dimension of the set of points of increase.

Let $Q$ be any II$_1$–factor. It is shown that any standard lattice ${\euscript G}$ can be realized as the standard invariant of a free product of (several) rescalings of $Q$. In particular, if $Q$ has fundamental group equal to the positive reals and if $P$ is the free product of infinitely many copies of $Q$, then $P$ has subfactors giving rise to all possible standard invariants. Similarly, given a II$_1$–subfactor $N\subset M$, it is shown there are subfactors ${\hat N}\subset{\hat M}$ having the same standard invariant as $N\subset M$ but where ${\hat M}$, respectively ${\hat N}$, is the free product of $M$, respectively $N$, with rescalings of $Q$.

Three kinds of proper actions of increasing strength are defined. We prove that the three definitions specialize to the definitions by Bourbaki, by Palais and by Baum, Connes and Higson in their respective settings. The third of these, which thus turns out to be the strongest, originally only concerns actions of second countable locally compact groups on metrizable spaces. In this situation, it is shown to coincide with the other two definitions if the total space locally has the Lindelöf property and the orbit space is regular.

We extend a result of M. Heins by showing that for any sequence of points $(z_n)$ in the unit disk ${\Bbb D}$ tending to the boundary, there is a Blaschke product $B$ which is universal for noneuclidian translates in the sense that the set $\{B((z\,{+}\,z_n)/(1\,{+}\,\overline{z}_nz))\,{:} n\,{\in}\,{\Bbb N}\}$ is locally uniformly dense in the set of all holomorphic functions bounded by one on ${\Bbb D}$. From this, we conclude that for every countable set ${\sp L}$ of hyperbolic/parabolic automorphisms of the unit disk there exists a Blaschke product which is a common cyclic vector in $H^2$ for the composition operators associated with the elements in ${\sp L}$. These results are obtained by transferring the associated approximation problems to interpolation problems on the corona of $H^\infty$.

The purpose of this note is to call attention to an error in the paper [1]. The main result was a new proof of the main theorem in [2]. In line 1 of page 446 of [1] one finds the following remark: “This is the result obtained in [2]”. However, a comparison of the two results shows that the generator which is called $d_1 d^{-1}_2$ in the notation of [1] appears in [2] but is missing from the list in [1].

A Kolmogorov-type inequality is derived from a Jackson-type inequality and a Bernstein-type inequality. Several new results follow as corollaries.

In this paper, we are concerned with the boundedness of all the solutions of the two-dimensional differential system $$ \left.\begin{array}{@{}l@{}}x'\,{=}\,{-}|y|^{p-2} y+f(t)\\ y'\,{=}\,|x|^{q-2} x+g(t) \end{array}\right\} $$ where $1\,{<}\,p$, $1\,{<}\,q\,{<}\,2$ and $5pq\,{>}\,3p+4q$, $pq/(p+q)\,{<}\,1$, $f, g$ are $C^5$ 1-periodic functions.

We investigate inclusion indices for general function spaces, not necessarily symmetric. Using them, we estimate the grade of proximity between two spaces $E\,{\hookrightarrow}\, F$ when we have certain information on the inclusion. The results are based on ideas from interpolation theory.

This paper considers various kinds of multiplicative transfers, namely Evens transfer, Kozlowski transfer and normalized Evens transfer in singular and Galois cohomologies, which have been introduced as tools for solving the problem of determining characteristic classes of direct image bundles for covering maps (Fulton–MacPherson) or Stiefel–Whitney classes of induced Galois representation for finite separable extension of commutative fields of characteristic not 2 (Kahn).

We show that normalized Evens transfer is covariant, when restricted to homogeneous elements or Stiefel–Whitney classes, in the case of singular cohomology. We deduce that, in the case of Galois cohomology, Kozlowski transfer and the transfer involved by Kahn coïncide on invertible elements. We conclude that the Kahn formula can be viewed as an inflation of the Fulton–MacPherson formula.

The multifractal formalism for functions has been proved to be valid for a large class of selfsimilar functions. All the functions that have been studied are (or turned out to be) associated with a family of contractions which satisfies some separation conditions. In this paper, we extend the validity in the presence of overlaps involved by the well-known $n$-scale dilation family. Our method of proof is based on wavelet analysis and some interesting properties of this family.

We investigate, for a locally compact group $G$, the operator amenability of the Fourier–Stieltjes algebra $B(G)$ and of the reduced Fourier–Stieltjes algebra $B_r(G)$. The natural conjecture is that any of these algebras is operator amenable if and only if $G$ is compact. We partially prove this conjecture with mere operator amenability replaced by operator $C$-amenability for some constant $C\,{<}\, 5$. In the process, we obtain a new decomposition of $B(G)$, which can be interpreted as the non-commutative counterpart of the decomposition of $M(G)$ into the discrete and the continuous measures. We further introduce a variant of operator amenability – called operator Connes-amenability – which also takes the dual space structure on $B(G)$ and $B_r(G)$ into account. We show that $B_r(G)$ is operator Connes-amenable if and only if G is amenable. Surprisingly, $B({\mathbb F}_2)$ is operator Connes-amenable although ${\mathbb F}_2$, the free group in two generators, fails to be amenable.