Every convolution operator on a space of ultradifferentiable functions of Beurling or Roumieu type and on the corresponding space of ultradistributions is hypercyclic and chaotic when it is not a multiple of the identity. The operator of differentiation is hypercyclic on the space A−∞, but it need not be hypercyclic on radial weighted algebras of entire functions.

We consider the classical universal covering $\exp: {{\mathbb C}}\To {{\mathbb C}}^*$ of the complex torus as an algebraic structure. The exponentiation is seen here as an abstract homomorphism from a divisible torsion-free group onto the multiplicative group of an algebraically closed field of characteristic zero, with cyclic kernel. We prove that any structure satisfying this description is isomorphic to the classical one provided that the cardinality of the underlying field is equal to that of ${{\mathbb C}}$. This can also be seen as a model-theoretic statement on the categoricity of a corresponding $L_{\omega_1,\omega}$-sentence. The proof is a combination of arithmetic and model-theoretic methods.

When $T\,{:}\, X\,{\longrightarrow}\,X$ is a one-sided topologically mixing subshift of finite type and $\varphi\,{:}\,X\,{\longrightarrow}\,R$ is a continuous function, one can define the Ruelle operator ${\cal L}_\varphi\,{:}\,C(X)\,{\longrightarrow}\,C(X)$ on the space $C(X)$ of real-valued continuous functions on $X$. The dual operator ${\cal L}^*_\varphi$ always has a probability measure $\nu$ as an eigenvector corresponding to a positive eigenvalue (${\cal L}^*_\varphi\nu\,{=}\,\lambda\nu$ with $\lambda\,{>}\,0$). Necessary and sufficient conditions on such an eigenmeasure $\nu$ are obtained for $\varphi$ to belong to two important spaces of functions, $W(X,T)$ and ${\rm Bow} (X,T)$. For example, $\varphi\,{\in}\,{\rm Bow}(X,T)$ if and only if $\nu$ is a measure with a certain approximate product structure. This is used to apply results of Bradley to show that the natural extension of the unique equilibrium state $\mu_\varphi$ of $\varphi\,{\in}\,{\rm Bow}(X,T)$ has the weak Bernoulli property and hence is measure-theoretically isomorphic to a Bernoulli shift. It is also shown that the unique equilibrium state of a two-sided Bowen function has the weak Bernoulli property. The characterizations mentioned above are used in the case of $g$-measures to obtain results on the ‘reverse’ of a $g$-measure.

The covariogram of a convex body $K$ provides the volumes of the intersections of $K$ with all its possible translates. Matheron conjectured in 1986 that this information determines $K$ among all convex bodies, up to certain known ambiguities. It is proved that this is the case if $K\subset{\mathbb R}^2$ is not ${\rm C}^1$, or it is not strictly convex, or its boundary contains two arbitrarily small ${\rm C}^2$ open portions ‘on opposite sides’. Examples are also constructed that show that this conjecture is false in ${\mathbb R}^n$ for any $n\geq 4$.

It is proved that a connected infinite graph has a normal spanning tree (the infinite analogue of a depth-first search tree) if and only if it has no minor obtained canonically from either an (ℵ0, ℵ1)-regular bipartite graph or an order-theoretic Aronszajn tree. This disproves Halin's conjecture that only the first of these obstructions was needed to characterize the graphs with normal spanning trees. As a corollary Halin's further conjecture is deduced, that a connected graph has a normal spanning tree if and only if all its minors have countable colouring number.

The precise classification of the (ℵ0, ℵ1)-regular bipartite graphs remains an open problem. One such class turns out to contain obvious infinite minor-antichains, so as an unexpected corollary Thomas's result that the infinite graphs are not well-quasi-ordered as minors is reobtained.

Given a computably enumerable set W, there is a Turing degree which is the least jump of any set in which W is computably enumerable, namely 0′. Remarkably, this is not a phenomenon of computably enumerable sets. It is shown that for every subset A of ℕ, there is a Turing degree, c′μ(A), which is the least degree of the jumps of all sets X for which A is [sum ]01(X). In addition this result provides an isomorphism invariant method for assigning Turing degrees to certain torsion-free abelian groups.

The relation between compactly generated groups of polynomial growth and FC−-groups is clarified. It follows that L1(G) is symmetric for G compactly generated and of polynomial growth.

Using simplicial methods developed in an earlier note, the paper constructs topological quantum field theories using an algebraic model of a homotopy n-type as initial data, generalising constructions of Yetter for n=1 and n=2.

We prove a general Borg-type result for reflectionless unitary CMV operators $U$ associated with orthogonal polynomials on the unit circle. The spectrum of $U$ is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle.

In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory functions to prove an infinite sequence of trace formulas connected with the CMV operator $U$.

We work in set theory without the axiom of choice: ZF. We show that the axiom BC: Compact Hausdorff spaces are Baire, is equivalent to the following axiom: Every tree has a subtree whose levels are finite, which was introduced by Blass (cf. [4]). This settles a question raised by Brunner (cf. [9, p. 438]). We also show that the axiom of Dependent Choices is equivalent to the axiom: In a Hausdorff locally convex topological vector space, convex-compact convex sets are Baire. Here convex-compact is the notion which was introduced by Luxemburg (cf. [16]).

Further properties of a group $\Gamma$ introduced by the first author in 1980 (sometimes called the first Grigorchuk group) are established. These are notably that any two infinite finitely generated subgroups of $\Gamma$ have isomorphic subgroups of finite index and that the generalized word problem for $\Gamma$ is soluble.

By using Morse theory, we study the existence and multiplicity of nontrivial solutions for a class of p-sublinear p-Laplacian elliptic equations.

We extend uncertainty principles which are valid for the Fourier transform to the setting of the ambiguity function. A general result is established for annihilating sets: strongly/weakly annihilating sets for the Fourier transform yield such sets for the ambiguity function, extending a result known for sets of finite measure. We apply this to sublevel sets of nondegenerate quadratic forms. Our main result is a sharp version of Beurling's uncertainty principle for the ambiguity function.

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), their full isometry groups Isom(M) are calculated. It is determined which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and it is verified that the inclusion of Isom(M) to Diff(M) is a bijection on path components.

Results from the character theory of symmetric groups are used to obtain an asymptotic estimate for the subgroup growth of fundamental groups of closed 2-manifolds. The main result implies an affirmative answer, for the class of groups investigated, to a question of Lubotzky's concerning the relationship between the subgroup growth of a one-relator group and that of a free group of appropriately chosen rank. As byproducts, an interesting statistical property of commutators in symmetric groups and the fact that in a ‘large’ surface group almost all finite index subgroups are maximal are obtained, among other things. The approach requires an asymptotic estimate for the sum $\Sigma 1/(\chi_\lambda(1))^s$ taken over all partitions $\lambda$ of $n$ with fixed $s \ge 1$ , which is also established.

An independence algebra is an algebra A in which the subalgebras satisfy the exchange axiom, and any map from a basis of A into A extends to an endomorphism. Independence algebras fall into two classes; the first is specified by a set X, a group G, and a G-space C. The second is much more restricted; it is shown that the subalgebra lattice is a projective or affine geometry, and a complete classification of the finite algebras is given.

S.-Y. Cheng, P. Li and S.-T. Yau proved comparison theorems for the volume of extrinsic balls in minimal submanifolds of space forms. These results were extended by S. Markvorsen for minimal submanifolds of a riemannian manifold with just an upper bound on the sectional curvature. In the paper an isoperimetric inequality for extrinsic balls in minimal submanifolds of a riemannian manifold N with sectional curvatures bounded from above by a non-positive constant is found. As a corollary of this result an alternative proof is obtained of the comparison for the volume of extrinsic balls stated by the preceding authors, but now the equality case is characterized when the upper bound for the sectional curvatures of the ambient manifold is strictly negative. Finally, when the sectional curvatures of N are bounded from above for any constant (positive or negative), it is proved that the ∞-isoperimetric quotient of the extrinsic balls is bounded from below by the mean curvature of the geodesic spheres in the m-dimensional real space forms.

Benford's law (to base $B$) for an infinite sequence $\{x_k: k \ge 1\}$ of positive quantities $x_k$ is the assertion that $\{ \log_B x_k : k \ge 1\}$ is uniformly distributed $(\bmod\ 1)$. The $3x+1$ function $T(n)$ is given by $T(n)=(3n+1)/{2}$ if $n$ is odd, and $T(n)= n/2$ if $n$ is even. This paper studies the initial iterates $x_k= T^{(k)}(x_0)$ for $1 \le k \le N$ of the $3x+1$ function, where $N$ is fixed. It shows that for most initial values $x_0$, such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence $\{\log_B x_k: 1 \le k \le N \}$ is small.

It is proved that the measure algebra M(G) of a locally compact group G is Connes-amenable if and only if G is amenable.

In this article we extend well-known results of Livsic on the regularity of measurable solutions to the cocycle equation. Livsic proved a regularity result for real valued cocycles, but for compact Lie group valued cocycles his result was restricted to coboundaries (that is, trivial or constant cocycles). In this article we prove the complete result.

We present several useful applications of these results.