The paper concerns sufficiently saturated structures M over a countable language with a unary predicate P. It is shown that if $P(M)$ is stably embedded and there are no Vaughtian pairs with respect to P, then an infinite group is interpretable over M (in an infinitary sense of ‘interpretable’). Also, it is shown that if M is $\omega$-categorical, $f\,{:}\,D\,{\longrightarrow}\,P$ is a 0-definable map with finite fibres, and $P(M)$ is stably embedded but D is not, then some infinite group is first-order interpretable over M.

Binz, Neukirch and Wenzel proved a profinite version of the Kurosh subgroup theorem. In the paper a subgroup theorem for certain profinite fundamental groups of graphs of profinite groups is shown. In particular, the case of free products with amalgamation is discussed and illustrated in the context of real function fields in one variable.

Let $p$ be a prime number, $F$ a number field, and ${\cal H}_F^n$ the set of all unramified cyclic extensions over $F$ of degree $p$ having a relative normal integral basis. When $\z_p \in F^{\times}$, Childs determined the set ${\cal H}_F^n$ in terms of Kummer generators. When $p=3$ and $F$ is an imaginary quadratic field, Brinkhuis determined this set in a form which is, in a sense, analogous to Childs's result. The paper determines this set for all $p \ge 3$ and $F$ with $\z_p \not\in F^{\times}$ (and satisfying an additional condition), using the result of Childs and a technique developed by Brinkhuis. Two applications are also given.

A set $A\subseteq \{1,2,\ldots,n\}$ is said to be $k$-separated if, when considered on the circle, any two elements of $A$ are separated by a gap of size at least $k$.

A conjecture due to Holroyd and Johnson that an analogue of the Erdős–Ko–Rado theorem holds for $k$-separated sets is proved. In particular, the result holds for the vertex-critical subgraph of the Kneser graph identified by Schrijver, the collection of separated sets. A version of the Erdős–Ko–Rado theorem for weighted $k$-separated sets is also given.

It is shown that a $\scriptstyle \overline {\mathbb Q}$-curve of genus $g$ and with stable reduction (in some generalized sense) at every finite place outside a finite set $S$ can be defined over a finite extension $L$ of its field of moduli $K$ depending only on $g$, $S$ and $K$. Furthermore, there exist $L$-models that inherit all places of good and stable reduction of the original curve (except possibly for finitely many exceptional places depending on $g$, $K$ and $S$). This descent result yields this moduli form of the Shafarevich conjecture: given $g$, $K$ and $S$ as above, only finitely many $K$-points on the moduli space ${\cal M}_g$ correspond to $\scriptstyle \overline {\mathbb Q}$-curves of genus $g$ and with good reduction outside $S$. Other applications to arithmetic geometry, like a modular generalization of the Mordell conjecture, are given.

The paper shows that for every positive integer $p > 2$, there exists a compact non-orientable surface of genus $p$ with at least $4p$ automorphisms if $p$ is odd, or at least $8\,(p-2)$ automorphisms if $p$ is even, with improvements for odd $p\not\equiv 3$ mod 12. Further, these bounds are shown to be sharp (in that no larger group of automorphisms exists with genus $p$) for infinitely many values of $p$ in each congruence class modulo 12, with the possible (but unlikely) exception of 3 mod 12.

It is shown that Alperin's weight conjecture is equivalent to the existence of contractible chain complexes whose entries have the right dimension coming from some of the alternating sum formulations. It is conjectured that for the other formulations and for Dade's ordinary conjecture, there also exist such contractible chain complexes.

Let $\mu$ be the self-similar measure for a linear function system $S_jx=\rho x+b_j$ ($j=1,2,\ldots,m$) on the real line with the probability weight $\{p_j\}_{j=1}^m$. Under the condition that $\{S_j\}_{j=1}^m$ satisfies the finite type condition, the $L^q$-spectrum $\tau(q)$ of $\mu$ is shown to be differentiable on $(0,\infty)$; as an application, $\mu$ is exact dimensional and satisfies the multifractal formalism.

The existence of $2\pi$-periodic solutions of the second-order differential equation \[ x''+f(x)x'+ax^+-bx^-+g(x)=p(t), \qquad n\in \mathbb{N},\] where $a, b$ satisfy $1/\sqrt{a}+1/\sqrt{b}=2/n$ and $p(t)=p(t+2\pi)$, $t\in \mathbb{R}$, is examined. Assume that limits $\lim_{x\to\pm\infty}F(x)=F(\pm\infty)$ ($F(x)=\int_0^xf(u) du$) and $\lim_{x\to\pm\infty}g(x)=g(\pm\infty)$ exist and are finite. It is proved that the equation has at least one $2\pi$-periodic solution provided that the zeros of the function $\Sigma_1$ are simple and the zeros of the functions $\Sigma_1, \Sigma_2$ are different and the signs of $\Sigma_2$ at the zeros of $\Sigma_1$ in $[0,2\pi/n)$ do not change or change more than two times, where $\Sigma_1$ and $\Sigma_2$ are defined as follows: \[\Sigma_1(\theta)=\frac{n}{\pi}\left[\frac{g(+\infty)}{a}-\frac{g(-\infty)}{b} \right]-\frac {1}{2\pi}\int_0^{2\pi}p(t)\varphi(t+\theta)\,dt,\qquad \theta\in [0, 2\pi/n],\]\[\Sigma_2(\theta)=\frac{n}{\pi}[F(+\infty)-F(-\infty)]-\frac{1}{2\pi} \int_0^{2\pi}p(t)\varphi'(t+\theta)\,dt, \qquad\theta\in [0, 2\pi/n].\] Moreover, it is also proved that the given equation has at least one $2\pi$-periodic solution provided that the following conditions hold: \begin{eqnarray*} -\infty&<&\liminf_{x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}\le \limsup_ {x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}<+\infty,\\ 0&<&\liminf_{x\to\pm\infty}\frac{g(x)}{|x|^{q-2}x}\leq \limsup_{x\to\pm\infty} \frac{g(x)}{|x|^{q-2}x}<+\infty, \end{eqnarray*} with $1\leq p<q<2$.

The paper shows that, for subanalytic stratifications, Lipschitz equisingularity as defined by Mostowski is preserved after intersection with generic wings, that is, $L$-regularity implies $L^*$-regularity. This was one of the conditions required of a good equisingularity notion by Teissier in his foundational 1974 Arcata paper.

Previous authors have shown that Lipschitz equisingularity is generic, implies bilipschitz triviality, and hence topological triviality, and implies equimultiplicity.

The paper first proves a result on the upper semi-continuity of the entropy map of random bundle maps, and then proves some large deviation results for random maps perturbations of an Axiom A diffeomorphism.

A new large class of rearrangement-invariant (symmetric) spaces is introduced which contains most classical spaces used in applications. It consists of spaces with arbitrary fundamental function $\f(t)$ which are interpolation spaces with respect to the corresponding extreme spaces $\Lambda_\varphi$ and $M_\varphi$ (the quasinorm case is also permitted). The (quasi)norms for these new spaces are of the form $\|\varphi(t)f^*(t)\|_{\widetilde E}$, where $\widetilde E$ is an arbitrary rearrangement-invariant function space with respect to the measure $dt/t$. Thus the considered spaces include and generalize Lorentz, Lorentz–Zygmund and other similar spaces as well as the spaces $L_{p,\alpha, E}$, previously studied by Pustylnik. In spite of their generality, these spaces can be investigated deeply and in detail with rather sharp results that open a simple way to various applications, extending results about their classical examples. For instance, their dual spaces, sharp conditions for separability and mutual embeddings, dilation functions, Boyd indices and so on are found. A theorem on optimal weak type interpolation in these spaces is also proved.

A sufficient condition is given under which the sum, product and indeed any polynomial combination of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. This generalizes a result of Gillespie for Hilbert space operators. It is shown in particular that if $X$ is a UMD space, then the sum of finitely many commuting real scalar-type spectral operators acting on $X$ is a well-bounded operator (a result which fails on general reflexive Banach spaces).

Projections are constructed in the rotation algebra that are orthogonal to their Fourier transform and which are fixed under the flip automorphism. Such projections are expected in a construction of an inductive limit structure for the irrational rotation algebra that is invariant under the Fourier transform (namely, as two circle algebras of the same dimension, which are swapped by the Fourier transform, plus a few points). The calculation is based on Rieffel's construction of the Schwartz space as an equivalence bimodule of rotation algebras as well as on the theory of theta functions.

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.

Various kinds of minimal 3-spheres with constant sectional curvature in the complex projective space $\mathbb{C}^n$ are studied.

It is known that the group of homeomorphisms of a non-compact, connected, metrizable manifold acts transitively on certain spaces of measures defined, among other properties, by their behaviour at the manifold's set of ‘ends’. As a generalization of previous work of Fathi in the realm of compact manifolds, it is shown that the given action admits a continuous section when restricted to ‘biregular’ homeomorphisms and measures. This result exhibits the isotropy groups of measure-preserving homeomorphisms as deformation retracts of the corresponding acting groups.

When a Fock-adapted Feller cocycle on a $C^*$-algebra is regular, completely positive and contractive, it possesses a stochastic generator that is necessarily completely bounded. Necessary and sufficient conditions are given, in the form of a sequence of identities, for a completely bounded map to generate a weakly multiplicative cocycle. These are derived from a product formula for iterated quantum stochastic integrals. Under two alternative assumptions, one of which covers all previously considered cases, the first identity in the sequence is shown to imply the rest.