For a real abelian number field F and for a prime p we study the relation between the p-parts of the class groups and of the quotients of global units modulo cyclotomic units along the cyclotomic p-extension of F. Assuming Greenberg's conjecture about the vanishing of the λ-invariant of the extension, a map between these groups has been constructed by several authors, and shown to be an isomorphism if p does not split in F. We focus in the split case, showing that there are, in general, non-trivial kernels and cokernels.

The inverse problem for Schrödinger operators on metric graphs is investigated in the presence of magnetic field. Graphs without loops and with Euler characteristic zero are considered. It is shown that the knowledge of the Titchmarsh–Weyl matrix function (Dirichlet-to-Neumann map) for just two values of the magnetic field allows one to reconstruct the graph and potential on it provided a certain additional no-resonance condition is satisfied.

Let k ∈ ℕ and let E be a Banach space such that every k-homogeneous polynomial defined on a subspace of E has an extension to E. We prove that every norm one k-homogeneous polynomial, defined on a subspace, has an extension with a uniformly bounded norm. The analogous result for holomorphic functions of bounded type is obtained. We also prove that given an arbitrary subspace F ⊂ E, there exists a continuous morphism φk, F: (kF) → (kE) satisfying φk, F(P)|F = P, if and only E is isomorphic to a Hilbert space.

We illustrate a minor error in the biadjointness result for 2-categories of traced monoidal categories and tortile monoidal categories stated by Joyal, Street and Verity. We also show that the biadjointness holds after suitably changing the definition of 2-cells.

We consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ + f(x)ẋ + g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m − 1)/2] limit cycles, where [·] denotes the integer part function.

The unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this paper we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if and only if the given knot appears in an unknotting sequence of a torus knot.

We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.

Here we determine the arithmetic data i.e. the totally real number field and the set of ramified places of the defining quaternion algebra, of all those commensurability classes of arithmetic Fuchsian groups which contain a surface group of genus 2, i.e. a group of signature (2;– –).

A Klein–Schottky group is an extended Kleinian group, containing no reflections and whose orientation-preserving half is a Schottky group. A dihedral-Klein–Schottky group is an extended Kleinian group generated by two different Klein–Schottky groups, both with the same orientation-preserving half. We provide a structural description of the dihedral-Klein–Schottky groups.

Let M be a handlebody of genus g, with a Schottky structure. An imaginary reflection τ of M is an orientation-reversing homeomorphism of M, of order two, whose restriction to its interior is an hyperbolic isometry having at most isolated fixed points. It is known that the number of fixed points of τ is at most g + 1; τ is called a maximal imaginary reflection if it has g + 1 fixed points. As a consequence of the structural description of the dihedral-Klein–Schottky groups, we are able to provide upper bounds for the cardinality of the set of fixed points of two or three different imaginary reflections acting on a handlebody with a Schottky structure. In particular, we show that maximal imaginary reflections are unique.

We consider iterations of smooth non-invertible maps on manifolds of real dimension 4, which are hyperbolic, conformal on stable manifolds and finite-to-one on basic sets. The dynamics of non-invertible maps can be very different than the one of diffeomorphisms, as was shown for example in [4, 7, 12, 17, 19], etc. In [13] we introduced a notion of inverse topological pressure P− which can be used for estimates of the stable dimension δs(x) (i.e the Hausdorff dimension of the intersection between the local stable manifold Wsr(x) and the basic set Λ, x ∈ Λ). In [10] it is shown that the usual Bowen equation is not always true in the case of non-invertible maps. By using the notion of inverse pressure P−, we showed in [13] that δs(x) ≤ ts(ϵ), where ts(ϵ) is the unique zero of the function t → P−(tφs, ϵ), for φs(y):= log|Dfs(y)|, y ∈ Λ and ϵ > 0 small. In this paper we prove that if Λ is not a repellor, then ts(ϵ) < 2 for any ϵ > 0 small enough. In [11] we showed that a holomorphic s-hyperbolic map on 2 has a global unstable set with empty interior. Here we show in a more general setting than in [11], that the Hausdorff dimension of the global unstable set Wu() is strictly less than 4 under some technical derivative condition. In the non-invertible case we may have (infinitely) many unstable manifolds going through a point in Λ, and the number of preimages belonging to Λ may vary. In [17], Qian and Zhang studied the case of attractors for non-invertible maps and gave a condition for a basic set to be an attractor in terms of the pressure of the unstable potential. In our case the situation is different, since the local unstable manifolds may intersect both inside and outside Λ and they do not form a foliation like the stable manifolds. We prove here that the upper box dimension of Wsr(x) ∩ Λ is less than ts(ϵ) for any point x ∈ Λ. We give then an estimate of the Hausdorff dimension of Wu() by a different technique, using the Holder continuity of the unstable manifolds with respect to their prehistories.

Multisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

We consider sets of real numbers in [0, 1) with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by 1/2, are given by a modified variational principle.