We provide sufficient conditions on a positive finite rotation invariant Borel measure on {𝔻}^n which guarantee that the analogue of the Carleson measure theorem remains valid for Bergman spaces of holomorphic and n-harmonic functions on {𝔻}^n generated by the measure.

This paper considers the long-time behavior of solutions for the Cauchy problem of a class of second-order nonlinear differential equations: -x''+ f(t,x,x')x'+g(x)=h(t). Under appropriate conditions it is shown that the solutions of the problem possess some dichotomy properties.

Let Ea(u,v) be the extremal algebra determined by two hermitians u and v with u^2=v^2=1. We show that: Ea(u,v)={f+gu:f,g\in C(𝕋)} , where [] is the unit circle; Ea(u,v) is C^*-equivalent to C^*({\cal G}), where {\cal G} is the infinite dihedral group; most of the hermitian elements k of Ea(u,v) have the property that k^n is hermitian for all odd n but for no even n; any two hermitian words in {\cal G} generate an isometric copy of Ea(u,v) in Ea(u,v).

We apply a previously obtained ansatz for extremal Kähler metrics to show that if a manifold admits a Hodge metric with constant scalar curvature, then the total space of the projectivization of a line bundle with first Chern class equal to the Kähler class of the metric admits a one-parameter family of extremal Kähler metrics. This generalizes earlier constructions.

A compact Klein surface X is a compact surface with a dianalytic structure. Such a surface is said to be q-hyperelliptic if it admits an involution \phi , that is an order two automorphism, such that X/ < \phi > has algebraic genus q. A Klein surface of genus 1 and k boundary components is a k-bordered torus.

By means of NEC groups, q-hyperelliptic k-bordered tori are studied and a geometrical description of their associated Teichmüller spaces is given.

We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).

We calculate the height of quantum determinantal ideals in the algebra of quantum matrices, and also the Gelfand-Kirillov, Krull and Classical Krull dimensions of the corresponding quantum determinantal factors.

Methods from fibrewise homology theory are illustrated by computations of cohomology rings of certain mapping spaces arising in the geometry of loop groups, specifically the spaces of maps from S^1 to the classifying space BSO(n) of SO(n) and maps from S{\hskip1}^2 to BSU(n).

In this note we use the Hopf map to construct a family of metrics in the 3-sphere parametrized on the space of positive smooth functions in the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. Also, the fibres are all geodesics if and only if the metric comes from a constant function and so, we have a Berger 3-sphere. Every geodesic in a 3-dimensional Riemannian manifold is a minimum for each elastic energy functional. Therefore, we characterize those functions on the 2-sphere that locally give metrics which have all the fibres being elastica, i.e., critical points of those functionals. Some applications are given including one to the Willmore-Chen variational problem.

Existence theorems for the equation F(x)=\varphi (x) are proved when F is a function with “good” surjective properties and \varphi satisfies certain compactness conditions on countable subsets of the space. Also results for certain homotopic perturbations of the equation are obtained. The results lead to various fixed point theorems of Darbo type for F=id, but they are also applicable if F acts between different spaces. Also the inclusions F(x)\in \varphi (x) (resp. F(x)\subseteq \varphi (x)) for multivalued functions \varphi (resp. F and \varphi) are studied. There are some connections with the theory of 0-epi maps.

A Banach space is called (almost) transitive if the isometry group acts (almost) transitively on the unit sphere. The main problems around transitivity are the Banach-Mazur conjecture that the only separable and transitive Banach spaces are the Hilbert ones (1930) and the Wood conjecture that C_0(L) cannot be almost transitive in its natural supremum norm unless L is a singleton (1982). In this note we give necessary and sufficient conditions on the locally compact space L for the (almost) transitivity of C_0(L). This will clarify the topological content of Wood's problem.

We derive a finite normal series for the group {\rm Aut}\nolimits _*(X\;\!) of self-homotopy equivalences which induce identity automorphisms of homology groups, where X is a countable, simply-connected and finite-dimensional CW-complex.

An operator T on a Banach space is called ‘semi B-Fredholm’ if for some n \in {\tf="times-b"N} the range R(T\;\!^n) of T\;\!^n is closed and the induced operator T_n on R(T\;\!^n) semi-Fredholm. Semi B-Fredholm operators are stable under finite rank perturbation, and subject to the spectral mapping theorem; on Hilbert spaces they decompose as sums of nilpotent and semi-Fredholm operators. In addition some recent generalizations of the punctured neighborhood theorem turn out to be consequences of Grabiner's theory of ‘topological uniform descent’.

In this paper, we show that Weyl's theorem holds for operators having the single valued extension property and quasisimilarity preserves Weyl's theorem for these operators under some assumptions for spectral subsets, respectively.

Let G be a finite group. The symmetric crosscap number \tilde \sigma (G) is the minimum topological genus of any compact non-orientable surface (with empty boundary) on which G acts effectively. We first survey some of the basic facts about the symmetric crosscap number; this includes relationships between this parameter and others. We obtain formulas for the symmetric crosscap number for three families of groups, the dicyclic groups, the abelian groups with most factors in the canonical form isomorphic to Z_(2), and the hamiltonian groups with no odd order part. We also determine \tilde \sigma (G) for each group G with order less than 16. The groups with symmetric crosscap numbers 1 and 2 have been classified. We show here that there are no groups with \tilde\sigma=3; this affirms a conjecture of Tucker.

All local solutions of the two dimensional Einstein-Weyl equations are found, and related to the compact examples obtained in [1].

We study the topological properties of cohomogeneity one flat manifolds and their orbits. Among other results we prove that principal orbits of R^n are isometric to R^{n-1} or S^k(c)\times R^{n-k-1}. We show that if M has one singular orbit, it is a totally geodesic submanifold of M and if M is orientable then there is at most one singular orbit.

We discuss classes of topological groups which can be approximated by p-adic Lie groups, and varieties of Hausdorff groups generated by classes of \hboxp-adic Lie groups (for a single or multiple p). We give several characterizations of locally compact pro-p-adic Lie groups and locally compact pro-discrete groups, and prove a “pro-version” of Cartan's Theorem: whenever a locally compact group is a pro-p-adic Lie group and a pro-q-adic Lie group for distinct primes p and q, it is pro-discrete. If a locally compact group can be approximated by p-adic Lie groups for variable primes p, then it is a pro-p-adic Lie group for some prime p.

AC-operators generalise normal operators on Hilbert space in the context of well-boundedness. In this paper we study AC-operators T=U+iV, where U and V are commuting well-bounded operators with decomposition of the identity of bounded variation. We also explore some properties of AC-operators by applying the theory of (Foiaş) decomposable operators.

We study the moduli space of pluriregular threefolds of general type with K3X=8, 10, 12 and p_g(X)=5 and whose canonical map is a finite morphism onto a smooth quadric Q⊂ ℙ4[Copf].