Reducibility of homogeneous Riemannian structures of the class ℐ3 in low dimension

We study naturally reductive spaces with an Ambrose-Singer connection having an algebraic curvature tensor field. We prove a decomposition theorem for spaces of dimension n6.This work was partially supported by M.P.I.     

On the convergence of the multivariate “homogeneous”qd-algorithm

The convergence of columns in the univariateqd-algorithm to reciprocals of polar singularities of meromorphic functions has often proved to be very useful. A multivariateqd-algorithm was discovered in 1982 for the construction of the so-called homogeneous Padé approximants.In the first section we repeat the univariate convergence results. In the second section we summarize the homogeneous multivariateqd-algorithm. In the third section a multivariate convergence result is proved by combining results from the previous sections. This convergence result is compared with another theorem for the general order multivariateqdg-algorithm. The main difference lies in the fact that the homogeneous form detects the polar singularities pointwise while the general form detects them curvewise.     

On 3-dimensional Riemannian Σ-spaces

The following result is proved: a 3-dimensional connected and simply connected Riemannian manifold admitting a reduced -structure (in the sense of O. Loos) is either a Riemannian symmetric space or it is isometric to a unimodular Lie group with a left-invariant Riemannian metric. At the same time, we give first nontrivial examples of Riemannian -spaces, which are not symmetric of finite order.     

On the character and π-weight of homogeneous compacta

UnderGCH, χ(X)≤π(X) for every homogenous compactumX. CH implies that a homogeneous compactum of countable π-weight is first countable. There is a compact space of countable π-weight and uncountable character which is homogeneous underMA+GCH, but not underCH.     

On Rough ℐ2-Convergence of Double Sequences

In this study, we introduce the notion of rough ?₂-convergence and the set of rough ?₂-limit points of a double sequence and obtained two rough ?₂-convergence criteria associated with this set. Later, we proved that this set is closed and convex. Finally, we examined the relationships between the set of cluster points and the set of rough ?₂-limit points of a double sequence.     

On the uniqueness of almost-Kähler structures

We show uniqueness up to sign of positive, orthogonal almost-Kähler structures on any non-scalar flat Kähler–Einstein surface.     

On the Gromov width of homogeneous Kähler manifolds

We compute the Gromov width of homogeneous Kähler manifolds with second Betti number equal to one. Our result is based on the recent preprint [4] and on the upper bound of the Gromov width for such manifolds obtained in [6].     

On the rigidity of Riemannian–Penrose inequality for asymptotically flat 3-manifolds with corners

Mathematische Zeitschrift - In this paper we prove a rigidity result for the equality case of the Penrose inequality on 3-dimensional asymptotically flat manifolds with nonnegative scalar curvature...     

On the shape of random Pólya structures

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton–Watson tree $C_n$ and many small forests, where with probability tending to one, as $n$ tends to infinity, any forest $F_n\left(v\right)$, that is attached to a node $v$ in $C_n$, is maximally of size $\left|F_n\left(v\right)\right|=O(logn)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities.In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for $\left|F_n\left(v\right)\right|$, namely $\left|F_n\left(v\right)\right|=\Theta (logn)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n\left(v\right)$ is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Pólya structures.     

On the essential spectrum of Schrödinger operators on Riemannian manifolds

The main result of this paper is a lower bound for the essential spectrum of Schrödinger operators −Δ+V on Riemannian manifolds. In particular, we obtain conditions on V which imply the discreteness of the spectrum, or equivalently, the compactness of the resolvent.     

On the homotopy type of Poincaré spaces

We study the homotopy type of finite-oriented Poincaré spaces (and, in particular, of closed topological manifolds) in even dimension. Our results relate polarized homotopy types over a stage of the Postnikov tower with the concept of CW-tower of categories due to Baues. This fact allows us to obtain a new formula for the top-dimensional obstruction for extending maps to homotopy equivalences. Then we complete the paper with an algebraic characterization of high-dimensional handlebodies. Received: April 14, 1999?Published online: October 2, 2001     

Localization of Chern–Simons type invariants of Riemannian foliations

We prove an Atiyah–Bott–Berline–Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern–Simons type invariants, for example the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations, to the union of closed leaves. Various examples are given to illustrate our method.     

On compact aspherical homogeneous manifolds of dimension ≤7

In the paper, a classification of aspherical compact homogeneous manifolds of dimension less than or equal to 7 is discussed.     

On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

On the primality of 2h·3n+1

We consider the primality test of Williams and Zarnke for rational integers of the form 2h·3ⁿ+1. We give an algebraic proof of the test, and we resolve a sign ambiguity. We also show that the conditions of the original test can be relaxed, especially if h is divisible by a power of 2.     

On mappings in the Orlicz–Sobolev classes on Riemannian manifolds

We study the problems of the continuous and homeomorphic extension to the boundary of lower Q-homeomorphisms between domains on Riemannian manifolds and formulate the corresponding consequences for homeomorphisms with finite distortion in the Orlicz–Sobolev classes W_loc^1,j W_{loc}^{1,varphi } under a condition of the Calderon type for the function φ and, in particular, in the Sobolev classes W_loc^1,p W_{loc}^{1,p} for p > n − 1.     

On the rank of 3×3×3-tensors

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U???V???W is the minimum dimension of a subspace of U???V???W containing τ and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.