Function Spaces on Singular Manifolds

It is shown that most of the well-known basic results for Sobolev-Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in $\mathbb{R}^n$, continue to be valid on a wide class of Riemannian manifolds with singularities and boundary, provided suitable weights, which reflect the nature of the singularities, are introduced. These results are of importance for the study of partial differential equations on piece-wise smooth domains.Comment: 37 pages, 1 figure, final version, augmented by additional references; to appear in Math. Nachrichte

Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type

Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gr{\"o}chenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and the first named author that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel spaces of various types of interest as coorbits. We obtain several old and new wavelet characterizations based on precise smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed smoothness, and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable.Comment: to appear in Journal of Functional Analysi

Collective classification for labeling of places and objects in 2D and 3D range data

In this paper, we present an algorithm to identify types of places and objects from 2D and 3D laser range data obtained in indoor environments. Our approach is a combination of a collective classification method based on associative Markov networks together with an instance-based feature extraction using nearest neighbor. Additionally, we show how to select the best features needed to represent the objects and places, reducing the time needed for the learning and inference steps while maintaining high classification rates. Experimental results in real data demonstrate the effectiveness of our approach in indoor environments

Homogeneous Equations of Algebraic Petri Nets

Algebraic Petri nets are a formalism for modeling distributed systems and algorithms, describing control and data flow by combining Petri nets and algebraic specification. One way to specify correctness of an algebraic Petri net model "N" is to specify a linear equation "E" over the places of "N" based on term substitution, and coefficients from an abelian group "G". Then, "E" is valid in "N" iff "E" is valid in each reachable marking of "N". Due to the expressive power of Algebraic Petri nets, validity is generally undecidable. Stable linear equations form a class of linear equations for which validity is decidable. Place invariants yield a well-understood but incomplete characterization of all stable linear equations. In this paper, we provide a complete characterization of stability for the subclass of homogeneous linear equations, by restricting ourselves to the interpretation of terms over the Herbrand structure without considering further equality axioms. Based thereon, we show that stability is decidable for homogeneous linear equations if "G" is a cyclic group

Hardy inequalities in function spaces

summary:Let $\Omega$ be a bounded $C^\infty$ domain in $\Bbb{R}^n$. The paper deals with inequalities of Hardy type related to the function spaces $B^s_{pq}(\Omega)$ and $F^s_{pq}(\Omega)$