On the dynamic asymptotic dimension of \'etale groupoids

We investigate the dynamic asymptotic dimension for \'etale groupoids introduced by Guentner, Willett and Yu. In particular, we establish several permanence properties, including estimates for products and unions of groupoids. We also establish invariance of the dynamic asymptotic dimension under Morita equivalence. In the second part of the article, we consider a canonical coarse structure on an \'etale groupoid and compare the asymptotic dimension of the resulting coarse space with the dynamic asymptotic dimension of the underlying groupoid.Comment: 17 page

Categorical approach to the Baum-Connes conjecture for \ue9tale groupoids

We consider the equivariant Kasparov category associated to an \ue9talegroupoid, and by leveraging its triangulated structure we study its localization atthe “weakly contractible” objects, extending previous work by Meyer and Nest.We prove that the subcategory of weakly contractible objects is complementary to thelocalizing subcategory of projective objects, which are defined in terms of “compactlyinduced” algebras with respect to certain proper subgroupoids related to isotropy.The resulting “strong” Baum–Connes conjecture implies the classical one, and itsformulation clarifies several permanence properties and other functorial statements.We present multiple applications, including consequences for the Universal CoefficientTheorem, a generalized “Going-Down” principle, injectivity results for groupoids thatare amenable at infinity, the Baum-Connes conjecture for group bundles, and a resultabout the invariance of K-groups of twisted groupoid C*-algebras under homotopy oftwists

K-theory and homotopies of twists on ample groupoids

This paper investigates the K-theory of twisted groupoid C*-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum–Connes conjecture with coefficients gives rise to an isomorphism between the K-theory groups of the respective twisted groupoid C*-algebras. The results are also interpreted in an inverse semigroup setting and applied to generalized Renault–Deaconu groupoids and P-graph algebras

Flexible und plattformunabhängige Entwicklung mikrocontrollerbasierter mechatronischer Systeme für Nutzer ohne Vorwissen

Der steigende Bedarf an mikrocontrollerbasierten mechatronischen Systemen ist heute nicht mehr zu übersehen. Zur Deckung des Bedarfs werden gewöhnlich modellbasierte Methoden eingesetzt, welche die Arbeit der Experten beschleunigen. Diese Methoden sind aber selbst für technikbegeisterte Laien oft nicht einsetzbar, da sie ein fundiertes Vorwissen voraussetzen. An dieser Stelle setzt die vorliegende Arbeit an. Sie stellt verschiedene Konzepte vor, mit welchen es möglich ist, die elektronische Hardware und die Software für mikrocontrollerbasierte mechatronische Systeme zu entwickeln, ohne dass hierfür Vorwissen auf den Gebieten der Elektronikentwicklung oder der Informatik benötigt wird. Eine solche Vereinfachung geschah bisher gewöhnlich nur durch die Kapselung hardwarenaher Funktionen in abstrakten Modulen. In der vorliegenden Arbeit wird jedoch ein anderer Weg gewählt. Die Nutzer sollen während der Nutzung ein Grundverständnis für die Funktionsweise mechatronischer Systeme erlangen.In this thesis new concepts for designing development systems for mechatronic systems are introduced. The concepts allow flexible and simple usage, even if the users don’t have prior expert knowledge. For this purpose, approaches are presented, which allow a transparent illustration of the mode of operation of sensors, actuators and used platforms, allowing users to understand related technical topics.In the first part of the thesis, basic knowledge and the state of the art are presented. After this, the concepts of the microsystems development systems “EasyKit” and “EasyKit macht Schule” are described. In these systems microcontrollers are used as platform, because they already contain a high functional integration. Because of this, novice users prefer to use them as platform of choice. The electronic circuits, including the microcontroller, are provided in shape modular hardware blocks. They are programmed graphically with modular software blocks. The approach of programming introduced, uses a combination of the advantages of sequential function charts and synchronous data flow charts which increases the flexibility. Tests showed that even users without prior technical knowledge were able to program the microcontroller with these languages.The EasyKit concept was advanced, to offer increased flexibility and simplicity during programming. Besides, there was the goal to give users without expert knowledge the capability of developing electronics on the circuitry level, which is far more flexible than developing on the modular level. For this purpose, requirements are analyzed and new approaches are presented in the thesis. The most important approach, to make the soft-ware development more flexible, is the introduction of a new additional programming level, which supports graphical and textual programming methods at the same time. For assisting the user during the hardware development on the circuitry level, approaches are presented, which allow modeling most sensors and actuators, by abstracting them to their types of interfaces. Through this, the user can be supported when developing driver circuits to be used between the sensors, actuators and the microcontroller. Besides, this approach allows a comprehensible visualization of the signal behavior and the signal transformation at the interface of the microcontroller. Further approaches to increase the usability during the development phase are also presented in the thesis.The most promising approaches were implemented to a development environment and tested with members of the target audience

A Going-Down principle for ample groupoids and the Baum-Connes conjecture

We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoids which are strongly amenable at infinity. This result in turn is then used to relate the Baum-Connes conjecture for an ample groupoid group bundle which is strongly amenable at infinity to the Baum-Connes conjecture for the fibres.Comment: minor corrections, final version to appear in Adv. Math., 59 page