45 research outputs found
Strong Shift Equivalence of -correspondences
We define a notion of strong shift equivalence for -correspondences and
show that strong shift equivalent -correspondences have strongly Morita
equivalent Cuntz-Pimsner algebras. Our analysis extends the fact that strong
shift equivalent square matrices with non-negative integer entries give stably
isomorphic Cuntz-Krieger algebras.Comment: 26 pages. Final version to appear in Israel Journal of Mathematic
Exchange Leavitt path algebras and stable rank
We characterize those Leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1, 2 or ∞. Concrete criteria in terms of properties of the underlying graph are given for each case
Twisted k-graph algebras associated to Bratteli diagrams
Given a system of coverings of k-graphs, we show that the cohomology of the
resulting (k+1)-graph is isomorphic to that of any one of the k-graphs in the
system. We then consider Bratteli diagrams of 2-graphs whose twisted
C*-algebras are matrix algebras over noncommutative tori. For such systems we
calculate the ordered K-theory and the gauge-invariant semifinite traces of the
resulting 3-graph C*-algebras. We deduce that every simple C*-algebra of this
form is Morita equivalent to the C*-algebra of a rank-2 Bratteli diagram in the
sense of Pask-Raeburn-R{\o}rdam-Sims.Comment: 28 pages, pictures prepared using tik
Learning to Learn in Collective Adaptive Systems: Mining Design Patterns for Data-driven Reasoning
Engineering collective adaptive systems (CAS) with learning capabilities is a challenging task due to their multidimensional and complex design space. Data-driven approaches for CAS design could introduce new insights enabling system engineers to manage the CAS complexity more cost-effectively at the design-phase. This paper introduces a systematic approach to reason about design choices and patterns of learning-based CAS. Using data from a systematic literature review, reasoning is performed with a novel application of data-driven methodologies such as clustering, multiple correspondence analysis and decision trees. The reasoning based on past experience as well as supporting novel and innovative design choices are demonstrated
The groupoid approach to Leavitt path algebras
When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graph’s boundary path groupoid. This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras