Strength of convergence and multiplicities in the spectrum of a C*-dynamical system

We consider separable $C^*$-dynamical systems $(A,G,\alpha)$ for which the induced action of the group $G$ on the spectrum $\hat A$ of the $C^*$-algebra $A$ is free. We study how the representation theory of the associated crossed-product $C^*$-algebra $A\rtimes_\alpha G$ depends on the representation theory of $A$ and the properties of the action of $G$ on $\hat A$. Our main tools involve computations of upper and lower bounds on multiplicity numbers associated to irreducible representations of $A\rtimes_\alpha G$. We apply our techniques to give necessary and sufficient conditions, in terms of $A$ and the action of $G$ on $\hat A$, for $A\rtimes_{\alpha}G$ to be (i) a continuous-trace $C^*$-algebra, (ii) a Fell $C^*$-algebra and (iii) a bounded-trace $C^*$-algebra. When $G$ is amenable, we also give necessary and sufficient conditions for the crossed-product $C^*$-algebra $A\rtimes_{\alpha}G$ to be (iv) a liminal $C^*$-algebra and (v) a Type I $C^*$-algebra. The results in (i), (iii)--(v) extend some earlier special cases in which $A$ was assumed to have the corresponding property.Comment: Publication version, to appear in Proc. London Math. So

Strength of convergence in the orbit space of a transformation group

Let (G, X) be a second-countable transformation group with G acting freely on X. It is shown that measure-theoretic accumulation of the action and topological strength of convergence in the orbit space X/G provide equivalent ways of quantifying the extent of non-properness of the action. These notions are linked via the representation theory of the transformation-group C*-algebra.Comment: 27 page

Strength of convergence in the orbit space of a groupoid

Let G be a second-countable locally-compact Hausdorff groupoid with a Haar system, and let {x_n} be a sequence in the unit space of G. We show that the notions of strength of convergence of {x_n} in the orbit space and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid C*-algebra.Comment: 34 page

Amortising the Cost of Mutation Based Fault Localisation using Statistical Inference

Mutation analysis can effectively capture the dependency between source code and test results. This has been exploited by Mutation Based Fault Localisation (MBFL) techniques. However, MBFL techniques suffer from the need to expend the high cost of mutation analysis after the observation of failures, which may present a challenge for its practical adoption. We introduce SIMFL (Statistical Inference for Mutation-based Fault Localisation), an MBFL technique that allows users to perform the mutation analysis in advance against an earlier version of the system. SIMFL uses mutants as artificial faults and aims to learn the failure patterns among test cases against different locations of mutations. Once a failure is observed, SIMFL requires either almost no or very small additional cost for analysis, depending on the used inference model. An empirical evaluation of SIMFL using 355 faults in Defects4J shows that SIMFL can successfully localise up to 103 faults at the top, and 152 faults within the top five, on par with state-of-the-art alternatives. The cost of mutation analysis can be further reduced by mutation sampling: SIMFL retains over 80% of its localisation accuracy at the top rank when using only 10% of generated mutants, compared to results obtained without sampling

Improving Christofides' Algorithm for the s-t Path TSP

We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP.Comment: 31 pages, 5 figure