6,405 research outputs found
Strength of convergence and multiplicities in the spectrum of a C*-dynamical system
We consider separable -dynamical systems for which the
induced action of the group on the spectrum of the -algebra
is free. We study how the representation theory of the associated
crossed-product -algebra depends on the representation
theory of and the properties of the action of on . Our main
tools involve computations of upper and lower bounds on multiplicity numbers
associated to irreducible representations of . We apply our
techniques to give necessary and sufficient conditions, in terms of and the
action of on , for to be (i) a
continuous-trace -algebra, (ii) a Fell -algebra and (iii) a
bounded-trace -algebra. When is amenable, we also give necessary and
sufficient conditions for the crossed-product -algebra
to be (iv) a liminal -algebra and (v) a Type I
-algebra. The results in (i), (iii)--(v) extend some earlier special cases
in which 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
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