9,819 research outputs found
Convergence between Categorical Representations of Reeb Space and Mapper
The Reeb space, which generalizes the notion of a Reeb graph, is one of the
few tools in topological data analysis and visualization suitable for the study
of multivariate scientific datasets. First introduced by Edelsbrunner et al.,
it compresses the components of the level sets of a multivariate mapping and
obtains a summary representation of their relationships. A related construction
called mapper, and a special case of the mapper construction called the Joint
Contour Net have been shown to be effective in visual analytics. Mapper and JCN
are intuitively regarded as discrete approximations of the Reeb space, however
without formal proofs or approximation guarantees. An open question has been
proposed by Dey et al. as to whether the mapper construction converges to the
Reeb space in the limit.
In this paper, we are interested in developing the theoretical understanding
of the relationship between the Reeb space and its discrete approximations to
support its use in practical data analysis. Using tools from category theory,
we formally prove the convergence between the Reeb space and mapper in terms of
an interleaving distance between their categorical representations. Given a
sequence of refined discretizations, we prove that these approximations
converge to the Reeb space in the interleaving distance; this also helps to
quantify the approximation quality of the discretization at a fixed resolution
Inverse problems for linear hyperbolic equations using mixed formulations
We introduce in this document a direct method allowing to solve numerically
inverse type problems for linear hyperbolic equations. We first consider the
reconstruction of the full solution of the wave equation posed in - a bounded subset of - from a partial
distributed observation. We employ a least-squares technique and minimize the
-norm of the distance from the observation to any solution. Taking the
hyperbolic equation as the main constraint of the problem, the optimality
conditions are reduced to a mixed formulation involving both the state to
reconstruct and a Lagrange multiplier. Under usual geometric optic conditions,
we show the well-posedness of this mixed formulation (in particular the inf-sup
condition) and then introduce a numerical approximation based on space-time
finite elements discretization. We prove the strong convergence of the
approximation and then discussed several examples for and . The
problem of the reconstruction of both the state and the source term is also
addressed
International Outsourcing and Individual Job Separations
This paper studies the effects of international outsourcing on individual transitions out of jobs in the Danish manufacturing sector for the period 1992-2001. Estimation of a single risk duration model, where no distinction is made between different types of transitions out of the job, shows that outsourcing has a clear significant positive effect on the job separation rate, but the effect corresponds to a limited number of lost jobs. A competing risks duration model that distinguishes between job-to-job and job-to-unemployment transitions is also estimated. Outsourcing is found to increase the unemployment risk of workers and in particular low-skilled workers, but again the quantitative impact is not dramatic. Outsourcing also increases the job change hazard rate and mostly so for high-skilled workers.international outsourcing; job separations; competing risks duration model
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