13,905 research outputs found
Similarity measures for mid-surface quality evaluation
Mid-surface models are widely used in engineering analysis to simplify the analysis of thin-walled parts, but it can be difficult to ensure that the mid-surface model is representative of the solid part from which it was generated. This paper proposes two similarity measures that can be used to evaluate the quality of a mid-surface model by comparing it to a solid model of the same part. Two similarity measures are proposed; firstly a geometric similarity evaluation technique based on the Hausdorff distance and secondly a topological similarity evaluation method which uses geometry graph attributes as the basis for comparison. Both measures are able to provide local and global similarity evaluation for the models. The proposed methods have been implemented in a software demonstrator and tested on a selection of representative models. They have been found to be effective for identifying geometric and topological errors in mid-surface models and are applicable to a wide range of practical thin-walled designs
Line transversals to disjoint balls
We prove that the set of directions of lines intersecting three disjoint
balls in in a given order is a strictly convex subset of . We then
generalize this result to disjoint balls in . As a consequence, we can
improve upon several old and new results on line transversals to disjoint balls
in arbitrary dimension, such as bounds on the number of connected components
and Helly-type theorems.Comment: 21 pages, includes figure
Polynomial Norms
In this paper, we study polynomial norms, i.e. norms that are the
root of a degree- homogeneous polynomial . We first show
that a necessary and sufficient condition for to be a norm is for
to be strictly convex, or equivalently, convex and positive definite. Though
not all norms come from roots of polynomials, we prove that any
norm can be approximated arbitrarily well by a polynomial norm. We then
investigate the computational problem of testing whether a form gives a
polynomial norm. We show that this problem is strongly NP-hard already when the
degree of the form is 4, but can always be answered by testing feasibility of a
semidefinite program (of possibly large size). We further study the problem of
optimizing over the set of polynomial norms using semidefinite programming. To
do this, we introduce the notion of r-sos-convexity and extend a result of
Reznick on sum of squares representation of positive definite forms to positive
definite biforms. We conclude with some applications of polynomial norms to
statistics and dynamical systems
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