726 research outputs found
Tropical polar cones, hypergraph transversals, and mean payoff games
We discuss the tropical analogues of several basic questions of convex
duality. In particular, the polar of a tropical polyhedral cone represents the
set of linear inequalities that its elements satisfy. We characterize the
extreme rays of the polar in terms of certain minimal set covers which may be
thought of as weighted generalizations of minimal transversals in hypergraphs.
We also give a tropical analogue of Farkas lemma, which allows one to check
whether a linear inequality is implied by a finite family of linear
inequalities. Here, the certificate is a strategy of a mean payoff game. We
discuss examples, showing that the number of extreme rays of the polar of the
tropical cyclic polyhedral cone is polynomially bounded, and that there is no
unique minimal system of inequalities defining a given tropical polyhedral
cone.Comment: 27 pages, 6 figures, revised versio
Fuzzy Toric Geometries
We describe a construction of fuzzy spaces which approximate projective toric
varieties. The construction uses the canonical embedding of such varieties into
a complex projective space: The algebra of fuzzy functions on a toric variety
is obtained by a restriction of the fuzzy algebra of functions on the complex
projective space appearing in the embedding. We give several explicit examples
for this construction; in particular, we present fuzzy weighted projective
spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction
is actually suited for arbitrary subvarieties of complex projective spaces, one
can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on
fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number
of available fuzzy spaces significantly, we show that the fuzzification of a
projective toric variety amounts to a quantization of its toric base.Comment: 1+25 pages, extended version, to appear in JHE
Spectral Generalized Multi-Dimensional Scaling
Multidimensional scaling (MDS) is a family of methods that embed a given set
of points into a simple, usually flat, domain. The points are assumed to be
sampled from some metric space, and the mapping attempts to preserve the
distances between each pair of points in the set. Distances in the target space
can be computed analytically in this setting. Generalized MDS is an extension
that allows mapping one metric space into another, that is, multidimensional
scaling into target spaces in which distances are evaluated numerically rather
than analytically. Here, we propose an efficient approach for computing such
mappings between surfaces based on their natural spectral decomposition, where
the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS
procedure enables efficient embedding by implicitly incorporating smoothness of
the mapping into the problem, thereby substantially reducing the complexity
involved in its solution while practically overcoming its non-convex nature.
The method is compared to existing techniques that compute dense correspondence
between shapes. Numerical experiments of the proposed method demonstrate its
efficiency and accuracy compared to state-of-the-art approaches
Knowledge Capture in CMM Inspection Planning: Barriers and Challenges
Coordinate Measuring Machines (CMM) have been widely used as a means of evaluating product quality and controlling quality manufacturing processes. Many techniques have been developed to facilitate the generation of CMM measurement plans. However, there are major gaps in the understanding of planning such strategies. This significant lack of explicitly available knowledge on how experts prepare plans and carry out measurements slows down the planning process, leading to the repetitive reinvention of new plans while preventing the automation or even semi-automation of the process. The objectives of this paper are twofold: (i) to provide a review of the existing inspection planning systems and discuss the barriers and challenges, especially from the aspect of knowledge capture and formalization; and (ii) to propose and demonstrate a novel digital engineering mixed reality paradigm which has the potential to facilitate the rapid capture of implicit inspection knowledge and explicitly represent this in a formalized way. An outline and the results of the development of an early stage prototype - which will form the foundation of a more complex system to address the aforementioned technological challenges identified in the literature survey - will be given
Learning Membership Functions in a Function-Based Object Recognition System
Functionality-based recognition systems recognize objects at the category
level by reasoning about how well the objects support the expected function.
Such systems naturally associate a ``measure of goodness'' or ``membership
value'' with a recognized object. This measure of goodness is the result of
combining individual measures, or membership values, from potentially many
primitive evaluations of different properties of the object's shape. A
membership function is used to compute the membership value when evaluating a
primitive of a particular physical property of an object. In previous versions
of a recognition system known as Gruff, the membership function for each of the
primitive evaluations was hand-crafted by the system designer. In this paper,
we provide a learning component for the Gruff system, called Omlet, that
automatically learns membership functions given a set of example objects
labeled with their desired category measure. The learning algorithm is
generally applicable to any problem in which low-level membership values are
combined through an and-or tree structure to give a final overall membership
value.Comment: See http://www.jair.org/ for any accompanying file
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