7,526 research outputs found
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Geometry, pregeometry and beyond
This article explores the overall geometric manner in which human beings make
sense of the world around them by means of their physical theories; in
particular, in what are nowadays called pregeometric pictures of Nature. In
these, the pseudo-Riemannian manifold of general relativity is considered a
flawed description of spacetime and it is attempted to replace it by
theoretical constructs of a different character, ontologically prior to it.
However, despite its claims to the contrary, pregeometry is found to
surreptitiously and unavoidably fall prey to the very mode of description it
endeavours to evade, as evidenced in its all-pervading geometric understanding
of the world. The question remains as to the deeper reasons for this human,
geometric predilection--present, as a matter of fact, in all of physics--and as
to whether it might need to be superseded in order to achieve the goals that
frontier theoretical physics sets itself at the dawn of a new century: a
sounder comprehension of the physical meaning of empty spacetime.Comment: 41 pages, Latex. v2: Date added. v3: Main arguments refined,
secondary discussions abridged; expands on the published versio
Discovery and recognition of motion primitives in human activities
We present a novel framework for the automatic discovery and recognition of
motion primitives in videos of human activities. Given the 3D pose of a human
in a video, human motion primitives are discovered by optimizing the `motion
flux', a quantity which captures the motion variation of a group of skeletal
joints. A normalization of the primitives is proposed in order to make them
invariant with respect to a subject anatomical variations and data sampling
rate. The discovered primitives are unknown and unlabeled and are
unsupervisedly collected into classes via a hierarchical non-parametric Bayes
mixture model. Once classes are determined and labeled they are further
analyzed for establishing models for recognizing discovered primitives. Each
primitive model is defined by a set of learned parameters.
Given new video data and given the estimated pose of the subject appearing on
the video, the motion is segmented into primitives, which are recognized with a
probability given according to the parameters of the learned models.
Using our framework we build a publicly available dataset of human motion
primitives, using sequences taken from well-known motion capture datasets. We
expect that our framework, by providing an objective way for discovering and
categorizing human motion, will be a useful tool in numerous research fields
including video analysis, human inspired motion generation, learning by
demonstration, intuitive human-robot interaction, and human behavior analysis
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