3,765 research outputs found
Geometry of iteration stable tessellations: Connection with Poisson hyperplanes
Since the seminal work by Nagel and Weiss, the iteration stable (STIT)
tessellations have attracted considerable interest in stochastic geometry as a
natural and flexible, yet analytically tractable model for hierarchical spatial
cell-splitting and crack-formation processes. We provide in this paper a
fundamental link between typical characteristics of STIT tessellations and
those of suitable mixtures of Poisson hyperplane tessellations using martingale
techniques and general theory of piecewise deterministic Markov processes
(PDMPs). As applications, new mean values and new distributional results for
the STIT model are obtained.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ424 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin
note: text overlap with arXiv:1001.099
Prototypes, Poles, and Topological Tessellations of Conceptual Spaces
Abstract. The aim of this paper is to present a topological method for constructing
discretizations (tessellations) of conceptual spaces. The method works for a class of
topological spaces that the Russian mathematician Pavel Alexandroff defined more than
80 years ago. Alexandroff spaces, as they are called today, have many interesting
properties that distinguish them from other topological spaces. In particular, they exhibit
a 1-1 correspondence between their specialization orders and their topological structures.
Recently, a special type of Alexandroff spaces was used by Ian Rumfitt to elucidate the
logic of vague concepts in a new way. According to his approach, conceptual spaces such
as the color spectrum give rise to classical systems of concepts that have the structure
of atomic Boolean algebras. More precisely, concepts are represented as regular open
regions of an underlying conceptual space endowed with a topological structure.
Something is subsumed under a concept iff it is represented by an element of the
conceptual space that is maximally close to the prototypical element p that defines that
concept. This topological representation of concepts comes along with a representation
of the familiar logical connectives of Aristotelian syllogistics in terms of natural settheoretical
operations that characterize regular open interpretations of classical Boolean
propositional logic.
In the last 20 years, conceptual spaces have become a popular tool of dealing with a
variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics
and philosophy, mainly due to the work of Peter Gärdenfors and his collaborators. By using
prototypes and metrics of similarity spaces, one obtains geometrical discretizations of
conceptual spaces by so-called Voronoi tessellations. These tessellations are extensionally
equivalent to topological tessellations that can be constructed for Alexandroff spaces.
Thereby, Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an
approach that works for a more general class of spaces, namely, for weakly scattered
Alexandroff spaces. This class of spaces provides a convenient framework for conceptual
spaces as used in epistemology and related disciplines in general. Alexandroff spaces are
useful for elucidating problems related to the logic of vague concepts, in particular they
offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the
logic of clearness (Bobzien) that overcomes certain problems of the concept of higher2
order vagueness. Moreover, these spaces help find a natural place for classical syllogistics
in the framework of conceptual spaces. The crucial role of order theory for Alexandroff
spaces can be used to refine the all-or-nothing distinction between prototypical and nonprototypical
stimuli in favor of a more fine-grained gradual distinction between more-orless
prototypical elements of conceptual spaces. The greater conceptual flexibility of the
topological approach helps avoid some inherent inadequacies of the geometrical approach,
for instance, the so-called “thickness problem” (Douven et al.) and problems of selecting
a unique metric for similarity spaces. Finally, it is shown that only the Alexandroff account can deal with an issue that is gaining more and more importance for the theory of conceptual spaces, namely, the role that digital conceptual spaces play in the area of artificial intelligence, computer science and related disciplines.
Keywords: Conceptual Spaces, Polar Spaces, Alexandroff Spaces, Prototypes, Topological Tessellations, Voronoi Tessellations, Digital Topology
Continuum Line-of-Sight Percolation on Poisson-Voronoi Tessellations
In this work, we study a new model for continuum line-of-sight percolation in
a random environment driven by the Poisson-Voronoi tessellation in the
-dimensional Euclidean space. The edges (one-dimensional facets, or simply
1-facets) of this tessellation are the support of a Cox point process, while
the vertices (zero-dimensional facets or simply 0-facets) are the support of a
Bernoulli point process. Taking the superposition of these two processes,
two points of are linked by an edge if and only if they are sufficiently
close and located on the same edge (1-facet) of the supporting tessellation. We
study the percolation of the random graph arising from this construction and
prove that a 0-1 law, a subcritical phase as well as a supercritical phase
exist under general assumptions. Our proofs are based on a coarse-graining
argument with some notion of stabilization and asymptotic essential
connectedness to investigate continuum percolation for Cox point processes. We
also give numerical estimates of the critical parameters of the model in the
planar case, where our model is intended to represent telecommunications
networks in a random environment with obstructive conditions for signal
propagation.Comment: 30 pages, 4 figures. Accepted for publication in Advances in Applied
Probabilit
Tessellations of homogeneous spaces of classical groups of real rank two
Let H be a closed, connected subgroup of a connected, simple Lie group G with
finite center. The homogeneous space G/H has a "tessellation" if there is a
discrete subgroup D of G, such that D acts properly discontinuously on G/H, and
the double-coset space D\G/H is compact. Note that if either H or G/H is
compact, then G/H has a tessellation; these are the obvious examples.
It is not difficult to see that if G has real rank one, then only the obvious
homogeneous spaces have tessellations. Thus, the first interesting case is when
G has real rank two. In particular, R.Kulkarni and T.Kobayashi constructed
examples that are not obvious when G = SO(2,2n) or SU(2,2n). H.Oh and D.Witte
constructed additional examples in both of these cases, and obtained a complete
classification when G = SO(2,2n). We simplify the work of Oh-Witte, and extend
it to obtain a complete classification when G = SU(2,2n). This includes the
construction of another family of examples.
The main results are obtained from methods of Y.Benoist and T.Kobayashi: we
fix a Cartan decomposition G = KAK, and study the intersection of KHK with A.
Our exposition generally assumes only the standard theory of connected Lie
groups, although basic properties of real algebraic groups are sometimes also
employed; the specialized techniques that we use are developed from a fairly
elementary level.Comment: 74 pages, 7 figure
Honeycomb tessellations and canonical bases for permutohedral blades
This paper studies two families of piecewise constant functions which are
determined by the -skeleta of collections of honeycomb tessellations of
with standard permutohedra. The union of the codimension
cones obtained by extending the facets which are incident to a vertex of such a
tessellation is called a blade. We prove ring-theoretically that such a
honeycomb, with 1-skeleton built from a cyclic sequence of segments in the root
directions , decomposes locally as a Minkowski sum of
isometrically embedded components of hexagonal honeycombs: tripods and
one-dimensional subspaces. For each triangulation of a cyclically oriented
polygon there exists such a factorization. This consequently gives resolution
to an issue proposed and developed by A. Ocneanu, to find a structure theory
for an object he discovered during his investigations into higher Lie theories:
permutohedral blades. We introduce a certain canonical basis for a vector space
spanned by piecewise constant functions of blades which is compatible with
various quotient spaces appearing in algebra, topology and scattering
amplitudes. Various connections to scattering amplitudes are discussed, giving
new geometric interpretations for certain combinatorial identities for one-loop
Parke-Taylor factors. We give a closed formula for the graded dimension of the
canonical blade basis. We conjecture that the coefficients of the generating
function numerators for the diagonals are symmetric and unimodal.Comment: Added references; new section on configuration space
Shape-Driven Nested Markov Tessellations
A new and rather broad class of stationary (i.e. stochastically translation
invariant) random tessellations of the -dimensional Euclidean space is
introduced, which are called shape-driven nested Markov tessellations. Locally,
these tessellations are constructed by means of a spatio-temporal random
recursive split dynamics governed by a family of Markovian split kernel,
generalizing thereby the -- by now classical -- construction of iteration
stable random tessellations. By providing an explicit global construction of
the tessellations, it is shown that under suitable assumptions on the split
kernels (shape-driven), there exists a unique time-consistent whole-space
tessellation-valued Markov process of stationary random tessellations
compatible with the given split kernels. Beside the existence and uniqueness
result, the typical cell and some aspects of the first-order geometry of these
tessellations are in the focus of our discussion
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