14,940 research outputs found
Extreme-value statistics from Lagrangian convex hull analysis for homogeneous turbulent Boussinesq convection and MHD convection
We investigate the utility of the convex hull of many Lagrangian tracers to
analyze transport properties of turbulent flows with different anisotropy. In
direct numerical simulations of statistically homogeneous and stationary
Navier-Stokes turbulence, neutral fluid Boussinesq convection, and MHD
Boussinesq convection a comparison with Lagrangian pair dispersion shows that
convex hull statistics capture the asymptotic dispersive behavior of a large
group of passive tracer particles. Moreover, convex hull analysis provides
additional information on the sub-ensemble of tracers that on average disperse
most efficiently in the form of extreme value statistics and flow anisotropy
via the geometric properties of the convex hulls. We use the convex hull
surface geometry to examine the anisotropy that occurs in turbulent convection.
Applying extreme value theory, we show that the maximal square extensions of
convex hull vertices are well described by a classic extreme value
distribution, the Gumbel distribution. During turbulent convection,
intermittent convective plumes grow and accelerate the dispersion of Lagrangian
tracers. Convex hull analysis yields information that supplements standard
Lagrangian analysis of coherent turbulent structures and their influence on the
global statistics of the flow.Comment: 18 pages, 10 figures, preprin
Convex Hull of Planar H-Polyhedra
Suppose are planar (convex) H-polyhedra, that is, $A_i \in
\mathbb{R}^{n_i \times 2}$ and $\vec{c}_i \in \mathbb{R}^{n_i}$. Let $P_i =
\{\vec{x} \in \mathbb{R}^2 \mid A_i\vec{x} \leq \vec{c}_i \}$ and $n = n_1 +
n_2$. We present an $O(n \log n)$ algorithm for calculating an H-polyhedron
with the smallest such that
p-convexity, p-plurisubharmonicity and the Levi problem
Three results in p-convex geometry are established. First is the analogue of
the Levi problem in several complex variables, namely: local p-convexity
implies global p-convexity. The second asserts that the support of a minimal
p-dimensional current is contained in the p-hull of the boundary union with the
"core" of the space. Lastly, the exteme rays in the convex cone of p-positive
matrices are characterized. This is a basic result with many applications
Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing
Motivated by the desire to cope with data imprecision, we study methods for
taking advantage of preliminary information about point sets in order to speed
up the computation of certain structures associated with them.
In particular, we study the following problem: given a set L of n lines in
the plane, we wish to preprocess L such that later, upon receiving a set P of n
points, each of which lies on a distinct line of L, we can construct the convex
hull of P efficiently. We show that in quadratic time and space it is possible
to construct a data structure on L that enables us to compute the convex hull
of any such point set P in O(n alpha(n) log* n) expected time. If we further
assume that the points are "oblivious" with respect to the data structure, the
running time improves to O(n alpha(n)). The analysis applies almost verbatim
when L is a set of line-segments, and yields similar asymptotic bounds. We
present several extensions, including a trade-off between space and query time
and an output-sensitive algorithm. We also study the "dual problem" where we
show how to efficiently compute the (<= k)-level of n lines in the plane, each
of which lies on a distinct point (given in advance).
We complement our results by Omega(n log n) lower bounds under the algebraic
computation tree model for several related problems, including sorting a set of
points (according to, say, their x-order), each of which lies on a given line
known in advance. Therefore, the convex hull problem under our setting is
easier than sorting, contrary to the "standard" convex hull and sorting
problems, in which the two problems require Theta(n log n) steps in the worst
case (under the algebraic computation tree model).Comment: 26 pages, 5 figures, 1 appendix; a preliminary version appeared at
SoCG 201
Archetypal analysis of galaxy spectra
Archetypal analysis represents each individual member of a set of data
vectors as a mixture (a constrained linear combination) of the pure types or
archetypes of the data set. The archetypes are themselves required to be
mixtures of the data vectors. Archetypal analysis may be particularly useful in
analysing data sets comprising galaxy spectra, since each spectrum is,
presumably, a superposition of the emission from the various stellar
populations, nebular emissions and nuclear activity making up that galaxy, and
each of these emission sources corresponds to a potential archetype of the
entire data set. We demonstrate archetypal analysis using sets of composite
synthetic galaxy spectra, showing that the method promises to be an effective
and efficient way to classify spectra. We show that archetypal analysis is
robust in the presence of various types of noise.Comment: 6 pages, 5 figures, 1 style-file. Accepted for publication by MNRA
On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape
We study the lengths of curves passing through a fixed number of points on
the boundary of a convex shape in the plane. We show that for any convex shape
, there exist four points on the boundary of such that the length of any
curve passing through these points is at least half of the perimeter of . It
is also shown that the same statement does not remain valid with the additional
constraint that the points are extreme points of . Moreover, the factor
cannot be achieved with any fixed number of extreme points. We
conclude the paper with few other inequalities related to the perimeter of a
convex shape.Comment: 7 pages, 8 figure
Random Convex Hulls and Extreme Value Statistics
In this paper we study the statistical properties of convex hulls of
random points in a plane chosen according to a given distribution. The points
may be chosen independently or they may be correlated. After a non-exhaustive
survey of the somewhat sporadic literature and diverse methods used in the
random convex hull problem, we present a unifying approach, based on the notion
of support function of a closed curve and the associated Cauchy's formulae,
that allows us to compute exactly the mean perimeter and the mean area enclosed
by the convex polygon both in case of independent as well as correlated points.
Our method demonstrates a beautiful link between the random convex hull problem
and the subject of extreme value statistics. As an example of correlated
points, we study here in detail the case when the points represent the vertices
of independent random walks. In the continuum time limit this reduces to
independent planar Brownian trajectories for which we compute exactly, for
all , the mean perimeter and the mean area of their global convex hull. Our
results have relevant applications in ecology in estimating the home range of a
herd of animals. Some of these results were announced recently in a short
communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special
issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting
On the forces that cable webs under tension can support and how to design cable webs to channel stresses
In many applications of Structural Engineering the following question arises:
given a set of forces applied at
prescribed points , under what
constraints on the forces does there exist a truss structure (or wire web) with
all elements under tension that supports these forces? Here we provide answer
to such a question for any configuration of the terminal points
in the two- and
three-dimensional case. Specifically, the existence of a web is guaranteed by a
necessary and sufficient condition on the loading which corresponds to a finite
dimensional linear programming problem. In two-dimensions we show that any such
web can be replaced by one in which there are at most elementary loops,
where elementary means the loop cannot be subdivided into subloops, and where
is the number of forces
applied at points strictly within the convex hull of
. In three-dimensions we show
that, by slightly perturbing ,
there exists a uniloadable web supporting this loading. Uniloadable means it
supports this loading and all positive multiples of it, but not any other
loading. Uniloadable webs provide a mechanism for distributing stress in
desired ways.Comment: 18 pages, 8 figure
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