955 research outputs found

### Clear and Compress: Computing Persistent Homology in Chunks

We present a parallelizable algorithm for computing the persistent homology
of a filtered chain complex. Our approach differs from the commonly used
reduction algorithm by first computing persistence pairs within local chunks,
then simplifying the unpaired columns, and finally applying standard reduction
on the simplified matrix. The approach generalizes a technique by G\"unther et
al., which uses discrete Morse Theory to compute persistence; we derive the
same worst-case complexity bound in a more general context. The algorithm
employs several practical optimization techniques which are of independent
interest. Our sequential implementation of the algorithm is competitive with
state-of-the-art methods, and we improve the performance through parallelized
computation.Comment: This result was presented at TopoInVis 2013
(http://www.sci.utah.edu/topoinvis13.html

### Finding Pairwise Intersections Inside a Query Range

We study the following problem: preprocess a set O of objects into a data
structure that allows us to efficiently report all pairs of objects from O that
intersect inside an axis-aligned query range Q. We present data structures of
size $O(n({\rm polylog} n))$ and with query time $O((k+1)({\rm polylog} n))$
time, where k is the number of reported pairs, for two classes of objects in
the plane: axis-aligned rectangles and objects with small union complexity. For
the 3-dimensional case where the objects and the query range are axis-aligned
boxes in R^3, we present a data structures of size $O(n\sqrt{n}({\rm polylog}
n))$ and query time $O((\sqrt{n}+k)({\rm polylog} n))$. When the objects and
query are fat, we obtain $O((k+1)({\rm polylog} n))$ query time using $O(n({\rm
polylog} n))$ storage

### Computing and reducing slope complexes

In this paper we provide a new characterization of cell de-
composition (called slope complex) of a given 2-dimensional continuous
surface. Each patch (cell) in the decomposition must satisfy that there
exists a monotonic path for any two points in the cell. We prove that any
triangulation of such surface is a slope complex and explain how to obtain
new slope complexes with a smaller number of slope regions decomposing
the surface. We give the minimal number of slope regions by counting
certain bounding edges of a triangulation of the surface obtained from
its critical points.Ministerio de EconomÃa y Competitividad MTM2015-67072-

### Semi-dynamic connectivity in the plane

Motivated by a path planning problem we consider the following procedure.
Assume that we have two points $s$ and $t$ in the plane and take
$\mathcal{K}=\emptyset$. At each step we add to $\mathcal{K}$ a compact convex
set that does not contain $s$ nor $t$. The procedure terminates when the sets
in $\mathcal{K}$ separate $s$ and $t$. We show how to add one set to
$\mathcal{K}$ in $O(1+k\alpha(n))$ amortized time plus the time needed to find
all sets of $\mathcal{K}$ intersecting the newly added set, where $n$ is the
cardinality of $\mathcal{K}$, $k$ is the number of sets in $\mathcal{K}$
intersecting the newly added set, and $\alpha(\cdot)$ is the inverse of the
Ackermann function

### Mind the Gap: A Study in Global Development through Persistent Homology

The Gapminder project set out to use statistics to dispel simplistic notions
about global development. In the same spirit, we use persistent homology, a
technique from computational algebraic topology, to explore the relationship
between country development and geography. For each country, four indicators,
gross domestic product per capita; average life expectancy; infant mortality;
and gross national income per capita, were used to quantify the development.
Two analyses were performed. The first considers clusters of the countries
based on these indicators, and the second uncovers cycles in the data when
combined with geographic border structure. Our analysis is a multi-scale
approach that reveals similarities and connections among countries at a variety
of levels. We discover localized development patterns that are invisible in
standard statistical methods

### Topological characteristics of oil and gas reservoirs and their applications

We demonstrate applications of topological characteristics of oil and gas
reservoirs considered as three-dimensional bodies to geological modeling.Comment: 12 page

### Categorification of persistent homology

We redevelop persistent homology (topological persistence) from a categorical
point of view. The main objects of study are diagrams, indexed by the poset of
real numbers, in some target category. The set of such diagrams has an
interleaving distance, which we show generalizes the previously-studied
bottleneck distance. To illustrate the utility of this approach, we greatly
generalize previous stability results for persistence, extended persistence,
and kernel, image and cokernel persistence. We give a natural construction of a
category of interleavings of these diagrams, and show that if the target
category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational
Geometr

### A Bichromatic Incidence Bound and an Application

We prove a new, tight upper bound on the number of incidences between points
and hyperplanes in Euclidean d-space. Given n points, of which k are colored
red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between
the k red points and m hyperplanes spanned by all n points provided that m =
\Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal
and Aronov.
We use this incidence bound to prove that a set of n points, no more than n-k
of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also
provide an infinite family of counterexamples to a conjecture of Purdy's on the
number of hyperplanes spanned by a set of points in dimensions higher than 3,
and present new conjectures not subject to the counterexample.Comment: 12 page

### On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs

We show that the maximum cardinality of an anti-chain composed of
intersections of a given set of n points in the plane with half-planes is close
to quadratic in n. We approach this problem by establishing the equivalence
with the problem of the maximum monotone path in an arrangement of n lines. For
a related problem on antichains in families of convex pseudo-discs we can
establish the precise asymptotic bound: it is quadratic in n. The sets in such
a family are characterized as intersections of a given set of n points with
convex sets, such that the difference between the convex hulls of any two sets
is nonempty and connected.Comment: 10 pages, 3 figures. revised version correctly attributes the idea of
Section 3 to Tverberg; and replaced k-sets by "linearly separable sets" in
the paper and the title. Accepted for publication in Israel Journal of
Mathematic

### Computation of protein geometry and its applications: Packing and function prediction

This chapter discusses geometric models of biomolecules and geometric
constructs, including the union of ball model, the weigthed Voronoi diagram,
the weighted Delaunay triangulation, and the alpha shapes. These geometric
constructs enable fast and analytical computaton of shapes of biomoleculres
(including features such as voids and pockets) and metric properties (such as
area and volume). The algorithms of Delaunay triangulation, computation of
voids and pockets, as well volume/area computation are also described. In
addition, applications in packing analysis of protein structures and protein
function prediction are also discussed.Comment: 32 pages, 9 figure

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