105,705 research outputs found
The monotonicity of f-vectors of random polytopes
Let K be a compact convex body in Rd, let Kn be the convex hull of n points
chosen uniformly and independently in K, and let fi(Kn) denote the number of
i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is
increasing in n. In dimension d>=3 we prove that if lim(E((f[d
-1](Kn))/(An^c)->1 when n->infinity for some constants A and c > 0 then the
function E(f[d-1](Kn)) is increasing for n large enough. In particular, the
number of facets of the convex hull of n random points distributed uniformly
and independently in a smooth compact convex body is asymptotically increasing.
Our proof relies on a random sampling argument
Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces
This paper transfers a randomized algorithm, originally used in geometric
optimization, to computational problems in commutative algebra. We show that
Clarkson's sampling algorithm can be applied to two problems in computational
algebra: solving large-scale polynomial systems and finding small generating
sets of graded ideals. The cornerstone of our work is showing that the theory
of violator spaces of G\"artner et al.\ applies to polynomial ideal problems.
To show this, one utilizes a Helly-type result for algebraic varieties. The
resulting algorithms have expected runtime linear in the number of input
polynomials, making the ideas interesting for handling systems with very large
numbers of polynomials, but whose rank in the vector space of polynomials is
small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange
Planar Visibility: Testing and Counting
In this paper we consider query versions of visibility testing and visibility
counting. Let be a set of disjoint line segments in and let
be an element of . Visibility testing is to preprocess so that we can
quickly determine if is visible from a query point . Visibility counting
involves preprocessing so that one can quickly estimate the number of
segments in visible from a query point .
We present several data structures for the two query problems. The structures
build upon a result by O'Rourke and Suri (1984) who showed that the subset,
, of that is weakly visible from a segment can be
represented as the union of a set, , of triangles, even though
the complexity of can be . We define a variant of their
covering, give efficient output-sensitive algorithms for computing it, and
prove additional properties needed to obtain approximation bounds. Some of our
bounds rely on a new combinatorial result that relates the number of segments
of visible from a point to the number of triangles in that contain .Comment: 22 page
First-principles molecular structure search with a genetic algorithm
The identification of low-energy conformers for a given molecule is a
fundamental problem in computational chemistry and cheminformatics. We assess
here a conformer search that employs a genetic algorithm for sampling the
low-energy segment of the conformation space of molecules. The algorithm is
designed to work with first-principles methods, facilitated by the
incorporation of local optimization and blacklisting conformers to prevent
repeated evaluations of very similar solutions. The aim of the search is not
only to find the global minimum, but to predict all conformers within an energy
window above the global minimum. The performance of the search strategy is: (i)
evaluated for a reference data set extracted from a database with amino acid
dipeptide conformers obtained by an extensive combined force field and
first-principles search and (ii) compared to the performance of a systematic
search and a random conformer generator for the example of a drug-like ligand
with 43 atoms, 8 rotatable bonds and 1 cis/trans bond
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