6,126 research outputs found
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
The P-Norm Push: A Simple Convex Ranking Algorithm that Concentrates at the Top of the List
We are interested in supervised ranking algorithms that perform especially well near the top of the
ranked list, and are only required to perform sufficiently well on the rest of the list. In this work,
we provide a general form of convex objective that gives high-scoring examples more importance.
This “push” near the top of the list can be chosen arbitrarily large or small, based on the preference
of the user. We choose â„“p-norms to provide a specific type of push; if the user sets p larger, the
objective concentrates harder on the top of the list. We derive a generalization bound based on
the p-norm objective, working around the natural asymmetry of the problem. We then derive a
boosting-style algorithm for the problem of ranking with a push at the top. The usefulness of the
algorithm is illustrated through experiments on repository data. We prove that the minimizer of the
algorithm’s objective is unique in a specific sense. Furthermore, we illustrate how our objective is
related to quality measurements for information retrieval
Uncertainty Relations for Angular Momentum
In this work we study various notions of uncertainty for angular momentum in
the spin-s representation of SU(2). We characterize the "uncertainty regions''
given by all vectors, whose components are specified by the variances of the
three angular momentum components. A basic feature of this set is a lower bound
for the sum of the three variances. We give a method for obtaining optimal
lower bounds for uncertainty regions for general operator triples, and evaluate
these for small s. Further lower bounds are derived by generalizing the
technique by which Robertson obtained his state-dependent lower bound. These
are optimal for large s, since they are saturated by states taken from the
Holstein-Primakoff approximation. We show that, for all s, all variances are
consistent with the so-called vector model, i.e., they can also be realized by
a classical probability measure on a sphere of radius sqrt(s(s+1)). Entropic
uncertainty relations can be discussed similarly, but are minimized by
different states than those minimizing the variances for small s. For large s
the Maassen-Uffink bound becomes sharp and we explicitly describe the
extremalizing states. Measurement uncertainty, as recently discussed by Busch,
Lahti and Werner for position and momentum, is introduced and a generalized
observable (POVM) which minimizes the worst case measurement uncertainty of all
angular momentum components is explicitly determined, along with the minimal
uncertainty. The output vectors for the optimal measurement all have the same
length r(s), where r(s)/s goes to 1 as s tends to infinity.Comment: 30 pages, 22 figures, 1 cut-out paper model, video abstract available
on https://youtu.be/h01pHekcwF
Decomposition of Geometric Set Systems and Graphs
We study two decomposition problems in combinatorial geometry. The first part
deals with the decomposition of multiple coverings of the plane. We say that a
planar set is cover-decomposable if there is a constant m such that any m-fold
covering of the plane with its translates is decomposable into two disjoint
coverings of the whole plane. Pach conjectured that every convex set is
cover-decomposable. We verify his conjecture for polygons. Moreover, if m is
large enough, we prove that any m-fold covering can even be decomposed into k
coverings. Then we show that the situation is exactly the opposite in 3
dimensions, for any polyhedron and any we construct an m-fold covering of
the space that is not decomposable. We also give constructions that show that
concave polygons are usually not cover-decomposable. We start the first part
with a detailed survey of all results on the cover-decomposability of polygons.
The second part investigates another geometric partition problem, related to
planar representation of graphs. The slope number of a graph G is the smallest
number s with the property that G has a straight-line drawing with edges of at
most s distinct slopes and with no bends. We examine the slope number of
bounded degree graphs. Our main results are that if the maximum degree is at
least 5, then the slope number tends to infinity as the number of vertices
grows but every graph with maximum degree at most 3 can be embedded with only
five slopes. We also prove that such an embedding exists for the related notion
called slope parameter. Finally, we study the planar slope number, defined only
for planar graphs as the smallest number s with the property that the graph has
a straight-line drawing in the plane without any crossings such that the edges
are segments of only s distinct slopes. We show that the planar slope number of
planar graphs with bounded degree is bounded.Comment: This is my PhD thesi
Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices
Approximating the set of reachable states of a dynamical system is an
algorithmic yet mathematically rigorous way to reason about its safety.
Although progress has been made in the development of efficient algorithms for
affine dynamical systems, available algorithms still lack scalability to ensure
their wide adoption in the industrial setting. While modern linear algebra
packages are efficient for matrices with tens of thousands of dimensions,
set-based image computations are limited to a few hundred. We propose to
decompose reach set computations such that set operations are performed in low
dimensions, while matrix operations like exponentiation are carried out in the
full dimension. Our method is applicable both in dense- and discrete-time
settings. For a set of standard benchmarks, it shows a speed-up of up to two
orders of magnitude compared to the respective state-of-the art tools, with
only modest losses in accuracy. For the dense-time case, we show an experiment
with more than 10.000 variables, roughly two orders of magnitude higher than
possible with previous approaches
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