63 research outputs found
The Complexity of Order Type Isomorphism
The order type of a point set in maps each -tuple of points to
its orientation (e.g., clockwise or counterclockwise in ). Two point sets
and have the same order type if there exists a mapping from to
for which every -tuple of and the
corresponding tuple in have the same
orientation. In this paper we investigate the complexity of determining whether
two point sets have the same order type. We provide an algorithm for
this task, thereby improving upon the algorithm
of Goodman and Pollack (1983). The algorithm uses only order type queries and
also works for abstract order types (or acyclic oriented matroids). Our
algorithm is optimal, both in the abstract setting and for realizable points
sets if the algorithm only uses order type queries.Comment: Preliminary version of paper to appear at ACM-SIAM Symposium on
Discrete Algorithms (SODA14
Kernelization of the Subset General Position Problem in Geometry
In this paper, we consider variants of the Geometric Subset General Position problem. In defining this problem, a geometric subsystem is specified, like a subsystem of lines, hyperplanes or spheres. The input of the problem is a set of n points in mathbb{R}^d and a positive integer k. The objective is to find a subset of at least k input points such that this subset is in general position with respect to the specified subsystem. For example, a set of points is in
general position with respect to a subsystem of hyperplanes in mathbb{R}^d if no d+1 points lie on the same
hyperplane. In this paper, we study the Hyperplane Subset General Position problem under two parameterizations.
When parameterized by k then we exhibit a polynomial kernelization for the problem. When parameterized by h=n-k,
or the dual parameter, then we exhibit polynomial kernels which are also tight, under standard complexity theoretic
assumptions.
We can also exhibit similar kernelization results for d-Polynomial Subset General Position, where a vector space of polynomials
of degree at most d are specified as the underlying subsystem such that the size of the basis for this vector space is b. The objective is to find a set of at least k input points, or in the dual delete at most h = n-k points, such that no b+1 points lie on the same polynomial. Notice that this is a generalization of many well-studied geometric variants of the Set Cover problem, such as Circle Subset General Position. We also study general projective variants of these problems. These problems are also related to other geometric problems like Subset Delaunay Triangulation problem
Gravitational-wave imprints of compact and galactic-scale environments in extreme-mass-ratio binaries
Circumambient and galactic-scale environments are intermittently present
around black holes that reside in active galactic nuclei. As supermassive black
holes impart energy on their host galaxy, so the galactic environment affects
the dynamics of solar-mass objects around black holes and the gravitational
waves emitted from non-vacuum asymmetric binaries. Only recently an exact
general-relativistic solution has been found that describes a Schwarzschild
black hole immersed in a dark matter halo of the Hernquist type. We perform an
extensive analysis of generic geodesics delving in such non-vacuum spacetimes
and compare our results with those obtained in Schwarzschild, as well as
calculate their gravitational-wave emission. Our findings indicate that the
radial and polar oscillation frequency ratios descend deeper into the strong
gravity region as the compactness of the halo increases. This translates to a
redshift of non-vacuum geodesics and their resulting waveforms with respect to
the vacuum ones. We calculate the overlap between waveforms resulting from
Schwarzschild and non-vacuum geometries and find that it decreases as the halo
compactness grows, meaning that dark matter environments should be
distinguishable by space-borne detectors. For compact environments, we find
that the apsidal precession is strongly affected due to the gravitational pull
of dark matter; the orbit's axis can rotate in the opposite direction as that
of the orbital motion, leading to a retrograde precession drift that depends on
the halo mass, as opposed to the typical prograde precession transpiring in
galactic-scale environments. Gravitational waves in retrograde-to-prograde
alterations demonstrate transient frequency phenomena around critical
non-precessing turning points, thus they may serve as `smoking guns' for the
presence of compact dark matter environments around supermassive black holes.Comment: 19 pages, 10 figures, revisions regarding detectability and addition
of new figures and sections, abstract reduced to fit arxiv limits, accepted
for publication in PR
Gravitational-Wave Tests of General Relativity with Ground-Based Detectors and Pulsar-Timing Arrays
This review is focused on tests of Einstein's theory of General Relativity
with gravitational waves that are detectable by ground-based interferometers
and pulsar timing experiments. Einstein's theory has been greatly constrained
in the quasi-linear, quasi-stationary regime, where gravity is weak and
velocities are small. Gravitational waves will allow us to probe a
complimentary, yet previously unexplored regime: the non-linear and dynamical
strong-field regime. Such a regime is, for example, applicable to compact
binaries coalescing, where characteristic velocities can reach fifty percent
the speed of light and compactnesses can reach a half. This review begins with
the theoretical basis and the predicted gravitational wave observables of
modified gravity theories. The review continues with a brief description of the
detectors, including both gravitational wave interferometers and pulsar timing
arrays, leading to a discussion of the data analysis formalism that is
applicable for such tests. The review ends with a discussion of gravitational
wave tests for compact binary systems.Comment: 123 pages, 5 figures, replaced with version accepted for publication
in the Living Reviews in Relativit
Experimental gravity with electromagnetic and gravitational waves
Electromagnetic and gravitational observations can be used to elucidate the nature of compact objects and the fundamental properties of the material in their vicinity. Our ability to extract information about the underlying physics from observations of both electromagnetic and gravitational spectra depends on our understanding of the gravity theory that describes the geometry around these compact objects. For electromagnetic observations, we must also understand the complex astrophysics that produces the observed radiation. In this dissertation, we describe our efforts to constrain and detect deviations from general relativity using: the electromagnetic radiation emitted by an accretion disk around a black hole; the gravitational waves produced when comparable-mass black holes collide; and we have also studied chaotic signatures that could appear when a small compact object falls into a supermassive object during an extreme mass-ratio inspiral. Our analyses combined relativistic ray-tracing and Markov Chain Monte Carlo sampling techniques, as well as analytical and numerical calculations of the motion of particles. We found that even when a simple astrophysical model for the accretion disk is assumed a priori, the uncertainties and covariances between the parameters of the model and the parameters that control a deviation from general relativity make tests of general relativity very challenging when applied to accretion disk spectrum observations. We also found that current gravitational wave observations place constraints on metric deformation parameters that are more stringent than what can be achieved with current X-ray instruments. Based on our numerical findings when studying extreme mass-ratio inspirals, we conjecture that the geodesics of the as-of-yet unknown exact solution for spinning black holes in a dynamical Chern-Simons theory is integrable. Consequently, we predict the existence a fourth integral of motion associated with the exact solution. The work presented in this thesis advances the development of both analytic calculations and computational simulations to test our understanding of gravity’s fundamental properties with electromagnetic and gravitational waves
New horizons for fundamental physics with LISA
The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of gravitational waves can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas
New horizons for fundamental physics with LISA
The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of gravitational waves can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas
New horizons for fundamental physics with LISA
ArtÃculo escrito por un elevado número de autores, solo se referencian el que aparece en primer lugar, el nombre del grupo de colaboración, si le hubiere, y los autores pertenecientes a la UAMThe Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of gravitational waves can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these area
New horizons for fundamental physics with LISA
The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of gravitational waves can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas
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