223,613 research outputs found
Quantifying Shape of Star-Like Objects Using Shape Curves and A New Compactness Measure
Shape is an important indicator of the physical and chemical behavior of natural and engineered particulate materials (e.g., sediment, sand, rock, volcanic ash). It directly or indirectly affects numerous microscopic and macroscopic geologic, environmental and engineering processes. Due to the complex, highly irregular shapes found in particulate materials, there is a perennial need for quantitative shape descriptions. We developed a new characterization method (shape curve analysis) and a new quantitative measure (compactness, not the topological mathematical definition) by applying a fundamental principle that the geometric anisotropy of an object is a unique signature of its internal spatial distribution of matter. We show that this method is applicable to “star-like” particles, a broad mathematical definition of shape fulfilled by most natural and engineered particulate materials. This new method and measure are designed to be mathematically intermediate between simple parameters like sphericity and full 3D shape descriptions.
For a “star-like” object discretized as a polyhedron made of surface planar elements, each shape curve describes the distribution of elemental surface area or volume. Using several thousand regular and highly irregular 3-D shape representations, built from model or real particles, we demonstrate that shape curves accurately encode geometric anisotropy by mapping surface area and volume information onto a pair of dimensionless 2-D curves. Each shape curve produces an intrinsic property (length of shape curve) that is used to describe a new definition of compactness, a property shown to be independent of translation, rotation, and scale. Compactness exhibits unique values for distinct shapes and is insensitive to changes in measurement resolution and noise. With increasing ability to rapidly capture digital representations of highly irregular 3-D shapes, this work provides a new quantitative shape measure for direct comparison of shape across classes of particulate materials
The baryonic Tully-Fisher relation for different velocity definitions and implications for galaxy angular momentum
We study the baryonic Tully-Fisher relation (BTFR) at z=0 using 153 galaxies
from the SPARC sample. We consider different definitions of the characteristic
velocity from HI and H-alpha rotation curves, as well as HI line-widths from
single-dish observations. We reach the following results: (1) The tightest BTFR
is given by the mean velocity along the flat part of the rotation curve. The
orthogonal intrinsic scatter is extremely small (6%) and the best-fit slope is
3.85+/-0.09, but systematic uncertainties may drive the slope from 3.5 to 4.0.
Other velocity definitions lead to BTFRs with systematically higher scatters
and shallower slopes. (2) We provide statistical relations to infer the flat
rotation velocity from HI line-widths or less extended rotation curves (like
H-alpha and CO data). These can be useful to study the BTFR from large HI
surveys or the BTFR at high redshifts. (3) The BTFR is more fundamental than
the relation between angular momentum and galaxy mass (the Fall relation). The
Fall relation has about 7 times more scatter than the BTFR, which is merely
driven by the scatter in the mass-size relation of galaxies. The BTFR is
already the "fundamental plane" of galaxy discs: no value is added with a
radial variable as a third parameter.Comment: 12 pages, 6 figures, accepted for publication in MNRA
On the Reliability of Cross Correlation Function Lag Determinations in Active Galactic Nuclei
Many AGN exhibit a highly variable luminosity. Some AGN also show a
pronounced time delay between variations seen in their optical continuum and in
their emission lines. In effect, the emission lines are light echoes of the
continuum. This light travel-time delay provides a characteristic radius of the
region producing the emission lines. The cross correlation function (CCF) is
the standard tool used to measure the time lag between the continuum and line
variations. For the few well-sampled AGN, the lag ranges from 1-100 days,
depending upon which line is used and the luminosity of the AGN. In the best
sampled AGN, NGC 5548, the H_beta lag shows year-to-year changes, ranging from
about 8.7 days to about 22.9 days over a span of 8 years. In this paper it is
demonstrated that, in the context of AGN variability studies, the lag estimate
using the CCF is biased too low and subject to a large variance. Thus the
year-to-year changes of the measured lag in NGC 5548 do not necessarily imply
changes in the AGN structure. The bias and large variance are consequences of
finite duration sampling and the dominance of long timescale trends in the
light curves, not due to noise or irregular sampling. Lag estimates can be
substantially improved by removing low frequency power from the light curves
prior to computing the CCF.Comment: To appear in the PASP, vol 111, 1999 Nov; 37 pages; 10 figure
Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group
We use a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean C2 -smooth surface in the Heisenberg group H away from characteristic points, and a notion of intrinsic signed geodesic curvature for Euclidean C2 -smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss–Bonnet theorem. An application to Steiner’s formula for the Carnot–Carathéodory distance in H is provided
Calculus on surfaces with general closest point functions
The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs
Tensor Minkowski Functionals for random fields on the sphere
We generalize the translation invariant tensor-valued Minkowski Functionals
which are defined on two-dimensional flat space to the unit sphere. We apply
them to level sets of random fields. The contours enclosing boundaries of level
sets of random fields give a spatial distribution of random smooth closed
curves. We obtain analytic expressions for the ensemble expectation values for
the matrix elements of the tensor-valued Minkowski Functionals for isotropic
Gaussian and Rayleigh fields. We elucidate the way in which the elements of the
tensor Minkowski Functionals encode information about the nature and
statistical isotropy (or departure from isotropy) of the field. We then
implement our method to compute the tensor-valued Minkowski Functionals
numerically and demonstrate how they encode statistical anisotropy and
departure from Gaussianity by applying the method to maps of the Galactic
foreground emissions from the PLANCK data.Comment: 1+23 pages, 5 figures, Significantly expanded from version 1. To
appear in JCA
On the classification of flaring states of blazar
The time evolution of the electromagnetic emission from blazars, in
particular high frequency peaked sources (HBLs), displays irregular activity
not yet understood. In this work we report a methodology capable of
characterizing the time behavior of these variable objects. The Maximum
Likelihood Blocks (MLBs) is a model-independent estimator which sub-divides the
light curve into time blocks, whose length and amplitude are compatible with
states of constant emission rate of the observed source. The MLBs yields the
statistical significance in the rate variations and strongly suppresses the
noise fluctuations in the light curves. We apply the MLBs for the first time on
the long term X-ray light curves (RXTE/ASM) of Mkn~421,Mkn~501, 1ES 1959+650
and 1ES 2155-304, which consist of more than 10 years of observational data
(1996-2007). Using the MLBs interpretation of RXTE/ASM data, the integrated
time flux distribution is determined for each single source considered. We
identify in these distributions the characteristic level as well as the flaring
states of the blazars. All the distributions show a significant component at
negative flux values, most probably caused by an uncertainty in the background
subtraction and by intrinsic fluctuations of RXTE/ASM. This effect interests in
particular short time observations. In order to quantify the probability that
the intrinsic fluctuations give rise to a false identification of a flare, we
study a population of very faint sources and their integrated time flux
distribution. We determine duty cycle or fraction of time a source spent in the
flaring state of the source Mkn~421, Mkn~501, 1ES 1959+650 and 1ES 2155-304.
Moreover, we study the random coincidences between flares and generic sporadic
events such as high energy neutrinos or flares in other wavelengths.Comment: Accepted to A&
Toward tight gamma-ray burst luminosity relations
The large scatters of luminosity relations of gamma-ray bursts (GRBs) have
been one of the most important reasons that prevent the extensive applications
of GRBs in cosmology. Many efforts have been made to seek tight luminosity
relations. With the latest sample of 116 GRBs with measured redshift and
spectral parameters, we investigate 6 two-dimensional (2D) correlations and 14
derived three-dimensional (3D) correlations of GRBs to explore the possibility
of decreasing the intrinsic scatters of the luminosity relations of GRBs. We
find the 3D correlation of ---- to be
evidently tighter (at the confidence level) than its corresponding
2D correlations, i.e., the -- and
-- correlations. In addition, the coefficients before
the logarithms of and in the
---- correlation are almost exact
opposites of each other. Inputting this situation as a prior reduces the
relation to , where and denote
the peak energy and minimum rise time in the GRB rest frame. We discuss how our
findings can be interpreted/understood in the framework of the definition of
the luminosity (energy released in units of time). Our argument about the
connection between the luminosity relations of GRBs and the definition of the
luminosity provides a clear direction for exploring tighter luminosity
relations of GRBs in the future.Comment: 9 pages, 1 figure, 1 table, added discussion and clarification, added
references, minor language edit, published in The Astrophysical Journa
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