223,613 research outputs found

    Quantifying Shape of Star-Like Objects Using Shape Curves and A New Compactness Measure

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    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

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    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

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    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

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    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

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    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

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    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

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    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

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    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 EpeakE_{\mathrm{peak}}--τRT\tau_{\mathrm{RT}}--LL to be evidently tighter (at the 2σ2 \sigma confidence level) than its corresponding 2D correlations, i.e., the EpeakE_{\mathrm{peak}}--LL and τRT\tau_{\mathrm{RT}}--LL correlations. In addition, the coefficients before the logarithms of EpeakE_{\mathrm{peak}} and τRT\tau_{\mathrm{RT}} in the EpeakE_{\mathrm{peak}}--τRT\tau_{\mathrm{RT}}--LL correlation are almost exact opposites of each other. Inputting this situation as a prior reduces the relation to L(Epeak/τRT)0.842±0.064L \propto (E_{\mathrm{peak}}^{'} / \tau_{\mathrm{RT}}^{'})^{0.842 \pm 0.064}, where EpeakE_{\mathrm{peak}}^{'} and τRT\tau_{\mathrm{RT}}^{'} 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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