6,677 research outputs found
The Discrete Frenet Frame, Inflection Point Solitons And Curve Visualization with Applications to Folded Proteins
We develop a transfer matrix formalism to visualize the framing of discrete
piecewise linear curves in three dimensional space. Our approach is based on
the concept of an intrinsically discrete curve, which enables us to more
effectively describe curves that in the limit where the length of line segments
vanishes approach fractal structures in lieu of continuous curves. We verify
that in the case of differentiable curves the continuum limit of our discrete
equation does reproduce the generalized Frenet equation. As an application we
consider folded proteins, their Hausdorff dimension is known to be fractal. We
explain how to employ the orientation of carbons of amino acids along
a protein backbone to introduce a preferred framing along the backbone. By
analyzing the experimentally resolved fold geometries in the Protein Data Bank
we observe that this framing relates intimately to the discrete
Frenet framing. We also explain how inflection points can be located in the
loops, and clarify their distinctive r\^ole in determining the loop structure
of foldel proteins.Comment: 14 pages 12 figure
Ricci Curvature of the Internet Topology
Analysis of Internet topologies has shown that the Internet topology has
negative curvature, measured by Gromov's "thin triangle condition", which is
tightly related to core congestion and route reliability. In this work we
analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin,
etc. Ricci curvature measures whether local distances diverge or converge. It
is a more local measure which allows us to understand the distribution of
curvatures in the network. We show by various Internet data sets that the
distribution of Ricci cuvature is spread out, suggesting the network topology
to be non-homogenous. We also show that the Ricci curvature has interesting
connections to both local measures such as node degree and clustering
coefficient, global measures such as betweenness centrality and network
connectivity, as well as auxilary attributes such as geographical distances.
These observations add to the richness of geometric structures in complex
network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201
Perceptually Motivated Shape Context Which Uses Shape Interiors
In this paper, we identify some of the limitations of current-day shape
matching techniques. We provide examples of how contour-based shape matching
techniques cannot provide a good match for certain visually similar shapes. To
overcome this limitation, we propose a perceptually motivated variant of the
well-known shape context descriptor. We identify that the interior properties
of the shape play an important role in object recognition and develop a
descriptor that captures these interior properties. We show that our method can
easily be augmented with any other shape matching algorithm. We also show from
our experiments that the use of our descriptor can significantly improve the
retrieval rates
Interplanetary Alfvenic fluctuations: A statistical study of the directional variations of the magnetic field
Magnetic field data from HELIOS 1 and 2 are used to test a stochastic model for Alfvenic fluctuations recently proposed. A reasonable matching between observations and predictions is found. A rough estimate of the correlation length of the observed fluctuations is inferred
Geodesic PCA in the Wasserstein space
We introduce the method of Geodesic Principal Component Analysis (GPCA) on
the space of probability measures on the line, with finite second moment,
endowed with the Wasserstein metric. We discuss the advantages of this
approach, over a standard functional PCA of probability densities in the
Hilbert space of square-integrable functions. We establish the consistency of
the method by showing that the empirical GPCA converges to its population
counterpart, as the sample size tends to infinity. A key property in the study
of GPCA is the isometry between the Wasserstein space and a closed convex
subset of the space of square-integrable functions, with respect to an
appropriate measure. Therefore, we consider the general problem of PCA in a
closed convex subset of a separable Hilbert space, which serves as basis for
the analysis of GPCA and also has interest in its own right. We provide
illustrative examples on simple statistical models, to show the benefits of
this approach for data analysis. The method is also applied to a real dataset
of population pyramids
Statistical Characterization of the Chandra Source Catalog
The first release of the Chandra Source Catalog (CSC) contains ~95,000 X-ray
sources in a total area of ~0.75% of the entire sky, using data from ~3,900
separate ACIS observations of a multitude of different types of X-ray sources.
In order to maximize the scientific benefit of such a large, heterogeneous
data-set, careful characterization of the statistical properties of the
catalog, i.e., completeness, sensitivity, false source rate, and accuracy of
source properties, is required. Characterization efforts of other, large
Chandra catalogs, such as the ChaMP Point Source Catalog (Kim et al. 2007) or
the 2 Mega-second Deep Field Surveys (Alexander et al. 2003), while
informative, cannot serve this purpose, since the CSC analysis procedures are
significantly different and the range of allowable data is much less
restrictive. We describe here the characterization process for the CSC. This
process includes both a comparison of real CSC results with those of other,
deeper Chandra catalogs of the same targets and extensive simulations of
blank-sky and point source populations.Comment: To be published in the Astrophysical Journal Supplement Series (Fig.
52 replaced with a version which astro-ph can convert to PDF without issues.
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