65,612 research outputs found
Geometry of the Prytz Planimeter
The Prytz planimeter is a simple example of a system governed by a
non-holonomic constraint. It is unique among planimeters in that it measures
something more subtle than area, combining the area, centroid and other moments
of the region being measured, with weights depending on the length of the
planimeter. As a tool for measuring area, it is most accurate for regions that
are small relative to its length.
The configuration space of the planimeter is a non-principal circle bundle
acted on by SU(1,1), (isom. to SL(2,R)). The motion of the planimeter is
realized as parallel translation for a connection on this bundle and for a
connection on a principal SU(1,1)-bundle. The holonomy group is SU(1,1). As a
consequence, the planimeter is an example of a system with a phase shift on the
circle that is not a simple rotation.
There is a qualitative difference in the holonomy when tracing large regions
as opposed to small ones. Generic elements of SU(1,1) act on S^1 with two fixed
points or with no fixed points. When tracing small regions, the holonomy acts
without fixed points. Menzin's conjecture states (roughly) that if a planimeter
of length L traces the boundary of a region with area A > pi L^2, then it
exhibits an asymptotic behavior and the holonomy acts with two fixed points,
one attracting and one repelling. This is obvious if the region is a disk, and
intuitively plausible if the region is convex and A >> pi L^2. A proof of this
conjecture is given for a special case, and the conjecture is shown to imply
the isoperimetric inequality.Comment: AmS-TeX, 23 pages, 12 figures in 2 *.gif files. To appear in Reports
on Mathematical Physics. Part of proceedings of Workshop on Non-holonomic
Constraints in Dynamics, Univ. of Calgary, Aug. 199
The strongest gravitational lenses: I. The statistical impact of cluster mergers
For more than a decade now, it has been controversial whether or not the high
rate of giant gravitational arcs and the largest observed Einstein radii are
consistent with the standard cosmological model. Recent studies indicate that
mergers provide an efficient mechanism to substantially increase the
strong-lensing efficiency of individual clusters. Based on purely semi-analytic
methods, we investigated the statistical impact of cluster mergers on the
distribution of the largest Einstein radii and the optical depth for giant
gravitational arcs of selected cluster samples. Analysing representative
all-sky realizations of clusters at redshifts z < 1 and assuming a constant
source redshift of z_s = 2.0, we find that mergers increase the number of
Einstein radii above 10 arcsec (20 arcsec) by ~ 35 % (~ 55 %). Exploiting the
tight correlation between Einstein radii and lensing cross sections, we infer
that the optical depth for giant gravitational arcs with a length-to-width
ratio > 7.5 of those clusters with Einstein radii above 10 arcsec (20 arcsec)
increases by ~ 45 % (85 %). Our findings suggest that cluster mergers
significantly influence in particular the statistical lensing properties of the
strongest gravitational lenses. We conclude that semi-analytic studies must
inevitably take these events into account before questioning the standard
cosmological model on the basis of the largest observed Einstein radii and the
statistics of giant gravitational arcs.Comment: 23 pages, 18 figures; accepted for publication in Astronomy and
Astrophysics; v2: minor corrections (added clarifying comments; added Fig.
19) to match the accepted versio
Characterizing the Cluster Lens Population
We present a detailed investigation into which properties of CDM halos make
them effective strong gravitational lenses. Strong lensing cross sections of
878 clusters from an N-body simulation are measured by ray tracing through
13,594 unique projections. We measure concentrations, axis ratios,
orientations, and the amount of substructure of each cluster, and compare the
lensing weighted distribution of each quantity to that of the cluster
population as a whole. The concentrations of lensing clusters are on average
34% larger than the typical cluster in the Universe. Despite this bias, the
anomalously high concentrations (c >14) recently measured by several groups,
appear to be inconsistent with the concentration distribution in our
simulations, which predict < 2% of lensing clusters should have concentrations
this high. No correlation is found between lensing cross section and the amount
of substructure. We introduce several types of simplified dark matter halos,
and use them to isolate which properties of CDM clusters make them effective
lenses. Projections of halo substructure onto small radii and the large scale
mass distribution of clusters do not significantly influence cross sections.
The abundance of giant arcs is primarily determined by the mass distribution
within an average overdensity of ~ 10,000. A multiple lens plane ray tracing
algorithm is used to show that projections of large scale structure increase
the giant arc abundance by a modest amount <7%. We revisit the question of
whether there is an excess of giant arcs behind high redshift clusters in the
RCS survey and find that the number of high redshift (z > 0.6) lenses is in
good agreement with LCDM, although our simulations predict more low redshift (z
< 0.6) lenses than were observed. (abridged)Comment: 19 pages, 15 figures. Submitted to Ap
Evolution of drainage system morphology at a land-terminating Greenland outlet glacier
This work was funded by the UK Natural Environment Research Council (through grants to Nienow, Mair, and Wadham, and a studentship to Bartholomew), the Edinburgh University Moss Centenary Scholarship (Cowton and Bartholomew), and a Carnegie Research Grant (Nienow). We thank Ian Willis, Tim Bartholomaus and an anonymous referee for valuable comments which significantly improved the manuscript.Peer reviewedPublisher PD
Designing Volumetric Truss Structures
We present the first algorithm for designing volumetric Michell Trusses. Our
method uses a parametrization approach to generate trusses made of structural
elements aligned with the primary direction of an object's stress field. Such
trusses exhibit high strength-to-weight ratios. We demonstrate the structural
robustness of our designs via a posteriori physical simulation. We believe our
algorithm serves as an important complement to existing structural optimization
tools and as a novel standalone design tool itself
Evidence for the accelerated expansion of the Universe from weak lensing tomography with COSMOS
We present a tomographic cosmological weak lensing analysis of the HST COSMOS
Survey. Applying our lensing-optimized data reduction, principal component
interpolation for the ACS PSF, and improved modelling of charge-transfer
inefficiency, we measure a lensing signal which is consistent with pure
gravitational modes and no significant shape systematics. We carefully estimate
the statistical uncertainty from simulated COSMOS-like fields obtained from
ray-tracing through the Millennium Simulation. We test our pipeline on
simulated space-based data, recalibrate non-linear power spectrum corrections
using the ray-tracing, employ photometric redshifts to reduce potential
contamination by intrinsic galaxy alignments, and marginalize over systematic
uncertainties. We find that the lensing signal scales with redshift as expected
from General Relativity for a concordance LCDM cosmology, including the full
cross-correlations between different redshift bins. For a flat LCDM cosmology,
we measure sigma_8(Omega_m/0.3)^0.51=0.75+-0.08 from lensing, in perfect
agreement with WMAP-5, yielding joint constraints Omega_m=0.266+0.025-0.023,
sigma_8=0.802+0.028-0.029 (all 68% conf.). Dropping the assumption of flatness
and using HST Key Project and BBN priors only, we find a negative deceleration
parameter q_0 at 94.3% conf. from the tomographic lensing analysis, providing
independent evidence for the accelerated expansion of the Universe. For a flat
wCDM cosmology and prior w in [-2,0], we obtain w<-0.41 (90% conf.). Our dark
energy constraints are still relatively weak solely due to the limited area of
COSMOS. However, they provide an important demonstration for the usefulness of
tomographic weak lensing measurements from space. (abridged)Comment: 26 pages, 25 figures, matches version accepted for publication by
Astronomy and Astrophysic
Development of a Computer Vision-Based Three-Dimensional Reconstruction Method for Volume-Change Measurement of Unsaturated Soils during Triaxial Testing
Problems associated with unsaturated soils are ubiquitous in the U.S., where expansive and collapsible soils are some of the most widely distributed and costly geologic hazards. Solving these widespread geohazards requires a fundamental understanding of the constitutive behavior of unsaturated soils. In the past six decades, the suction-controlled triaxial test has been established as a standard approach to characterizing constitutive behavior for unsaturated soils. However, this type of test requires costly test equipment and time-consuming testing processes. To overcome these limitations, a photogrammetry-based method has been developed recently to measure the global and localized volume-changes of unsaturated soils during triaxial test. However, this method relies on software to detect coded targets, which often requires tedious manual correction of incorrectly coded target detection information. To address the limitation of the photogrammetry-based method, this study developed a photogrammetric computer vision-based approach for automatic target recognition and 3D reconstruction for volume-changes measurement of unsaturated soils in triaxial tests. Deep learning method was used to improve the accuracy and efficiency of coded target recognition. A photogrammetric computer vision method and ray tracing technique were then developed and validated to reconstruct the three-dimensional models of soil specimen
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