5,136 research outputs found
L^2 torsion without the determinant class condition and extended L^2 cohomology
We associate determinant lines to objects of the extended abelian category
built out of a von Neumann category with a trace. Using this we suggest
constructions of the combinatorial and the analytic L^2 torsions which, unlike
the work of the previous authors, requires no additional assumptions; in
particular we do not impose the determinant class condition. The resulting
torsions are elements of the determinant line of the extended L^2 cohomology.
Under the determinant class assumption the L^2 torsions of this paper
specialize to the invariants studied in our previous work. Applying a recent
theorem of D. Burghelea, L. Friedlander and T. Kappeler we obtain a Cheeger -
Muller type theorem stating the equality between the combinatorial and the
analytic L^2 torsions.Comment: 39 page
Low Velocity Granular Drag in Reduced Gravity
We probe the dependence of the low velocity drag force in granular materials
on the effective gravitational acceleration (geff) through studies of spherical
granular materials saturated within fluids of varying density. We vary geff by
a factor of 20, and we find that the granular drag is proportional to geff,
i.e., that the granular drag follows the expected relation Fprobe = {\eta}
{\rho}grain geff dprobe hprobe^2 for the drag force, Fprobe on a vertical
cylinder with depth of insertion, hprobe, diameter dprobe, moving through
grains of density {\rho}grain, and where {\eta} is a dimensionless constant.
This dimensionless constant shows no systematic variation over four orders of
magnitude in effective grain weight, demonstrating that the relation holds over
that entire range to within the precision of our data
Topology of parametrised motion planning algorithms
We introduce and study a new concept of parameterised topological complexity, a topological invariant motivated by the motion planning problem of robotics. In the parametrised setting, a motion planning algorithm has high degree of universality and flexibility, it can function under a variety of external conditions (such as positions of the obstacles etc). We explicitly compute the parameterised topological complexity of obstacle-avoiding collision-free motion of many particles (robots) in 3-dimensional space. Our results show that the parameterised topological complexity can be significantly higher than the standard (nonparametrised) invariant
MIRAGE: A Model for Ultra-High-Speed Protocol Analysis and Design
Current protocols are expected to become inefficient if used at speeds in excess of 1 Gigabit per second. While this premise is widely accepted, no model exists to explain the phenomenon. We define a model for understanding protocols which is aimed at explaining why such a barrier exists, and indicates alternate designs which do not have this limit.
Existing protocols are akin to classical mechanics; 1 Gigabit/second is the speed near which relativistic effects emerge. In order to account for these effects, we need to express knowledge at a distance, latent measurement, and uncertainty as real entities, not negligible estimates. The result is a model which expresses not only existing protocols, and may contribute to a better understanding of the Gigabit communications domain
Genetic and epigenetic regulation of abdominal aortic aneurysms
Abdominal aortic aneurysms (AAAs) are focal dilations of the aorta that develop from degenerative changes in the media and adventitia of the vessel. Ruptured AAAs have a mortality of up to 85%, thus it is important to identify patients with AAA at increased risk for rupture who would benefit from increased surveillance and/or surgical repair. Although the exact genetic and epigenetic mechanisms regulating AAA formation are not completely understood, Mendelian cases of AAA, which result from pathologic variants in a single gene, have helped provide a basic understanding of AAA pathophysiology. More recently, genome wide associated studies (GWAS) have identified additional variants, termed single nucleotide polymorphisms, in humans that may be associated with AAAs. While some variants may be associated with AAAs and play causal roles in aneurysm pathogenesis, it should be emphasized that the majority of SNPs do not actually cause disease. In addition to GWAS, other studies have uncovered epigenetic causes of disease that regulate expression of genes known to be important in AAA pathogenesis. This review describes many of these genetic and epigenetic contributors of AAAs, which altogether provide a deeper insight into AAA pathogenesis.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/155527/1/cge13705.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/155527/2/cge13705_am.pd
Critical points and resonance of hyperplane arrangements
If F is a master function corresponding to a hyperplane arrangement A and a
collection of weights y, we investigate the relationship between the critical
set of F, the variety defined by the vanishing of the one-form w = d log F, and
the resonance of y. For arrangements satisfying certain conditions, we show
that if y is resonant in dimension p, then the critical set of F has
codimension at most p. These include all free arrangements and all rank 3
arrangements.Comment: revised version, Canadian Journal of Mathematics, to appea
Geometric and homological finiteness in free abelian covers
We describe some of the connections between the Bieri-Neumann-Strebel-Renz
invariants, the Dwyer-Fried invariants, and the cohomology support loci of a
space X. Under suitable hypotheses, the geometric and homological finiteness
properties of regular, free abelian covers of X can be expressed in terms of
the resonance varieties, extracted from the cohomology ring of X. In general,
though, translated components in the characteristic varieties affect the
answer. We illustrate this theory in the setting of toric complexes, as well as
smooth, complex projective and quasi-projective varieties, with special
emphasis on configuration spaces of Riemann surfaces and complements of
hyperplane arrangements.Comment: 30 pages; to appear in Configuration Spaces: Geometry, Combinatorics
and Topology (Centro De Giorgi, 2010), Edizioni della Normale, Pisa, 201
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